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\begin{document}

%\date{}

\title{Just Do It: A Collection of Hartshorne Problems}
  \author{D. Zack Garza} 
\maketitle

\tableofcontents

\newpage

\hypertarget{i-varieties}{%
\section{I: Varieties}\label{i-varieties}}

\hypertarget{i.1-affine-varieties}{%
\subsection{I.1: Affine Varieties}\label{i.1-affine-varieties}}

\hypertarget{section}{%
\subsubsection{1.1}\label{section}}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\item
  Let \(Y\) be the plane curve \(y=x^2\) (i.e., \(Y\) is the zero set of
  the polynomial \(f=\) \(y-x^2\) ). Show that \(A(Y)\) is isomorphic to
  a polynomial ring in one variable over \(k\).
\item
  Let \(Z\) be the plane curve \(x y=1\). Show that \(A(Z)\) is not
  isomorphic to a polynomial ring in one variable over \(k\).
\item
  * Let \(f\) be any irreducible quadratic polynomial in \(k[x, y]\),
  and let \(W\) be the conic defined by \(f\). Show that \(A(W)\) is
  isomorphic to \(A(Y)\) or \(A(Z)\). Which one is it when?
\end{enumerate}

\hypertarget{the-twisted-cubic-curve}{%
\subsubsection{1.2 The Twisted Cubic
Curve}\label{the-twisted-cubic-curve}}

Let \(Y \subseteq \mathbf{A}^3\) be the set
\(Y = \left\{{(t, t^2,t^3) {~\mathrel{\Big\vert}~}t\in k}\right\}\).

\begin{itemize}
\tightlist
\item
  Show that \(Y\) is an affine variety of dimension 1.
\item
  Find generators for the ideal \(I\left(Y\right)\).
\item
  Show that \(A(Y)\) is isomorphic to a polynomial ring in one variable
  over \(k\).
\end{itemize}

We say that \(Y\) is given by the \emph{parametric representation}
\(x=t . y=t^2, z=t^3\).

\begin{quote}
Useful facts: \(\sqrt{I} = \sqrt{\prod p_i^{a_i}} = \prod p_i\) in a UFD
when \(I\) is a principal ideal factored into irreducibles. An ideal is
also radical iff the quotient is reduced, and
\(\left\langle{f}\right\rangle\) is radical when \(f\) is irreducible.
\end{quote}

\hypertarget{section-1}{%
\subsubsection{1.3}\label{section-1}}

Let \(Y\) be the algebraic set in \(\mathbf{A}^3\) defined by the two
polynomials \(x^2-yz\) and \(x z-x\). Show that \(Y\) is a union of
three irreducible components. Describe them and find their prime ideals.

\hypertarget{section-2}{%
\subsubsection{1.4}\label{section-2}}

If we identify \(\mathbf{A}^2\) with
\(\mathbf{A}^1 \times \mathbf{A}^1\) in the natural way, show that the
Zariski topology on \(\mathbf{A}^2\) is not the product topology of the
Zariski topologies on the two copies of \(\mathbf{A}^1\).

\hypertarget{section-3}{%
\subsubsection{1.5}\label{section-3}}

Show that a \(k\)-algebra \(B\) is isomorphic to the affine coordinate
ring of some algebraic set in \(\mathbf{A}^n\). for some \(n\), if and
only if \(B\) is a finitely generated \(k\)-algebra with no nilpotent
elements.

\hypertarget{section-4}{%
\subsubsection{1.6}\label{section-4}}

Any nonempty open subset of an irreducible topological space is dense
and irreducible. If \(Y\) is a subset of a topological space \(X\),
which is irreducible in its induced topology, then the closure
\(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\) is also
irreducible.

\hypertarget{section-5}{%
\subsubsection{1.7}\label{section-5}}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  Show that the following conditions are equivalent for a topological
  space \(X\) :

  \begin{itemize}
  \tightlist
  \item
    \(X\) is noetherian:
  \item
    Every nonempty family of closed subsets has a minimal element:
  \item
    \(X\) satisfies the ascending chain condition for open subsets:
  \item
    Every nonempty family of open subsets has a maximal element.
  \end{itemize}
\item
  A noetherian topological space is quasi-compact, i.e., every open
  cover has a finite subcover.
\item
  Any subset of a noetherian topological space is noetherian in its
  induced topology.
\item
  A noetherian space which is also Hausdorff must be a finite set with
  the discrete topology.
\end{enumerate}

\hypertarget{section-6}{%
\subsubsection{1.8}\label{section-6}}

Let \(Y\) be an affine variety of dimension \(r\) in \(\mathbf{A}^n\).
Let \(H\) be a hypersurface in \(\mathbf{A}^n\), and assume that
\(Y \nsubseteq H\). Then every irreducible component of \(Y \cap H\) has
dimension \(r-1\).\footnote{Use (b) above.}

\hypertarget{section-7}{%
\subsubsection{1.9}\label{section-7}}

Let \(a \subseteq A=k\left[x_1, \ldots, x_n\right]\) be an ideal which
can be generated by \(r\) elements. Then every irreducible component of
\(Z(a)\) has dimension \(\geqslant n-r\).

\hypertarget{section-8}{%
\subsubsection{1.10}\label{section-8}}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  If \(Y\) is any subset of a topological space \(X\), then
  \(\operatorname{dim} Y \leqslant \operatorname{dim} X\).
\item
  If \(X\) is a topological space which is covered by a family of open
  subsets \(\left\{L_1 ;\right.\), then
  \(\operatorname{dim} X=\sup \operatorname{dim} U_i\).
\item
  Give an example of a topological space \(X\) and a dense open subset
  \(U\) with \(\operatorname{dim} L^{\prime}<\operatorname{dim} X\).
\item
  If \(Y\) is a closed subset of an irreducible finite-dimensional
  topological space \(X\), and if
  \(\operatorname{dim} Y=\operatorname{dim} X\), then \(Y=X\).
\item
  Give an example of a noetherian topological space of infinite
  dimension.
\end{enumerate}

\hypertarget{section-9}{%
\subsubsection{1.11 *}\label{section-9}}

Let \(Y \subseteq \mathbf{A}^3\) be the curve given parametrically by
\(x=t^3, y=t^4, z=t^5\). Show that \(I(Y)\) is a prime ideal of height 2
in \(k[x, y ;-]\) which cannot be generated by 2 elements. We say \(Y\)
is not a local complete intersection-cf.~(Ex. 2.17).

\hypertarget{section-10}{%
\subsubsection{1.12}\label{section-10}}

Give an example of an irreducible polynomial \(f \in \mathbf{R}[x, y]\).
whose zero set \(Z(f)\) in \(\mathbf{A}_{\mathbf{R}}^2\) is not
irreducible (cf.~1.4.2).

\hypertarget{i.2-projective-varieties}{%
\subsection{I.2: Projective Varieties}\label{i.2-projective-varieties}}

\hypertarget{section-11}{%
\subsubsection{2.1}\label{section-11}}

Prove the ``homogeneous Nullstellensatz,'' which says if
\(a \subseteq S\) is a homogeneous ideal, and if \(f \in S\) is a
homogeneous polynomial with deg \(f>0\), such that \(f(P)=0\) for all
\(P \in Z(a)\) in \(\mathbf{P}^n\), then \(f^u \in a\) for some \(q>0\).
\footnote{Hint: Interpret the problem in terms of the affine
  \((n+1)\)-space whose affine coordinate ring is \(S\), and use the
  usual Nullstellensatz, (1.3A).}

\hypertarget{section-12}{%
\subsubsection{2.2}\label{section-12}}

For a homogeneous ideal \(a \subseteq S\), show that the following
conditions are equivalent:

\begin{enumerate}
\def\labelenumi{(\roman{enumi})}
\tightlist
\item
  \(Z(a) = \varnothing\) (the empty set);
\item
  \(\sqrt{a}=\) either \(S\) or the ideal \(S_{+}=\bigoplus_{d>0} S_d\);
\item
  \(a \supseteq S_d\) for some \(d>0\).
\end{enumerate}

\hypertarget{section-13}{%
\subsubsection{2.3}\label{section-13}}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  If \(T_1 \subseteq T_2\) are subsets of \(S^h\). then
  \(Z\left(T_1\right) \supseteq Z\left(T_2\right)\).
\item
  If \(Y_1 \subseteq Y_2\) are subsets of \(\mathbf{P}^n\), then
  \(I\left(Y_1\right) \supseteq I\left(Y_2\right)\).
\item
  For any two subsets \(Y_1, Y_2\) of
  \(\mathbf{P}^n, I\left(Y_1 \cup Y_2\right)=I\left(Y_1\right) \cap I\left(Y_2\right)\).
\item
  If \(a \subseteq S\) is a homogeneous ideal with
  \(Z(a) \neq \varnothing\). then \(I(Z(a))=\sqrt a\).
\item
  For any subset
  \(Y \subseteq \mathbf{P}^n, Z(I(Y))=\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\).
\end{enumerate}

\hypertarget{section-14}{%
\subsubsection{2.4}\label{section-14}}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  There is a 1-1 inclusion-reversing correspondence between algebraic
  sets in \(\mathbf{P}^n\). and homogeneous radical ideals of \(S\) not
  equal to \(S_{+}\) given by \(Y \mapsto I(Y)\) and \(a \mapsto Z(a)\).
  \footnote{Note: Since \(S_{+}\)does not occur in this correspondence,
    it is sometimes called the \textbf{irrelevant maximal ideal of
    \(S\)}.}
\item
  An algebraic set \(Y \subseteq \mathbf{P}^n\) is irreducible if and
  only if \(I\left(Y^{\prime}\right)\) is a prime ideal.
\item
  Show that \(\mathbf{P}^n\) itself is irreducible.
\end{enumerate}

\hypertarget{section-15}{%
\subsubsection{2.5}\label{section-15}}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  \(\mathbf{P}^n\) is a noetherian topological space.
\item
  Every algebraic set in \(\mathrm{P}^n\) can be written uniquely as a
  finite union of irreducible algebraic sets. no one containing another.
  These are called its irreducible components.
\end{enumerate}

\hypertarget{section-16}{%
\subsubsection{2.6}\label{section-16}}

If \(Y\) is a projective variety with homogeneous coordinate ring
\(S(Y)\), show that
\(\operatorname{dim} S(Y)=\operatorname{dim} Y+1\).\footnote{Hint: Let
  \(\varphi_i: U_i \rightarrow \mathbf{A}^n\) be the homeomorphism of
  (2.2), let \(Y_t\) be the affine variety
  \(\varphi_t\left(Y \cap U_i\right)\), and let \(A\left(Y_i\right)\) be
  its affine coordinate ring. Show that \(A\left(Y_t\right)\) can be
  identified with the subring of elements of degree 0 of the localized
  ring \(S(Y)_{x_i}\). Then show that
  \(S(Y)_{x_i} \cong A\qty{Y_i}[x_i, x_i^{-1}]\). Now use (1.7), (1.8A),
  and (Ex 1.10), and look at transcendence degrees. Conclude also that
  \(\operatorname{dim} Y=\operatorname{dim} Y_i\) whenever \(Y_i\) is
  nonempty.}

\hypertarget{section-17}{%
\subsubsection{2.7}\label{section-17}}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  \(\operatorname{dim} \mathbf{P}^n=n\).
\item
  If \(Y \subseteq \mathbf{P}^n\) is a quasi-projective variety, then
  \(\operatorname{dim} Y=\operatorname{dim} \mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\).\footnote{Hint:
    Use (Ex. 2.6) to reduce to (1.10).}
\end{enumerate}

\hypertarget{section-18}{%
\subsubsection{2.8}\label{section-18}}

A projective variety \(Y \subseteq \mathbf{P}^n\) has dimension \(n-1\)
if and only if it is the zero set of a single irreducible homogeneous
polynomial \(f\) of positive degree. \(Y\) is called a hypersurface in
\(\mathbf{P}^n\).

\hypertarget{projective-closure-of-an-affine-variety}{%
\subsubsection{2.9 Projective Closure of an Affine
Variety}\label{projective-closure-of-an-affine-variety}}

If \(Y \subseteq \mathbf{A}^n\) is an affine variety, we identify
\(\mathbf{A}^n\) with an open set \(U_0 \subseteq \mathbf{P}^n\) by the
homeomorphism \(\varphi_0\). Then we can speak of
\(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\), the
closure of \(Y\) in \(\mathbf{P}^n\), which is called the projective
closure of \(Y\).

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  Show that
  \(I(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu)\) is
  the ideal generated by \(\beta(I(Y))\), using the notation of the
  proof of \((2.2)\).
\item
  Let \(Y \subseteq \mathbf{A}^3\) be the twisted cubic of (Ex. 1.2).
  Its projective closure
  \(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu \subseteq \mathbf{P}^3\)
  is called the twisted cubic curve in \(\mathbf{P}^3\). Find generators
  for \(I(Y)\) and
  \(I(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu)\),
  and use this example to show that if \(f_1, \ldots, f_r\) generate
  \(I(Y)\), then
  \(\beta\left(f_1\right), \ldots, \beta\left(f_r\right)\) do not
  necessarily generate
  \(I(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu)\).
\end{enumerate}

\hypertarget{the-cone-over-a-projective-variety-fig.-1}{%
\subsubsection{2.10 The Cone Over a Projective Variety (Fig.
1)}\label{the-cone-over-a-projective-variety-fig.-1}}

Let \(Y \subseteq \mathbf{P}^n\) be a nonempty algebraic set, and let
\(\theta: \mathbf{A}^{n+1}-\{(0, \ldots, 0)\} \rightarrow \mathbf{P}^n\)
be the map which sends the point with affine coordinates
\(\left(a_0, \ldots, a_n\right)\) to the point with homogeneous
coordinates \(\left(a_0, \ldots, a_n\right)\). We define the affine cone
over \(Y\) to be

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  Show that \(C(Y)\) is an algebraic set in \(\mathbf{A}^{n+1}\), whose
  ideal is equal to \(I(Y)\), considered as an ordinary ideal in
  \(k\left[x_0, \ldots, x_n\right]\).
\item
  \(C(Y)\) is irreducible if and only if \(Y\) is.
\item
  \(\operatorname{dim} C(Y)=\operatorname{dim} Y+1\).
\end{enumerate}

Sometimes we consider the projective closure \(\overline{C(Y)}\) of
\(C(Y)\) in \(\mathbf{P}^{n+1}\). This is called the \textbf{projective
cone} over \(Y\).

\includegraphics{attachments/2022-09-17_22-10-43.png}

\hypertarget{linear-varieties-in-mathbfpn-to_work}{%
\subsubsection{\texorpdfstring{2.11 Linear Varieties in
\(\mathbf{P}^n\)}{2.11 Linear Varieties in \textbackslash mathbf\{P\}\^{}n}}\label{linear-varieties-in-mathbfpn-to_work}}

A hypersurface defined by a linear polynomial is called a hyperplane.

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\item
  Show that the following two conditions are equivalent for a variety
  \(Y\) in \(\mathbf{P}^n\) :

  \begin{enumerate}
  \def\labelenumii{(\roman{enumii})}
  \tightlist
  \item
    \(I(Y)\) can be generated by linear polynomials.
  \item
    \(Y\) can be written as an intersection of hyperplanes.
  \end{enumerate}

  In this case we say that \(Y\) is a \textbf{linear variety} in
  \(\mathbf{P}^n\).
\item
  If \(Y\) is a linear variety of dimension \(r\) in \(\mathbf{P}^n\),
  show that \(I(Y)\) is minimally generated by \(n-r\) linear
  polynomials.
\item
  Let \(Y, Z\) be linear varieties in \(\mathbf{P}^n\), with
  \(\operatorname{dim} Y=i, \operatorname{dim} Z=\) s. If
  \(r+s-n \geqslant 0\), then \(Y \cap Z \neq \varnothing\).
  Furthermore, if \(Y \cap Z \neq \varnothing\), then \(Y \cap Z\) is a
  linear variety of dimension \(\geqslant r+s-n\). \footnote{Think of
    \(\mathbf{A}^{n+1}\) as a vector space over \(k\), and work with its
    subspaces.}
\end{enumerate}

\hypertarget{the-d-uple-embedding-to_work}{%
\subsubsection{\texorpdfstring{2.12 The \(d\)-uple
Embedding}{2.12 The d-uple Embedding}}\label{the-d-uple-embedding-to_work}}

For given \(n, d>0\), let \(M_0, M_1, \ldots, M_N\) be all the monomials
of degree \(d\) in the \(n+1\) variables \(x_0, \ldots, x_n\), where
\(N={n+d\choose n}-1\). We define a mapping
\(\rho_d: \mathbf{P}^n \rightarrow \mathbf{P}^{N}\) by sending the point
\(P=\left(a_0, \ldots, a_n\right)\) to the point
\(\rho_d(P)=\left(M_0(a), \ldots, M_N(a)\right)\) obtained by
substituting the \(a_t\) in the monomials \(M_J\). This is called the
\(d\)-uple embedding of \(\mathbf{P}^n\) in \(\mathbf{P}^N\). For
example, if \(n=1, d=2\), then \(N=2\), and the image \(Y\) of the
2-uple embedding of \(\mathbf{P}^1\) in \(\mathbf{P}^2\) is a conic.

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  Let
  \(\theta: k\left[y_0, \ldots, y_v\right] \rightarrow k\left[x_0, \ldots, x_n\right]\)
  be the homomorphism defined by sending \(y_i\) to \(M_i\), and let a
  be the kernel of \(\theta\). Then \(a\) is a homogeneous prime ideal,
  and so \(Z\) (a) is a projective variety in \(\mathbf{P}^{N}\).
\item
  Show that the image of \(\rho_d\) is exactly \(Z(a)\). \footnote{One
    inclusion is easy. The other will require some calculation.}
\item
  Now show that \(\rho_d\) is a homeomorphism of \(\mathbf{P}^n\) onto
  the projective variety \(Z\) (a).
\item
  Show that the twisted cubic curve in \(\mathbf{P}^3\) (Ex. 2.9) is
  equal to the 3-uple embedding of \(\mathbf{P}^1\) in \(\mathbf{P}^3\),
  for suitable choice of coordinates.
\end{enumerate}

\hypertarget{section-19}{%
\subsubsection{2.13}\label{section-19}}

Let \(Y\) be the image of the 2-uple embedding of \(\mathbf{P}^2\) in
\(\mathbf{P}^5\). This is the Veronese surface. If \(Z \subseteq Y\) is
a closed curve (a \textbf{curve} is a variety of dimension 1), show that
there exists a hypersurface \(V \subseteq \mathbf{P}^5\) such that
\(V \cap Y = Z\).

\hypertarget{the-segre-embedding}{%
\subsubsection{2.14 The Segre Embedding}\label{the-segre-embedding}}

Let
\(\psi: \mathbf{P}^r \times \mathbf{P}^{s} \rightarrow \mathbf{P}^{N}\)
be the map defined by sending the ordered pair
\(\left(a_0, \ldots, a_r\right) \times\left(b_0, \ldots, b_s\right)\) to
\(\left(\ldots, a_i b_j, \ldots\right)\) in lexicographic order. where
\(N=r s+r+s\). Note that \(\psi\) is well-defined and injective. It is
called the Segre embedding. Show that the image of \(\psi\) is a
subvariety of \(\mathbf{P}^N\). \footnote{Hint: Let the homogeneous
  coordinates of \(\mathbf{P}^N\) be
  \(\left\{{z_{ij} {~\mathrel{\Big\vert}~}0\leq i, j\leq r}\right\}\)
  and let \(a\) be the kernel of the homomorphism
  \(k[\left\{{z_{ij}}\right\}] \to k[x_0,\cdots, x_r, y_0, \cdots, y_s]\),
  which sends \(z_{ij}\) to \(x_i y_j\). Then show that
  \(\operatorname{im}\psi = Z(a)\).}

\hypertarget{the-quadric-surface-in-mathbfp3-fig.-2-to_work}{%
\subsubsection{\texorpdfstring{2.15 The Quadric Surface in
\(\mathbf{P}^3\) (Fig.
2)}{2.15 The Quadric Surface in \textbackslash mathbf\{P\}\^{}3 (Fig. 2)}}\label{the-quadric-surface-in-mathbfp3-fig.-2-to_work}}

Consider the surface \(Q\) (a surface is a variety of dimension 2) in
\(\mathbf{P}^3\) defined by the equation \(x y-zw =0\).

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  Show that \(Q\) is equal to the Segre embedding of
  \(\mathbf{P}^1 \times \mathbf{P}^1\) in \(\mathbf{P}^3\). for suitable
  choice of coordinates.
\item
  Show that \(Q\) contains two families of lines (a line is a linear
  variety of dimension 1) \(\left\{L_t\right\},\left\{{M_t}\right\}\),
  each parametrized by \(t \in \mathbf{P}^1\). with the properties that
  if \(L_t \neq L_u\). then \(L_t \cap L_u=\varnothing\) : if
  \(M_t \neq M_u, M_t \cap M_u=\varnothing\), and for all \(t,u\),
  \(L_t \cap M_u=\) one point.
\item
  Show that \(Q\) contains other curves besides these lines, and deduce
  that the Zariski topology on \(Q\) is not homeomorphic via \(\psi\) to
  the product topology on \(\mathbf{P}^1 \times \mathbf{P}^1\) (where
  each \(\mathbf{P}^1\) has its Zariski topology).
\end{enumerate}

\includegraphics{attachments/2022-09-17_22-24-16.png}

\hypertarget{section-20}{%
\subsubsection{2.16}\label{section-20}}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  The intersection of two varieties need not be a variety. For example,
  let \(Q_1\) and \(Q_2\) be the quadric surfaces in \(\mathbf{P}^3\)
  given by the equations \(x^2-y w=0\) and \(x y-z w=0\), respectively.
  Show that \(Q_1 \cap Q_2\) is the union of a twisted cubic curve and a
  line.
\item
  Even if the intersection of two varieties is a variety, the ideal of
  the intersection may not be the sum of the ideals. For example, let
  \(C\) be the conic in \(\mathbf{P}^2\) given by the equation
  \(xy-zw=0\). Let \(L\) be the line given by \(y=0\). Show that
  \(C \cap L\) consists of one point \(P\), but that
  \(I(C)+I(L) \neq I(P)\).
\end{enumerate}

\hypertarget{complete-intersections}{%
\subsubsection{2.17 Complete
intersections}\label{complete-intersections}}

A variety \(Y\) of dimension \(r\) in \(\mathbf{P}^n\) is a (strict)
complete intersection if \(I(Y)\) can be generated by \(n-r\) elements.
\(Y\) is a set-theoretic complete intersection if \(Y\) can be written
as the intersection of \(n-r\) hypersurfaces.

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  Let \(Y\) be a variety in \(\mathbf{P}^n\), let \(Y=Z(a)\); and
  suppose that a can be generated by \(q\) elements. Then show that
  \(\operatorname{dim} Y \geqslant n-q\).
\item
  Show that a strict complete intersection is a set-theoretic complete
  intersection.
\item
  * The converse of (b) is false. For example let \(Y\) be the twisted
  cubic curve in \(\mathbf{P}^3\) (Ex. 2.9). Show that \(I(Y)\) cannot
  be generated by two elements. On the other hand, find hypersurfaces
  \(\mathrm{H}_1, \mathrm{H}_2\) of degrees 2,3 respectively, such that
  \(Y=H_1 \cap H_2\).
\item
  ** It is an unsolved problem whether every closed irreducible curve in
  \(\mathbf{P}^3\) is a set-theoretic intersection of two surfaces. See
  Hartshorne \([1]\) and Hartshorne \([5.III, \text{section } 5]\) for
  commentary.
\end{enumerate}

\hypertarget{i.3-morphisms}{%
\subsection{I.3: Morphisms}\label{i.3-morphisms}}

\hypertarget{section-21}{%
\subsubsection{3.1}\label{section-21}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that any conic in \(\mathbf{A}^2\) is isomorphic either to
  \(\mathbf{A}^1\) or \(\mathbf{A}^1\setminus\left\{{0}\right\}\)
  (cf.~Ex.1.1).
\item
  Show that \(\mathbf{A}^1\) is not isomorphic to any proper open subset
  of itself.\footnote{Use (b) above.}
\item
  Any conic in \({\mathbb{P}}^2\) is isomorphic to \({\mathbb{P}}^1\).
\item
  We will see later (Ex. 4.8) that any two curves are homeomorphic. But
  show now that \(\mathbf{A}^2\) is not even homeomorphic to
  \({\mathbb{P}}^2\).
\item
  If an affine variety is isomorphic to a projective variety, then it
  consists of only one point.
\end{enumerate}

\hypertarget{section-22}{%
\subsubsection{3.2}\label{section-22}}

A morphism whose underlying map on the topological spaces is a
homeomorphism need not be an isomorphism.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  For example, let \(\varphi: \mathbf{A}^1 \rightarrow \mathbf{A}^2\) be
  defined by \(t \mapsto\left(t^2, t^3\right)\). Show that \(\varphi\)
  defines a bijective bicontinuous morphism of \(\mathbf{A}^1\) onto the
  curve \(y^2=x^3\), but that \(\varphi\) is not an isomorphism.
\item
  For another example. let the characteristic of the base field \(k\) be
  \(p>0\), and define a map
  \(\rho: \mathbf{A}^1 \rightarrow \mathbf{A}^1\) by \(t \mapsto t^p\).
  Show that \(\varphi\) is bijective and bicontinuous but not an
  isomorphism. This is called the Frobenius morphism.
\end{enumerate}

\hypertarget{section-23}{%
\subsubsection{3.3}\label{section-23}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Let \(\varphi: X \rightarrow Y\) be a morphism. Then for each
  \(P \in X, \varphi\) induces a homomorphism of local rings
  \(\varphi_P^*: {\mathcal{O}}_{\phi(P), Y} \rightarrow {\mathcal{O}}_{P, Y}\).
\item
  Show that a morphism \(\varphi\) is an isomorphism if and only if
  \(\varphi\) is a homeomorphism, and the induced map \(\varphi_P^*\) on
  local rings is an isomorphism, for all \(P \in X\).
\item
  Show that if \(\varphi(X)\) is dense in \(Y\), then the map
  \(\rho_P^*\) is injective for all \(P \in X\).
\end{enumerate}

\hypertarget{section-24}{%
\subsubsection{3.4}\label{section-24}}

Show that the \(d{\hbox{-}}\)uple embedding of
\({\mathbb{P}}^n(\mathrm{Ex} .2 .12)\) is an isomorphism onto its image.

\hypertarget{section-25}{%
\subsubsection{3.5}\label{section-25}}

By abuse of language, we will say that a variety ``is affine'' if it is
isomorphic to an affine variety. If \(H \subseteq {\mathbb{P}}^n\) is
any hypersurface. show that \({\mathbb{P}}^n-H\) is affine.\footnote{Use
  (c) above.}

\hypertarget{there-are-quasi-affine-varieties-which-are-not-affine.}{%
\subsubsection{3.6 There are quasi-affine varieties which are not
affine.}\label{there-are-quasi-affine-varieties-which-are-not-affine.}}

For example, show that
\(\mathrm{I}=\mathbf{A}^2\setminus\left\{{ (0, 0) }\right\}\) is not
affine.\footnote{Use induction on \(\operatorname{dim} X\).}

\hypertarget{section-26}{%
\subsubsection{3.7}\label{section-26}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that any two curves in \({\mathbb{P}}^2\) have a nonempty
  intersection.
\item
  More generally, show that if \(Y \subseteq {\mathbb{P}}^n\) is a
  projective variety of dimension \(\geqslant 1\). and if \(H\) is a
  hypersurface. then \(Y \cap H \neq \varnothing\).\footnote{Note: We
    will give another proof of this result using sheaves of ideals later
    (V.10).}
\end{enumerate}

\hypertarget{section-27}{%
\subsubsection{3.8}\label{section-27}}

Let \(H_1\) and \(H\), be the hyperplanes in \({\mathbb{P}}^n\) defined
by \(x_1=0\) and \(x_{,}=0\), with \(i \neq j\). Show that any regular
function on \({\mathbb{P}}^n-\left(H_1 \cap H_1\right)\) is
constant.\footnote{This gives an alternate proof of \((3.4 \mathrm{a})\)
  in the case \(Y={\mathbb{P}}^n\).}

\hypertarget{section-28}{%
\subsubsection{3.9}\label{section-28}}

The homogeneous coordinate ring of a projective variety is not invariant
under isomorphism. For example, let \(X={\mathbb{P}}^1\). and let \(Y\)
be the 2-uple embedding of \({\mathbb{P}}^1\) in \({\mathbb{P}}^2\).
Then \(X \cong Y(\) Ex. 3.4). But show that \(S(X) \equiv S(Y)\).

\hypertarget{subvarieties.}{%
\subsubsection{3.10 Subvarieties.}\label{subvarieties.}}

A subset of a topological space is locally closed if it is an open
subset of its closure. or. equivalently. if it is the intersection of an
open set with a closed set.

If \(X\) is a quasi-affine or quasi-projective variety and \(Y\) is an
irreducible locally closed subset. then \(I\) is also a quasi-affine
(respectively, quasi-projective) variety by virtue of being a locally
closed subset of the same affine or projective space. We call this the
induced structure on Y. and we call \(Y\) a subvariety of \(X\).

Now let \(\varphi: X \rightarrow Y\) he a morphism. let
\(X^{\prime} \subseteq X\) and \(Y^{\prime} \subseteq Y\) be irreducible
locally closed subsets such that
\(\varphi\left(X^{\prime}\right) \subseteq Y^{\prime}\). Show that
\(\left.\varphi\right|_{X}: X^{\prime } \rightarrow Y^{\prime}\) is a
morphism.

\hypertarget{section-29}{%
\subsubsection{3.11}\label{section-29}}

Let \(X\) be any variety and let \(P \in X\). Show there is a 1-1
correspondence between the prime ideals of the local ring
\({\mathcal{O}}_P\) and the closed subvarieties of \(X\) containing
\(P\).

\hypertarget{section-30}{%
\subsubsection{3.12}\label{section-30}}

If \(P\) is a point on a variety \(X\), then
\(\operatorname{dim} {\mathcal{O}}_P =\operatorname{dim} X\).\footnote{Hint:
  Reduce 10 the affine case and use (3.2c)}

\hypertarget{the-local-ring-of-a-subvariety}{%
\subsubsection{3.13 The Local Ring of a
Subvariety}\label{the-local-ring-of-a-subvariety}}

Let \(Y \subseteq X\) be a subvariety. Let \({\mathcal{O}}_{Y,X}\) be
the set of equivalence classes \(\langle L, f\rangle\) where
\(L \subseteq X\) is open. \(L \cap Y \neq \varnothing\), and \(f\) is a
regular function on \(L\). We say \(\langle L , f\rangle\) is equivalent
to \(\left\langle{V, g}\right\rangle\) if \(f=g\) on \(U \cap V\).

Show that \({\mathcal{O}}_{Y , X}\) is a local ring, with residue field
\(K(Y)\) and \(\operatorname{dimension}=\operatorname{dim} \mathrm{X}-\)
\(\operatorname{dim} Y\). It is the local ring of \(Y\) on \(X\). Note
if \(Y=P\) is a point we get \({\mathcal{O}}_P\). and if \(Y=X\) we get
\(K(X)\). Note also that if \(Y\) is not a point, then \(K(Y)\) is not
algebraically closed, so in this way we get local rings whose residue
fields are not algebraically closed.

\hypertarget{projection-from-a-point.}{%
\subsubsection{3.14 Projection from a
Point.}\label{projection-from-a-point.}}

Let \({\mathbb{P}}^n\) be a hyperplane in \({\mathbb{P}}^{n+1}\) and let
\(P \in {\mathbb{P}}^{n+1}-{\mathbb{P}}^n\). Define a mapping
\(\varphi: {\mathbb{P}}^{n+1}\setminus\left\{{ P }\right\}\to {\mathbb{P}}^n\)
by \(\varphi(Q)=\) the intersection of the unique line containing \(P\)
and \(Q\) with \({\mathbb{P}}^n\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \(\varphi\) is a morphism.
\item
  Let \(Y \subseteq {\mathbb{P}}^3\) be the twisted cubic curve which is
  the image of the 3-uple embedding of \({\mathbb{P}}^1\) (Ex. 2.12). If
  \(t,u\) are the homogeneous coordinates on \({\mathbb{P}}^1\). we say
  that \(Y\) is the curve given parametrically by
  \((x, y, z, w)=\left(t^3, t^2 u, t u^2, u^3\right)\). Let
  \(P=(0,0,1,0)\), and let \({\mathbb{P}}^2\) be the hyperplane \(z=0\).
  Show that the projection of \(Y\) from \(P\) is a cuspidal cubic curve
  in the plane, and find its equation.
\end{enumerate}

\hypertarget{products-of-affine-varieties.}{%
\subsubsection{3.15 Products of Affine
Varieties.}\label{products-of-affine-varieties.}}

Let \(X \subseteq \mathbf{A}^n\) and \(Y \subseteq \mathbf{A}^m\) be
affine varieties.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \(X \times Y \subseteq \mathbf{A}^{n+m}\) with its induced
  topology is irreducible.\footnote{Hint: Suppose that \(X \times Y\) is
    a union of two closed subsets \(Z_1 \cup Z_2\). Let \(X_1=\)
    \(\left\{x \in X \mathrel{\Big|}x \times Y \subseteq Z_1\right\}, i=1,2\).
    Show that \(X=X_1 \cup X_2\) and \(X_1, X_2\) are closed. Then
    \(X=X_1\) or \(X_2\) so \(X \times Y=Z_1\) or \(Z_2\).} The affine
  variety \(X \times Y\) is called the product of \(X\) and \(Y\). Note
  that its topology is in general not equal to the product topology (Ex.
  1.4).
\item
  Show that \(A(X \times Y) \cong A(X) \otimes_k A(Y)\).
\item
  Show that \(X \times Y\) is a product in the category of varieties,
  i.e., show

  \begin{itemize}
  \tightlist
  \item
    the projections \(X \times Y \rightarrow X\) and
    \(X \times Y \rightarrow Y\) are morphisms, and
  \item
    given a variety \(Z\), and the morphisms
    \(Z \rightarrow X, Z \rightarrow Y\). there is a unique morphism
    \(Z \rightarrow X \times Y\) making a commutative diagram
  \end{itemize}
\end{enumerate}

\begin{tikzcd}
    Z && {X\times Y} \\
    \\
    & X && Y
    \arrow[from=1-1, to=3-2]
    \arrow[from=1-1, to=3-4]
    \arrow[from=1-3, to=3-2]
    \arrow[from=1-3, to=3-4]
    \arrow[from=1-1, to=1-3]
\end{tikzcd}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJaIl0sWzIsMCwiWFxcdGltZXMgWSJdLFsxLDIsIlgiXSxbMywyLCJZIl0sWzAsMl0sWzAsM10sWzEsMl0sWzEsM10sWzAsMV1d}{Link
to Diagram}
\end{quote}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\setcounter{enumi}{3}
\tightlist
\item
  Show that
  \(\operatorname{dim} X \times Y=\operatorname{dim} X+\operatorname{dim} Y\).
\end{enumerate}

\hypertarget{products-of-quasi-projective-varieties.}{%
\subsubsection{3.16 Products of Quasi-Projective
Varieties.}\label{products-of-quasi-projective-varieties.}}

Use the Segre embedding (Ex. 2.14) to identify
\({\mathbb{P}}^n \times {\mathbb{P}}^m\) with its image and hence give
it a structure of projective varieties. Now for any two quasi-projective
varieties \(X \subseteq {\mathbb{P}}^n\) and
\(Y \subseteq {\mathbb{P}}^m\), consider
\(X \times Y \subseteq {\mathbb{P}}^n \times {\mathbb{P}}^m\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \(X \times Y\) is a quasi-projective variety.
\item
  If \(X, Y\) are both projective, show that \(X \times Y\) is
  projective.
\item
  Show that \(X \times Y\) is a product in the category of varieties.
\end{enumerate}

\hypertarget{normal-varieties.}{%
\subsubsection{3.17 Normal Varieties.}\label{normal-varieties.}}

A variety \(Y\) is normal at a point \(P \in Y\) if \({\mathcal{O}}_P\)
is an integrally closed ring. \(Y\) is normal if it is normal at every
point.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that every conic in \({\mathbb{P}}^2\) is normal.
\item
  Show that the quadric surfaces \(Q_1, Q_2\) in \(\mathrm{P}^3\) given
  by equations \(Q_1: x y=zw\); \(Q_2: xy=z^2\) are normal. (cf.~(II.
  Ex. 6.4) for the latter.)
\item
  Show that the cuspidal cubic \(y^2=x^3\) in \(\mathbf{A}^2\) is not
  normal.
\item
  If \(Y\) is affine, then \(Y\) is normal \(\Leftrightarrow A(Y)\) is
  integrally closed.
\item
  Let \(Y\) be an affine variety. Show that there is a normal affine
  variety \(\tilde{Y}\), and a morphism
  \(\pi: \tilde{Y} \rightarrow Y\), with the property that whenever
  \(Z\) is a normal variety, and \(\varphi: Z \rightarrow Y\) is a
  \textbf{dominant} morphism (i.e., \(\varphi(Z)\) is dense in \(Y\)),
  then there is a unique morphism \(\theta: Z \rightarrow \tilde{Y}\)
  such that \(\varphi=\pi \quad \theta\). \(\tilde{Y}\) is called the
  \textbf{normalization} of \(Y\). You will need \((3.9 \mathrm{~A})\)
  above.
\end{enumerate}

\hypertarget{projectively-normal-varieties.}{%
\subsubsection{3.18 Projectively Normal
Varieties.}\label{projectively-normal-varieties.}}

A projective variety \(Y \subseteq \mathrm{P}^n\) is projectively normal
(with respect to the given embedding) if its homogeneous coordinate ring
\(S\left(Y \right)\) is integrally closed.

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  If \(Y\) is projectively normal, then \(Y\) is normal.
\item
  There are normal varieties in projective space which are not
  projectively normal. For example, let \(Y\) be the twisted quartic
  curve in \({\mathbb{P}}^3\) given parametrically by
  \((x, y: z, w)=\left(t^4, t^3 u, t u^3, u^4\right)\). Then \(Y\) is
  normal but not projectively normal. See (III, Ex. 5.6) for more
  examples.
\item
  Show that the twisted quartic curve \(Y\) above is isomorphic to
  \({\mathbb{P}}^1\). which is projectively normal. Thus projective
  normality depends on the embedding.
\end{enumerate}

\hypertarget{automorphisms-of-mathbfan.-to_work}{%
\subsubsection{\texorpdfstring{3.19 Automorphisms of
\(\mathbf{A}^n\).}{3.19 Automorphisms of \textbackslash mathbf\{A\}\^{}n.}}\label{automorphisms-of-mathbfan.-to_work}}

Let \(\varphi: \mathbf{A}^n \rightarrow \mathbf{A}^n\) be a morphism of
\(\mathbf{A}^n\) to \(\mathbf{A}^n\) given by \(n\) polynomials
\(f_1 \ldots . f_n\) of \(n\) variables \(x_1, \ldots x_n\). Let
\(J=\operatorname{det}\left[ {\frac{\partial f_i}{\partial x_j}\,} \right]\)
be the Jacobian polynomial of \(\varphi\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  If \(\varphi\) is an isomorphism (in which case we call \(\varphi\) an
  automorphism of \(\mathbf{A}^n\) ) show that \(J\) is a nonzero
  constant polynomial.
\item
  ** The converse of 1.is an unsolved problem, even for \(n=2\). See,
  for example, Vitushkin.
\end{enumerate}

\hypertarget{section-31}{%
\subsubsection{3.20}\label{section-31}}

Let \(Y\) be a variety of dimension \(\geqslant 2\), and let \(P \in Y\)
be a normal point. Let \(f\) be a regular function on \(Y-P\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \(f\) extends to a regular function on \(Y\).
\item
  Show this would be false for \(\operatorname{dim} Y=1\). See (III. Ex.
  3.5) for generalization.
\end{enumerate}

\hypertarget{group-varieties.}{%
\subsubsection{3.21. Group Varieties.}\label{group-varieties.}}

A group variety consists of a variety Y together with a morphism
\(\mu: Y \times Y \rightarrow Y\). such that the set of points of \(Y\)
with the operation given by \(\mu\) is a group. and such that the
inverse map \(y^{-y^{-1}}\) is also a morphism of \(Y \rightarrow Y\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  The additive group \(\mathbf{G}_a\) is given by the variety
  \(\mathbf{A}^1\) and the morphism
  \(\mu: \mathbf{A}^2 \rightarrow \mathbf{A}^1\) defined by
  \(\mu(a, b) = a+b\). Show it is a group variety.
\item
  The multiplicative croup \(\mathbf{G}_m\) is given by the variety
  \(\mathbf{A}^1\setminus\left\{{0}\right\}\), and the morphism
  \(\mu(a, b) = ab\). Show \(|\) in a group variety.
\item
  If \(G\) is a group variety, and \(X\) is any variety. show that the
  set \(\operatorname{Hom}(X, G)\) has a natural group structure.
\item
  For any variety \(X\), show that
  \(\operatorname{Hom}\left(X, \mathbf{G}_a\right)\) is isomorphic to ('
  (X) as a group under addition.
\item
  For any variety \(X\), show that
  \(\operatorname{Hom}\left(X, \mathbf{G}_m\right)\) is isomorphic to
  the group of units in \({\mathcal{O}}(X)\), under multiplication.
\end{enumerate}

\hypertarget{i.4-rational-maps}{%
\subsection{I.4: Rational Maps}\label{i.4-rational-maps}}

\hypertarget{section-32}{%
\subsubsection{4.1.}\label{section-32}}

If \(f\) and \(g\) are regular functions on open subsets \(U\) and \(V\)
of a variety \(X\), and if \(f=g\) on \(U \cap V\). show that the
function which is \(f\) on \(U\) and \(g\) on \(V\) is a regular
function on \(U \cup V\). Conclude that if \(f\) is a rational function
on \(X\), then there is a largest open subset \(U\) of \(X\) on which
\(f\) is represented by a regular function. We say that \(f\) is defined
at the points of \(U\).

\hypertarget{same-problem-for-rational-maps.}{%
\subsubsection{4.2. Same problem for rational
maps.}\label{same-problem-for-rational-maps.}}

If \(\varphi\) is a rational map of \(X\) to \(Y\), show there is a
largest open set on which \(\varphi\) is represented by a morphism. We
say the rational map is defined at the points of that open set.

\hypertarget{section-33}{%
\subsubsection{4.3.}\label{section-33}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Let \(f\) be the rational function on \({\mathbb{P}}^2\) given by
  \(f=x_1 / x_0\). Find the set of points where \(f\) is defined and
  describe the corresponding regular function.
\item
  Now think of this function as a rational map from \({\mathbb{P}}^2\)
  to \(\mathbf{A}^1\). Embed \(\mathbf{A}^1\) in \({\mathbb{P}}^1\), and
  let \(\varphi: {\mathbb{P}}^2 \rightarrow {\mathbb{P}}^1\) be the
  resulting rational map. Find the set of points where \(\varphi\) is
  defined, and describe the corresponding morphism.
\end{enumerate}

\hypertarget{section-34}{%
\subsubsection{4.4.}\label{section-34}}

A variety \(Y\) is rational if it is birationally equivalent to
\({\mathbb{P}}^n\) for some \(n\) (or, equivalently by (4.5), if
\(K(Y)\) is a pure transcendental extension of \(k\) ).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Any conic in \({\mathbb{P}}^2\) is a rational curve.
\item
  The cuspidal cubic \(y^2=x^3\) is a rational curve.
\item
  Let \(Y\) be the nodal cubic curve \(y^2 z=x^2(x+z)\) in
  \({\mathbb{P}}^2\). Show that the projection \(\varphi\) from the
  point \(P=(0,0,1)\) to the line \(z=0\) (Ex. 3.14) induces a
  birational map from \(Y\) to \({\mathbb{P}}^1\). Thus \(Y\) is a
  rational curve.
\end{enumerate}

\hypertarget{section-35}{%
\subsubsection{4.5.}\label{section-35}}

Show that the quadric surface \(Q: x y=z w\) in \({\mathbb{P}}^3\) is
birational to \({\mathbb{P}}^2\), but not isomorphic to
\({\mathbb{P}}^2\) (cf.~Ex. 2.15).

\hypertarget{plane-cremona-transformations.}{%
\subsubsection{4.6. Plane Cremona
Transformations.}\label{plane-cremona-transformations.}}

A birational map of \({\mathbb{P}}^2\) into itself is called a plane
Cremona transformation. We give an example, called a quadratic
transformation. It is the rational map
\(\varphi: {\mathbb{P}}^2 \rightarrow {\mathbb{P}}^2\) given by
\(\left(a_0, a_1, a_2\right) \rightarrow\left(a_1 a_2, a_0 a_2, a_0 a_1\right)\)
when no two of \(a_0, a_1, a_2\) are \(0\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \(\varphi\) is birational, and is its own inverse.
\item
  Find open sets \(U, V \subseteq {\mathbb{P}}^2\) such that
  \(\varphi: U \rightarrow V\) is an isomorphism.
\item
  Find the open sets where \(\varphi\) and \(\varphi^{-1}\) are defined.
  and describe the corresponding morphisms. See also (Chapter V, 4.2.3).
\end{enumerate}

\hypertarget{section-36}{%
\subsubsection{4.7.}\label{section-36}}

Let \(X\) and \(Y\) be two varieties. Suppose there are points
\(P \in X\) and \(Q \in Y\) such that the local rings
\({\mathcal{O}}_{P, X}\) and \({\mathcal{O}}_{Q, Y}\) are isomorphic as
\(k{\hbox{-}}\)algebras. Then show that there are open sets
\(P \in U \subseteq X\) and \(Q \in V \subseteq Y\) and an isomorphism
of \(U\) to \(V\) which sends \(P\) to \(Q\).

\hypertarget{section-37}{%
\subsubsection{4.8.}\label{section-37}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that any variety of positive dimension over \(k\) has the same
  cardinality as \(k\).\footnote{Use (b) above.}
\item
  Deduce that any two curves over \(k\) are homeomorphic (cf.~Ex. 3.1).
\end{enumerate}

\hypertarget{section-38}{%
\subsubsection{4.9.}\label{section-38}}

Let \(X\) be a projective variety of dimension \(r\) in \(P^n\). with
\(n \geqslant r+2\). Show that for suitable choice of \(P \notin X\).
and a linear \({\mathbb{P}}^{n-1} \subseteq {\mathbb{P}}^n\). the
projection from \(P\) to \({\mathbb{P}}^{n-1}\) (Ex. 3.14) induces a
birational morphism of \(X\) onto its image
\(X' \subseteq {\mathbb{P}}^{n-1}\). You will need to use (4.6A).
(4.7A). and (4.8A). This shows in particular that the birational map of
(4.9) can be obtained by a finite number of such projections.

\hypertarget{section-39}{%
\subsubsection{4.10.}\label{section-39}}

Let \(Y\) be the cuspidal cubic curve \(y^2=x^{3}\) in \(\mathbf{A}^2\).
Blow up the point \(O=(0.0)\). Let \(E\) be the exceptional curve. and
let \(\tilde{Y}\) be the strict transform of \(Y\). Show that \(E\)
meets \(\tilde{Y}\) in one point. and that
\(\tilde{Y} \cong \mathbf{A}^1\). In this case the morphism
\(\rho: \tilde{Y} \rightarrow Y\) is bijective and bicontinuous. but it
is not an isomorphism.

\hypertarget{i.5-nonsingular-varieties}{%
\subsection{I.5: Nonsingular
Varieties}\label{i.5-nonsingular-varieties}}

\hypertarget{section-40}{%
\subsubsection{5.1.}\label{section-40}}

Locate the singular points and sketch the following curves in
\({\mathbb{A}}^2\) (assume char \(k \neq 2\) ). Which is which in Figure
4 ?

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(x^2=x^4+y^4\) :
\item
  \(x y=x^6+y^6\) :
\item
  \(x^3=y^2+x^4+y^4\) :
\item
  \(x^2 y+x y^2=x^4+y^4\).
\end{enumerate}

\includegraphics{attachments/2022-09-21_00-14-42.png}

\hypertarget{section-41}{%
\subsubsection{5.2.}\label{section-41}}

Locate the singular points and describe the singularities of the
following surfaces in \({\mathbb{A}}^3\) (assume char \(k \neq 2\) ).
Which is which in Figure 5?

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  \(x y^2=z^2\)
\item
  \(x^2+y^2=z^2\)
\item
  \(xy + x^3 + y^3 = 0\).
\end{enumerate}

\includegraphics{attachments/2022-09-21_00-15-32.png}

\hypertarget{multiplicities.}{%
\subsubsection{5.3. Multiplicities.}\label{multiplicities.}}

Let \(Y \subseteq {\mathbb{A}}^2\) be a curve defined by the equation
\(f(x, y)=0\). Let \(P=(a, b)\) be a point of \({\mathbb{A}}^2\). Make a
linear change of coordinates so that \(P\) becomes the point \((0,0)\).
Then write \(f\) as a sum \(f=f_0+f_1+\ldots+f_d\), where \(f_i\) is a
homogeneous polynomial of degree \(i\) in \(x\) and \(y\). Then we
define the multiplicity of \(P\) on \(Y\), denoted \(\mu_P(Y)\), to be
the least \(r\) such that \(f_r \neq 0\). (Note that
\(P \in Y \Leftrightarrow \mu_P(Y)>0\).) The linear factors of \(f_r\)
are called the tangent directions at \(P\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \(\mu_P(Y) =1 \iff P\) is a nonsingular point of \(Y\).
\item
  Find the multiplicity of each of the singular points in (Ex. 5.1)
  above.
\end{enumerate}

\hypertarget{intersection-multiplicity.}{%
\subsubsection{5.4. Intersection
Multiplicity.}\label{intersection-multiplicity.}}

If \(Y, Z \subseteq {\mathbb{A}}^2\) are two distinct curves, given by
equations \(f=0, g=0\), and if \(P \in Y \cap Z\), we define the
intersection multiplicity \((Y \cdot Z)_P\) of \(Y\) and \(Z\) at \(P\)
to be the length of the \({\mathcal{O}}_P\)-module
\({\mathcal{O}}_P /\left\langle{f, g}\right\rangle\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \((Y \cdot Z)_P\) is finite, and
  \((Y \cdot Z)_P \geqslant \mu_P(Y) \cdot \mu_P(Z)\).
\item
  If \(P \in Y\), show that for almost all lines \(L\) through \(P\)
  (i.e., all but a finite number), \((L \cdot Y)_P=\mu_P(Y)\).
\item
  If \(Y\) is a curve of degree \(d\) in \({\mathbb{P}}^2\), and if
  \(L\) is a line in \({\mathbb{P}}^2, L \neq Y\), show that
  \((L \cdot Y)=d\). Here we define \((L \cdot Y)=\sum(L \cdot Y)_P\)
  taken over all points \(P \in\) \(L \cap Y\), where \((L \cdot Y)_p\)
  is defined using a suitable affine cover of \({\mathbb{P}}^2\).
\end{enumerate}

\hypertarget{section-42}{%
\subsubsection{5.5.}\label{section-42}}

For every degree \(d>0\), and every \(p=0\) or a prime number, give the
equation of a nonsingular curve of degree \(d\) in \({\mathbb{P}}^2\)
over a field \(k\) of characteristic \(p\).

\hypertarget{blowing-up-curve-singularities.}{%
\subsubsection{5.6. Blowing Up Curve
Singularities.}\label{blowing-up-curve-singularities.}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Let \(Y\) be the cusp or node of (Ex. 5.1). Show that the curve
  \(\tilde{Y}\) obtained by blowing up \(Y\) at \(O=(0,0)\) is
  nonsingular (cf.~(4.9.1) and (Ex. 4.10)).
\item
  We define a node (also called ordinary double point) to be a double
  point (i.e., a point of multiplicity 2 ) of a plane curve with
  distinct tangent directions (Ex. 5.3). If \(P\) is a node on a plane
  curve \(Y\), show that \(\varphi^{-1}(P)\) consists of two distinct
  nonsingular points on the blown-up curve \(\tilde{Y}\). We say that
  ``blowing up \(P\) resolves the singularity at \(P\)''.
\item
  Let \(P \in Y\) be the tacnode of \((\) Ex. 5.1). If
  \(\varphi: \tilde{Y} \rightarrow Y\) is the blowing-up at \(P\). show
  that \(\rho^{-1}(P)\) is a node. Using 2. we see that the tacnode can
  be resolved by two successive blowings-up.
\item
  Let \(Y\) be the plane curve \(y^3=x^5\), which has a ``higher order
  cusp'' at \(O\). Show that \(O\) is a triple point: that blowing up
  \(O\) gives rise to a double point (what kind?) and that one further
  blowing up resolves the singularity.
\end{enumerate}

Note: We will see later \((\mathrm{V}, 3.8)\) that any singular point of
a plane curve can be resolved by a finite sequence of successive
blowings-up.

\hypertarget{section-43}{%
\subsubsection{5.7.}\label{section-43}}

Let \(Y \subseteq {\mathbb{P}}^2\) be a nonsingular plane curve of
degree \(>1\), defined by the equation \(f(x, y, z)=0\). Let
\(X \subseteq {\mathbb{A}}^3\) be the affine variety defined by \(f\)
(this is the cone over \(Y\); see (Ex. 2.10) ). Let \(P\) be the point
\((0,0,0)\), which is the vertex of the cone. Let
\(\varphi: \tilde{X} \rightarrow X\) be the blowing-up of \(X\) at
\(P\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \(X\) has just one singular point, namely \(P\).
\item
  Show that \(\tilde{X}\) is nonsingular (cover it with open affines).
\item
  Show that \(\varphi^{-1}(P)\) is isomorphic to \(Y\).
\end{enumerate}

\hypertarget{section-44}{%
\subsubsection{5.8.}\label{section-44}}

Let \(Y \subseteq {\mathbb{P}}^n\) be a projective variety of dimension
\(r\). Let \(f_1, \ldots, f_t \in S=\)
\(k\left[x_0, \ldots, x_n\right]\) be homogeneous polynomials which
generate the ideal of \(Y\). Let \(P \in Y\) be a point, with
homogeneous coordinates \(P=\left(a_0, \ldots, a_n\right)\). Show that
\(P\) is nonsingular on \(Y\) if and only if the rank of the matrix
\(\left[ {\frac{\partial f_i}{\partial x_j}\,}(a_0,\cdots, a_n)\right]\)
is \(n-r\).\footnote{Use (b) above.}

\hypertarget{section-45}{%
\subsubsection{5.9.}\label{section-45}}

Let \(f \in k[x, y ; z]\) be a homogeneous polynomial, let
\(Y=Z(f) \subseteq {\mathbb{P}}^2\) be the algebraic set defined by
\(f\), and suppose that for every \(P \in Y\), at least one of
\({\frac{\partial f}{\partial x}\,}(P), {\frac{\partial f}{\partial y}\,}(P), {\frac{\partial f}{\partial z}\,}(P)\)
is nonzero. Show that \(f\) is irreducible (and hence that \(Y\) is a
nonsingular variety). \footnote{Use (c) above.}

\hypertarget{section-46}{%
\subsubsection{5.10.}\label{section-46}}

For a point \(P\) on a variety \(X\). let \({\mathfrak{m}}\) be the
maximal ideal of the local ring \({\mathcal{O}}_P\). We define the
Zariski tangent space \(T_P(X)\) of \(X\) at \(P\) to be the dual
\(k\)-vector space of \({\mathfrak{m}}/{\mathfrak{m}}^2\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  For any point \(P \in X\).
  \(\operatorname{dim} T_P(X) \geqslant \operatorname{dim} X\). with
  equality if and only if \(P\) is nonsingular.
\item
  For any morphism \(\varphi: X \rightarrow Y\), there is a natural
  induced \(k\)-linear map \(T_P(\varphi)\) :
  \(T_P(X) \rightarrow T_{\varphi(P)}(Y)\)
\item
  If \(\varphi\) is the vertical projection of the parabola \(x=y^2\)
  onto the \(x\)-axis, show that the induced map \(T_0(\varphi)\) of
  tangent spaces at the origin is the zero map.
\end{enumerate}

\hypertarget{the-elliptic-quartic-curve-in-mathbbp3.-to_work}{%
\subsubsection{\texorpdfstring{5.11. The Elliptic Quartic Curve in
\({\mathbb{P}}^3\).}{5.11. The Elliptic Quartic Curve in \{\textbackslash mathbb\{P\}\}\^{}3.}}\label{the-elliptic-quartic-curve-in-mathbbp3.-to_work}}

Let \(Y\) be the algebraic set in \({\mathbb{P}}^3\) defined by the
equations \(x^2-x z- yw=0\) and \(yz -xw - zw = 0\). Let \(P\) be the
point \((x, y, z, w)=(0,0,0,1)\). and let \(\varphi\) denote the
projection from \(P\) to the plane \(w=0\). Show that \(\varphi\)
induces an isomorphism of \(Y-P\) with the plane cubic curve
\(y^2 z-x^3+x z^2=0\) minus the point \((1,0,-1)\). Then show that \(Y\)
is an irreducible nonsingular curve. It is called the elliptic quartic
curve in \({\mathbb{P}}^3\). Since it is defined by two equations it is
another example of a complete intersection (Ex. 2.17).

\hypertarget{quadric-hypersurfaces.}{%
\subsubsection{5.12. Quadric
Hypersurfaces.}\label{quadric-hypersurfaces.}}

Assume char \(k \neq 2\). and let \(f\) be a homogeneous polynomial of
degree 2 in \(x_0 \ldots \ldots x_n\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that after a suitable linear change of variables, \(f\) can be
  brought into the form \(f=x_0^2+\ldots+x_r^2\) for some
  \(0 \leqslant r \leqslant n\).
\item
  Show that \(f\) is irreducible if and only if \(r \geqslant 2\).
\item
  Assume \(r \geqslant 2\), and let \(Q\) be the quadric hypersurface in
  \({\mathbb{P}}^n\) defined by \(f\). Show that the singular locus
  \(Z=\operatorname{Sing} Q\) of \(Q\) is a linear variety (Ex. 2.11) of
  dimension \(n-r-1\). In particular, \(Q\) is nonsingular if and only
  if \(r=n\).
\item
  In case \(r<n\), show that \(Q\) is a cone with axis \(Z\) over a
  nonsingular quadric hypersurface
  \(Q^{\prime} \subseteq {\mathbb{P}}^r\).\footnote{Use induction on
    \(\operatorname{dim} X\).}
\end{enumerate}

\hypertarget{section-47}{%
\subsubsection{5.13.}\label{section-47}}

It is a fact that any regular local ring is an integrally closed domain
(Matsumura \([2. \mathrm{Th}. 36, p. 121]\)). Thus we see from (5.3)
that any variety has a nonempty open subset of normal points (Ex. 3.17).
In this exercise, show directly (without using (5.3)) that the set of
nonnormal points of a variety is a proper closed subset (you will need
the finiteness of integral closure: see (3.9A)).

\hypertarget{analytically-isomorphic-singularities.}{%
\subsubsection{5.14. Analytically Isomorphic
Singularities.}\label{analytically-isomorphic-singularities.}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  If \(P \in Y\) and \(Q \in Z\) are analytically isomorphic plane curve
  singularities, show that the multiplicities \(\mu_P(Y)\) and
  \(\mu_Q(Z)\) are the same (Ex. 5.3).
\item
  Generalize the example in the text \((5.6 .3)\) to show that if
  \(f=f_r+f_{r+1}+\ldots \in\) \(k{\left[\left[ x, y \right]\right] }\),
  and if the leading form \(f_r\) of \(f\) factors as \(f_r=g_s h_t\),
  where \(g_s, h_t\) are homogeneous of degrees \(s\) and \(t\)
  respectively, and have no common linear factor, then there are formal
  power series in \(k{\left[\left[ x, y \right]\right] }\) such that
  \(f=gh\).
\item
  Let \(Y\) be defined by the equation \(f(x, y)=0\) in
  \({\mathbb{A}}^2\), and let \(P=(0,0)\) be a point of multiplicity
  \(r\) on \(Y\), so that when \(f\) is expanded as a polynomial in
  \(x\) and \(y\), we have \(f=f_r+\) higher terms. We say that \(P\) is
  an ordinary \(r{\hbox{-}}\)fold point if \(f_r\) is a product of \(r\)
  distinct linear factors. Show that any two ordinary double points are
  analytically isomorphic. Ditto for ordinary triple points. But show
  that there is a one-parameter family of mutually nonisomorphic
  ordinary 4-fold points.
\item
  * Assume char \(k \neq 2\). Show that any double point of a plane
  curve is analytically isomorphic to the singularity at \((0,0)\) of
  the curve \(y^{2}=x^r\), for a uniquely determined \(r \geqslant 2\).
  If \(r=2\) it is a node (Ex. 5.6). If \(r=3\) we call it a cusp: if
  \(r=4\) a tacnode. See \((\mathrm{V}, 3.9 .5)\) for further
  discussion.
\end{enumerate}

\hypertarget{families-of-plane-curves.}{%
\subsubsection{5.15. Families of Plane
Curves.}\label{families-of-plane-curves.}}

A homogeneous polynomial \(f\) of degree \(d\) in three variables
\(x, y, z\) has \({d+2\choose 2}\) coefficients. Let these coefficients
represent a point in \({\mathbb{P}}^N\). where
\(N= {d+2\choose 2} - 1 = {1\over 2}d(d+3)\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that this gives a correspondence between points of
  \({\mathbb{P}}^{N}\) and algebraic sets in \({\mathbb{P}}^2\) which
  can be defined by an equation of degree \(d\). The correspondence is
  1-1 except in some cases where \(f\) has a multiple factor.
\item
  Show under this correspondence that the (irreducible) nonsingular
  curves of degree \(d\) correspond 1-1 to the points of a nonempty
  Zariski-open subset of \({\mathbb{P}}^N\). \footnote{Note: We will
    give another proof of this result using sheaves of ideals later
    (V.10).}
\end{enumerate}

\hypertarget{i.6-nonsingular-curves}{%
\subsection{I.6: Nonsingular Curves}\label{i.6-nonsingular-curves}}

\hypertarget{section-48}{%
\subsubsection{6.1.}\label{section-48}}

Recall that a curve is rational if it is birationally equivalent to
\(\mathbf{P}^1(\mathrm{Ex} .4 .4)\). Let \(Y\) be a nonsingular rational
curve which is not isomorphic to \(\mathbf{P}^1\).

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  Show that \(Y\) is isomorphic to an open subset of \(\mathbf{A}^1\).
\item
  Show that \(Y\) is affine.
\item
  Show that \(A(Y)\) is a unique factorization domain.
\end{enumerate}

\hypertarget{an-elliptic-curve.}{%
\subsubsection{6.2. An Elliptic Curve.}\label{an-elliptic-curve.}}

Let \(Y\) be the curve \(y^2=x^3-x\) in \(\mathbf{A}^2\), and assume
that the characteristic of the base field \(k\) is \(\neq 2\). In this
exercise we will show that \(Y\) is not a rational curve, and hence
\(K(Y)\) is not a pure transcendental extension of \(k\).

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  Show that \(Y\) is nonsingular, and deduce that
  \(A=A(Y) \simeq k[x, y] /\left(y^2-x^3+x\right)\) is an integrally
  closed domain.
\item
  Let \(k[x]\) be the subring of \(K=K(Y)\) generated by the image of
  \(x\) in \(A\). Show that \(k[x]\) is a polynomial ring, and that
  \(A\) is the integral closure of \(k[x]\) in \(K\).
\item
  Show that there is an automorphism \(\sigma: A \rightarrow A\) which
  sends \(y\) to \(-y\) and leaves \(x\) fixed. For any \(a \in A\),
  define the norm of \(a\) to be \(N(a)=a \cdot \sigma(a)\). Show that
  \(N(a) \in k[x], N(1)=1\), and \(N(a b)=N(a) \cdot N(b)\) for any
  \(a, b \in A\).
\item
  Using the norm, show that the units in \(A\) are precisely the nonzero
  elements of \(k\). Show that \(x\) and \(y\) are irreducible elements
  of \(A\). Show that \(A\) is not a unique factorization domain.
\item
  Prove that \(Y\) is not a rational curve (Ex. 6.1). See (II, 8.20.3)
  and (III, Ex. 5.3) for other proofs of this important result.
\end{enumerate}

\hypertarget{section-49}{%
\subsubsection{6.3.}\label{section-49}}

Show by example that the result of \((6.8)\) is false if either (a)
\(\operatorname{dim} X \geqslant 2\), or (b) \(Y\) is not projective.

\hypertarget{section-50}{%
\subsubsection{6.4.}\label{section-50}}

Let \(Y\) be a nonsingular projective curve. Show that every nonconstant
rational function \(f\) on \(Y\) defines a surjective morphism
\(\varphi: Y \rightarrow \mathbf{P}^1\), and that for every
\(P \in \mathbf{P}^1\), \(\varphi^{-1}(P)\) is a finite set of points.

\hypertarget{section-51}{%
\subsubsection{6.5.}\label{section-51}}

Let \(X\) be a nonsingular projective curve. Suppose that \(X\) is a
(locally closed) subvariety of a variety \(Y\) (Ex. 3.10). Show that
\(X\) is in fact a closed subset of \(Y\). See (II, Ex. 4.4) for
generalization.

\hypertarget{automorphisms-of-mathbfp1.-to_work}{%
\subsubsection{\texorpdfstring{6.6. Automorphisms of
\(\mathbf{P}^1\).}{6.6. Automorphisms of \textbackslash mathbf\{P\}\^{}1.}}\label{automorphisms-of-mathbfp1.-to_work}}

Think of \(\mathbf{P}^1\) as
\(\mathbf{A}^1 \cup\left\{{\infty}\right\}\). Then we define a
fractional linear transformation of \(\mathbf{P}^1\) by sending
\(x \mapsto(a x+b)/(c x+d)\), for \(a, b, c, d \in k\), and
\(ad-b c \neq 0\).

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
  Show that a fractional linear transformation induces an automorphism
  of \(\mathbf{P}^1\) (i.e., an isomorphism of \(\mathbf{P}^1\) with
  itself). We denote the group of all these fractional linear
  transformations by \(\operatorname{PGL}(1)\).
\item
  Let Aut \(\mathbf{P}^1\) denote the group of all automorphisms of
  \(\mathbf{P}^1\). Show that Aut \(\mathbf{P}^1 \simeq\) Aut \(k(x)\),
  the group of \(k\)-automorphisms of the field \(k(x)\).
\item
  Now show that every automorphism of \(k(x)\) is a fractional linear
  transformation, and deduce that \(\mathrm{PGL}(1) \rightarrow\) Aut
  \(\mathbf{P}^1\) is an isomorphism.
\end{enumerate}

Note: We will see later (II. 7.1.1) that a similar result holds for
\(\mathbf{P}^n\) : every automorphism is given by a linear
transformation of the homogeneous coordinates.

\hypertarget{section-52}{%
\subsubsection{6.7.}\label{section-52}}

Let \(P_1, \ldots, P_r, Q_1, \ldots, Q_s\) be distinct points of
\(\mathbf{A}^1\). If \(\mathbf{A}^1-\left\{P_1, \ldots, P_r\right\}\) is
isomorphic to \(\mathbf{A}^1-\left\{Q_1, \ldots, Q_{\diamond}\right\}\),
show that \(r=s\). Is the converse true? Cf. (Ex. 3.1).

\hypertarget{i.7-intersections-in-projective-space}{%
\subsection{I.7: Intersections in Projective
Space}\label{i.7-intersections-in-projective-space}}

\hypertarget{section-53}{%
\subsubsection{7.1.}\label{section-53}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  Find the degree of the \(d\)-uple embedding of \(\mathbf{P}^n\) in
  \(\mathbf{P}^N\left(\right.\) Ex. 2.12).\footnote{Answer: \(d^n\).}
\item
  Find the degree of the Segre embedding of
  \(\mathbf{P}^r \times \mathbf{P}^s\) in \(\mathbf{P}^{\prime}(\) Ex.
  2.14).\footnote{Answer: \({r+s\choose s}\).}
\end{enumerate}

\hypertarget{section-54}{%
\subsubsection{7.2.}\label{section-54}}

Let \(Y\) be a variety of dimension \(r\) in \(\mathbf{P}^n\), with
Hilbert polynomial \(P_Y\). We define the arithmetic genus of \(Y\) to
be \(p_a(Y) = (-1)^r\left(P_Y(0)-1\right)\). This is an important
invariant which (as we will see later in (III, Ex. 5.3)) is independent
of the projective embedding of \(Y\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \(p_a\left(\mathbf{P}^n\right)=0\).
\item
  If \(Y\) is a plane curve of degree \(d\), show that
  \(p_a(Y)=\frac{1}{2}(d-1)(d-2)\).
\item
  More generally, if \(H\) is a hypersurface of degree \(d\) in
  \(\mathbf{P}^n\), then \(p_a(H)={d-1\choose n}\).
\item
  If \(Y\) is a complete intersection (Ex. 2.17) of surfaces of degrees
  \(a, h\) in \(\mathbf{P}^3\), then
  \(p_a(Y)=\frac{1}{2} a b(a+b-4)+1\).
\end{enumerate}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\setcounter{enumi}{4}
\tightlist
\item
  Let \(Y^r \subseteq \mathbf{P}^n, Z^{s} \subseteq \mathbf{P}^m\) be
  projective varieties, and embed
  \(Y \times Z \subseteq \mathbf{P}^n \times\)
  \(\mathbf{P}^m \rightarrow \mathbf{P}^N\) by the Segre embedding. Show
  that
\end{enumerate}

\hypertarget{the-dual-curve.}{%
\subsubsection{7.3. The Dual Curve.}\label{the-dual-curve.}}

Let \(Y \subseteq \mathbf{P}^2\) be a curve. We regard the set of lines
in \(\mathbf{P}^2\) as another projective space,
\(\left(\mathbf{P}^2\right)^*\). by taking \((a_0, a_1, a_2)\) as
homogeneous coordinates of the line \(L: a_0 x_0+a_1 x_1+a_2 x_2=0\).
For each nonsingular point \(P \in Y\), show that there is a unique line
\(T_P(Y)\) whose intersection multiplicity with \(Y\) at \(P\) is
\(>1\). This is the tangent line to \(Y\) at \(P\).

Show that the mapping \(P \mapsto T_P(Y)\) defines a morphism of Reg
\(Y\) (the set of nonsingular points of \(Y)\) into
\(\left(\mathbf{P}^2\right)^*\). The closure of the image of this
morphism is called the dual curve
\(Y^* \subseteq\left(\mathbf{P}^2\right)^*\) of \(Y\).

\hypertarget{section-55}{%
\subsubsection{7.4.}\label{section-55}}

Given a curve \(Y\) of degree \(d\) in \(\mathbf{P}^2\), show that there
is a nonempty open subset \(U\) of \(\left(\mathbf{P}^2\right)^*\) in
its Zariski topology such that for each \(L \in U, L\) meets \(Y\) in
exactly \(d\) points. \footnote{Hint: Show that the set of lines in
  \(\left(\mathbf{P}^2\right)^*\) which are either tangent to \(Y\) or
  pass through a singular point of \(Y\) is contained in a proper closed
  subset.}

This result shows that we could have defined the degree of \(Y\) to be
the number \(d\) such that almost all lines in \(\mathbf{P}^2\) meet
\(Y\) in \(d\) points, where ``almost all'' refers to a nonempty open
set of the set of lines, when this set is identified with the dual
projective space \(\left(\mathbf{P}^2\right)^*\)

\hypertarget{section-56}{%
\subsubsection{7.5.}\label{section-56}}

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that an irreducible curve \(Y\) of degree \(d>1\) in
  \(\mathbf{P}^2\) cannot have a point of multiplicity \(\geqslant d(\)
  Ex. 5.3).
\item
  If \(Y\) is an irreducible curve of degree \(d>1\) having a point of
  multiplicity \(d-1\). then \(Y\) is a rational curve (Ex. 6.1).
\end{enumerate}

\hypertarget{linear-varieties.}{%
\subsubsection{7.6. Linear Varieties.}\label{linear-varieties.}}

Show that an algebraic set \(Y\) of pure dimension \(r\) (i.e., every
irreducible component of \(Y\) has dimension \(r\) ) has degree 1 if and
only if \(Y\) is a linear variety (Ex. 2.11).\footnote{Hint: First, use
  (7.7) and treat the case \(\operatorname{dim} Y=1\). Then do the
  general case by cutting with a hyperplane and using induction.}

\hypertarget{section-57}{%
\subsubsection{7.7.}\label{section-57}}

Let \(Y\) be a variety of dimension \(r\) and degree \(d>1\) in
\(\mathbf{P}^n\). Let \(P \in Y\) be a nonsingular point. Define \(X\)
to be the closure of the union of all lines \(P Q\), where
\(Q \in Y, Q \neq P\).

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  Show that \(X\) is a variety of dimension \(r+1\).
\item
  Show that \(\operatorname{deg} X<d\). \footnote{Hint: Use induction on
    \(\operatorname{dim} Y\).}
\end{enumerate}

\hypertarget{section-58}{%
\subsubsection{7.8.}\label{section-58}}

Let \(Y^r \subseteq \mathbf{P}^n\) be a variety of degree 2. Show that
\(Y\) is contained in a linear subspace \(L\) of dimension \(r+1\) in
\(\mathbf{P}^n\). Thus \(Y\) is isomorphic to a quadric hypersurface in
\(\mathbf{P}^{r+1}(\mathrm{Ex} .5 .12)\)

\hypertarget{ii-schemes}{%
\section{II: Schemes}\label{ii-schemes}}

\hypertarget{ii.1-sheaves}{%
\subsection{II.1: Sheaves}\label{ii.1-sheaves}}

\hypertarget{ii.1.1.}{%
\subsubsection{II.1.1.}\label{ii.1.1.}}

Let \(A\) be an abelian group, and define the \textbf{constant presheaf}
associated to \(A\) on the topological space \(X\) to be the presheaf
\(U \mapsto A\) for all \(U \neq \varnothing\). with restriction maps
the identity. Show that the constant sheaf \(\mathscr{A}\) defined in
the text is the sheaf associated to this presheaf.

\hypertarget{ii.1.2.}{%
\subsubsection{II.1.2.}\label{ii.1.2.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  For any morphism of sheaves
  \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\), show that for each
  point \(P,(\operatorname{ker} \varphi)_P=\)
  \(\operatorname{ker}\left(\varphi_P\right)\) and
  \((\operatorname{im} \varphi)_P=\operatorname{im}\left(\varphi_P\right)\).
\item
  Show that \(\varphi\) is injective (respectively, surjective) if and
  only if the induced map on the stalks \(\varphi_P\) is injective
  (respectively, surjective) for all \(P\).
\item
  Show that a sequence of sheaves and morphisms is exact if and only if
  for each \(P \in X\) the corresponding sequence of stalks is exact as
  a sequence of abelian groups.
\end{enumerate}

\hypertarget{ii.1.3.}{%
\subsubsection{II.1.3.}\label{ii.1.3.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\) be a morphism of
  sheaves on \(X\). Show that \(\varphi\) is surjective if and only if
  the following condition holds: for every open set \(U \subseteq X\),
  and for every \(s \in \mathscr{G}\left(U \right)\), there is a
  covering \(\left\{U_i\right\}\) of \(U\), and there are elements
  \(t_i \in {\mathcal{F}}\left(U_i\right)\), such that
  \(\varphi\left(t_i\right)={ \left.{{s}} \right|_{{U_i}} }\) for all
  \(i\).
\item
  Give an example of a surjective morphism of sheaves
  \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\), and an open set
  \(U\) such that
  \(\varphi(U): {\mathcal{F}}(U) \rightarrow \mathscr{G}(U)\) is not
  surjective.
\end{enumerate}

\hypertarget{ii.1.4.}{%
\subsubsection{II.1.4.}\label{ii.1.4.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\) be a morphism of
  presheaves such that
  \(\varphi(U): {\mathcal{F}}(U) \rightarrow \mathscr{G}(U)\) is
  injective for each \(U\). Show that the induced map
  \(\varphi^{+}: {\mathcal{F}}^{+} \rightarrow \mathscr{G}^{+}\)of
  associated sheaves is injective.
\item
  Use part (a) to show that if
  \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\) is a morphism of
  sheaves, then \(\operatorname{im}\varphi\) can be naturally identified
  with a subsheaf of \(\mathscr{G}\). as mentioned in the text.
\end{enumerate}

\hypertarget{ii.1.5.}{%
\subsubsection{II.1.5.}\label{ii.1.5.}}

Show that a morphism of sheaves is an isomorphism if and only if it is
both injective and surjective.

\hypertarget{ii.1.6.}{%
\subsubsection{II.1.6.}\label{ii.1.6.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \({\mathcal{F}}^{\prime}\) be a subsheaf of a sheaf
  \(\tilde{{\mathcal{F}}}\). Show that the natural map of
  \(\tilde{{\mathcal{F}}}\) to the quotient sheaf
  \({\mathcal{F}}/{\mathcal{F}}'\) is surjective, and has kernel
  \({\mathcal{F}}^{\prime}\). Thus there is an exact sequence
\item
  Conversely, if is an exact sequence, show that
  \({\mathcal{F}}^{\prime}\) is isomorphic to a subsheaf of
  \({\mathcal{F}}\), and that \({\mathcal{F}}^{\prime \prime}\) is
  isomorphic to the quotient of \({\mathcal{F}}\) by this subsheaf.
\end{enumerate}

\hypertarget{ii.1.7.}{%
\subsubsection{II.1.7.}\label{ii.1.7.}}

Let \(\varphi: {\mathcal{F}}\rightarrow \mathscr{G}\) be a morphism of
sheaves.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that
  \(\operatorname{im}\varphi \cong {\mathcal{F}}/ \operatorname{ker} \varphi\).
\item
  Show that
  \(\operatorname{coker} \varphi \cong \mathscr{G} / \operatorname{im} \varphi\).
\end{enumerate}

\hypertarget{ii.1.8.}{%
\subsubsection{II.1.8.}\label{ii.1.8.}}

For any open subset \(U \subseteq X\), show that the functor
\(\Gamma(U . \cdot)\) from sheaves on \(X\) to abelian groups is a left
exact functor, i.e.. if is an exact sequence of sheaves, then is an
exact sequence of groups.

The functor \(\Gamma(U, {-})\) need not be exact: see \((\) Ex. 1.21)
below.

\hypertarget{ii.1.9.-direct-sum.}{%
\subsubsection{II.1.9. Direct Sum.}\label{ii.1.9.-direct-sum.}}

Let \({\mathcal{F}}\) and \(\mathscr{G}\) be sheaves on \(X\). Show that
the presheaf \(U \mapsto {\mathcal{F}}(U) \oplus\) \(\mathscr{G}(U)\) is
a sheaf. It is called the \textbf{direct sum} of \({\mathcal{F}}\) and
\(\mathscr{G}\), and is denoted by \({\mathcal{F}}\oplus \mathscr{G}\).

Show that it plays the role of direct sum and of direct product in the
category of sheaves of abelian groups on \(X\).

\hypertarget{ii.1.10.-direct-limit.}{%
\subsubsection{II.1.10. Direct Limit.}\label{ii.1.10.-direct-limit.}}

Let \(\left\{{{\mathcal{F}}_i}\right\}\) be a direct system of sheaves
and morphisms on \(X\). We define the direct limit of the system
\(\left\{{{\mathcal{F}}_i}\right\}\), denoted \(\lim {\mathcal{F}}_i\),
to be the sheaf associated to the presheaf
\(U \mapsto \varinjlim {{\mathcal{F}}_i}(U)\).

Show that this is a direct limit in the category of sheaves on \(X\),
i.e., that it has the following universal property: given a sheaf
\(\mathscr{G}\), and a collection of morphisms
\({\mathcal{F}}_i \rightarrow \mathscr{G}\). compatible with the maps of
the direct system, then there exists a unique map
\(\varinjlim {\mathcal{F}}_i \to \mathscr{G}\) such that for each \(i\),
the original map \(\mathscr{F_i} \rightarrow \mathscr{G}\) is obtained
by composing the maps
\({\mathcal{F}}_i \rightarrow \varinjlim {\mathcal{F}}_i \rightarrow \mathscr{G}\).

\hypertarget{ii.1.11.}{%
\subsubsection{II.1.11.}\label{ii.1.11.}}

Let \(\left\{{\mathcal{F}}_i\right\}\) be a direct system of sheaves on
a noetherian topological space \(X\). In this case show that the
presheaf \(U \mapsto \varinjlim {\mathcal{F}}_i(U)\) is already a sheaf.
In particular,
\(\Gamma\left(X, \varinjlim {{\mathcal{F}}}_i\right) = \varinjlim \Gamma\left(X, {\mathcal{F}}_i\right)\)

\hypertarget{ii.1.12.-inverse-limit.}{%
\subsubsection{II.1.12. Inverse Limit.}\label{ii.1.12.-inverse-limit.}}

Let, \(\left\{{{\mathcal{F}}_i}\right\}\) be an inverse system of
sheaves on \(X\). Show that the presheaf
\(U \mapsto \varprojlim {\mathcal{F}}_i\left(U \right)\) is a sheaf. It
is called the \textbf{inverse limit} of the system
\(\left\{{\mathcal{F}}_i\right\}\), and is denoted by
\(\varprojlim {{\mathcal{F}}}_i\). Show that it has the universal
property of an inverse limit in the category of sheaves.

\hypertarget{ii.1.13.-espace-etale-of-a-presheaf.}{%
\subsubsection{II.1.13. Espace Etale of a
Presheaf.}\label{ii.1.13.-espace-etale-of-a-presheaf.}}

Given a presheaf \({\mathcal{F}}\) on \(X\), we define a topological
space \(\operatorname{\operatorname{Spe}}({\mathcal{F}})\), called the
\textbf{espace etale} of \({\mathcal{F}}\), as follows\footnote{This
  exercise is included only to establish the connection between our
  definition of a sheaf and another definition often found in the
  literature. See for example Godement (1. Ch. II, 1.2).}. As a set,
\(\operatorname{Spe} = \bigcup_{p\in X} {\mathcal{F}}_P\). We define a
projection map
\(\pi: \operatorname{Spe} ({{\mathcal{F}}}) \rightarrow X\) by sending
\(s \in {\mathcal{F}}_p\) to \(P\). For each open set \(U \subseteq X\)
and each section \(s \in {\mathcal{F}}\left(U \right)\), we obtain a map
\(\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu: U \rightarrow \operatorname{Spe}(\left.{\mathcal{F}}\right)\)
by sending \(P \mapsto s_P\), its germ at \(P\).

This map has the property that
\(\pi \circ \mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu= \operatorname{id}\),
in other words, it is a ``section'' of \(\pi\) over \(U\). We now make
\(\operatorname{Spe}({\mathcal{F}})\) into a topological space by giving
it the strongest topology such that all the maps
\(\mkern 1.5mu\overline{\mkern-1.5mus\mkern-1.5mu}\mkern 1.5mu: U \rightarrow \operatorname{Spe}({\mathcal{F}})\)
for all \({U}\). and all \(s \in {\mathcal{F}}\left(U\right)\), are
continuous.

Now show that the sheaf \({\mathcal{F}}^{+}\)associated to
\({\mathcal{F}}\) can be described as follows: for any open set
\(U \subseteq X, {\mathcal{F}}^{+}\left(U \right)\) is the set of
continuous sections of \(\operatorname{Spe}({\mathcal{F}})\) over \(U\).

In particular, the original presheaf \({\mathcal{F}}\) was a sheaf if
and only if for each \(U, {\mathcal{F}}(U)\) is equal to the set of all
continuous sections of \(\operatorname{Spe}({\mathcal{F}})\) over \(U\).

\hypertarget{ii.1.14.-support.}{%
\subsubsection{II.1.14. Support.}\label{ii.1.14.-support.}}

Let \({\mathcal{F}}\) be a sheaf on \(X\), and let
\(s \in {\mathcal{F}}(U)\) be a section over an open set \(U\). The
\textbf{support of \(s\)}, denoted \(\mathop{\mathrm{supp}}s\), is
defined to be \(\left\{P \in U \mathrel{\Big|}s_P \neq 0\right\}\),
where \(s_P\) denotes the germ of \(s\) in the stalk
\({\mathcal{F}}_P\). Show that Supp \(s\) is a closed subset of \(U\).

We define the support of \({\mathcal{F}}\),
\(\mathop{\mathrm{supp}}{\mathcal{F}}\), to be
\(\left\{P \in X \mathrel{\Big|}\cdot {\mathcal{F}}_P \neq 0\right\}\).
It need not be a closed subset.

\hypertarget{ii.1.15.-sheaf-mathopmathcalh-mathitom.-to_work}{%
\subsubsection{\texorpdfstring{II.1.15. Sheaf
\(\mathop{\mathcal{H}\! \mathit{om}}\).}{II.1.15. Sheaf \textbackslash mathop\{\textbackslash mathcal\{H\}\textbackslash! \textbackslash mathit\{om\}\}.}}\label{ii.1.15.-sheaf-mathopmathcalh-mathitom.-to_work}}

Let \({\mathcal{F}}, \mathscr{G}\) be sheaves of abelian groups on
\(X\). For any open set \(U \subseteq X\), show that the set
\(\mathop{\mathrm{Hom}}({ \left.{{{\mathcal{F}}}} \right|_{{U}} }, { \left.{{\mathscr G}} \right|_{{U}} })\)
of morphisms of the restricted sheaves has a natural structure of
abelian group. Show that the presheaf
\(U \mapsto \operatorname{Hom}\left(\left.{\mathcal{F}}\right|_U,\left.\mathscr{G}\right|_U\right)\)
is a sheaf. It is called the \textbf{sheaf of local morphisms of
\({\mathcal{F}}\) into \(\mathscr{G}\), ``sheaf hom'' for short}, and is
denoted
\(\mathop{\mathcal{H}\! \mathit{om}}({\mathcal{F}}, \mathscr{G})\).

\hypertarget{ii.1.16.-flasque-sheaves.}{%
\subsubsection{II.1.16. Flasque
Sheaves.}\label{ii.1.16.-flasque-sheaves.}}

A sheaf \({\mathcal{F}}\) on a topological space \(X\) is
\textbf{flasque} if for every inclusion \(V \subseteq U\) of open sets,
the restriction map \({\mathcal{F}}(U) \rightarrow {\mathcal{F}}(V)\) is
surjective.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that a constant sheaf on an irreducible topological space is
  flasque. See (I, §1) for irreducible topological spaces.
\item
  If is an exact sequence of sheaves, and if \({\mathcal{F}}^{\prime}\)
  is flasque, then for any open set \(L\), the sequence of abelian
  groups is also exact.
\item
  If is an exact sequence of sheaves, and if \({\mathcal{F}}^{\prime}\)
  and \({\mathcal{F}}\) are flasque, then
  \({\mathcal{F}}^{\prime \prime}\) is flasque.
\item
  If \(f: X \rightarrow Y\) is a continuous map, and if
  \({\mathcal{F}}\) is a flasque sheaf on \(X\), then
  \(f_* {\mathcal{F}}\) is a flasque sheaf on \(Y\).
\item
  Let \({\mathcal{F}}\) be any sheaf on \(X\). We define a new sheaf
  \(\mathscr{G}\), called the sheaf of \textbf{discontinuous sections}
  of \({\mathcal{F}}\) as follows. For each open set
  \(U \subseteq X, \mathscr{G}(U)\) is the set of maps
  \(s: U \rightarrow \bigcup_{P \in U} {\mathcal{F}}_P\) such that for
  each \(P \in U, s(P) \in {\mathcal{F}}_P\). Show that \(\mathscr{G}\)
  is a flasque sheaf, and that there is a natural injective morphism of
  \({\mathcal{F}}\) to \(\mathscr{G}\).
\end{enumerate}

\hypertarget{ii.1.17.-skyscraper-sheaves.}{%
\subsubsection{II.1.17. Skyscraper
Sheaves.}\label{ii.1.17.-skyscraper-sheaves.}}

Let \(X\) be a topological space, let \(P\) be a point, and let \(A\) be
an abelian group. Define a sheaf \(i_P(A)\) on \(X\) as follows:
\(i_P(A)\left(U\right)=A\) if \(P \in U^{\prime}, 0\) otherwise. Verify
that the stalk of \(i_P(A)\) is \(A\) at every point \(Q \in\{P\}^{-}\),
and 0 elsewhere, where \(\{P\}^{-}\)denotes the closure of the set
consisting of the point \(P\). Hence the name ``skyscraper sheaf.''

Show that this sheaf could also be described as \(i_*(A)\), where \(A\)
denotes the constant sheaf \(A\) on the closed subspace \(\{P\}^{-}\),
and \(i:\{P\}^{-} \rightarrow X\) is the inclusion.

\hypertarget{ii.1.18.-adjoint-property-of-f-1.-to_work}{%
\subsubsection{\texorpdfstring{II.1.18. Adjoint Property of
\(f^{-1}\).}{II.1.18. Adjoint Property of f\^{}\{-1\}.}}\label{ii.1.18.-adjoint-property-of-f-1.-to_work}}

Let \(f: X \rightarrow Y\) be a continuous map of topological spaces.
Show that for any sheaf \({\mathcal{F}}\) on \(X\) there is a natural
map \(f^{-1} f_* {\mathcal{F}}\rightarrow {\mathcal{F}}\), and for any
sheaf \(\mathscr{G}\) on \(Y\) there is a natural map
\(\mathscr{G} \rightarrow f_* f^{-1} \mathscr{G}\).

Use these maps to show that there is a natural bijection of sets, for
any sheaves \({\mathcal{F}}\) on \(X\) and \(\mathscr{G}\) on \(Y\),
Hence we say that \(f^{-1}\) is a \textbf{left adjoint} of \(f_*\), and
that \(f_*\) is a \textbf{right adjoint} of \(f^{-1}\).

\hypertarget{ii.1.19.-extending-a-sheaf-by-zero.}{%
\subsubsection{II.1.19. Extending a Sheaf by
Zero.}\label{ii.1.19.-extending-a-sheaf-by-zero.}}

Let \(X\) be a topological space, let \(Z\) be a closed subset, let
\(i: Z \rightarrow X\) be the inclusion, let \(U=X-Z\) be the
complementary open subset, and let \(j: U \rightarrow X\) be its
inclusion.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \({\mathcal{F}}\) be a sheaf on \(Z\). Show that the stalk
  \(\left(i_* {\mathcal{F}}\right)_P\) of the direct image sheaf on
  \(X\) is \({\mathcal{F}}_P\) if \(P \in Z, 0\) if \(P \notin Z\).
  Hence we call \(i_* \cdot {\mathcal{F}}\) the sheaf obtained by
  \textbf{extending \({\mathcal{F}}\) by zero outside \(Z\).} By abuse
  of notation we will sometimes write \({\mathcal{F}}\) instead of
  \(i_* {\mathcal{F}}\), and say ``consider \({\mathcal{F}}\) as a sheaf
  on \(X\),'' when we mean "consider \(i_* {\mathcal{F}}\).
\item
  Now let \({\mathcal{F}}\) be a sheaf on \(U\). Let
  \(j_!({{\mathcal{F}}})\) be the sheaf on \(X\) associated to the
  presheaf \(V \mapsto {\mathcal{F}}(V)\) if
  \(V \subseteq U, V \mapsto 0\) otherwise. Show that the stalk
  \((j_!({\mathcal{F}}))_P\) is equal to \({\mathcal{F}}_P\) if
  \(P \in U, 0\) if \(P \notin U\), and show that \(j_! {\mathcal{F}}\)
  is the only sheaf on \(X\) which has this property, and whose
  restriction to \(U\) is \({\mathcal{F}}\). We call
  \(j_! {\mathcal{F}}\) the sheaf obtained by \textbf{extending
  \({\mathcal{F}}\) by zero outside \(U\).}
\item
  Now let \({\mathcal{F}}\) be a sheaf on \(X\). Show that there is an
  exact sequence of sheaves on \(X\),
\end{enumerate}

\hypertarget{ii.1.20.-subsheaf-with-supports.}{%
\subsubsection{II.1.20. Subsheaf with
Supports.}\label{ii.1.20.-subsheaf-with-supports.}}

Let \(Z\) be a closed subset of \(X\), and let \({\mathcal{F}}\) be a
sheaf on \(X\). We define \(\Gamma_Z(X, {\mathcal{F}})\) to be the
subgroup of \(\Gamma(X, {\mathcal{F}})\) consisting of all sections
whose support (Ex. 1.14) is contained in \(Z\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that the presheaf
  \(V \mapsto \Gamma_{\mathrm{Z} \cap V}\left(V,\left.{\mathcal{F}}\right|_V\right)\)
  is a sheaf. It is called the \textbf{subsheaf of \({\mathcal{F}}\)
  with supports in \(Z\)}, and is denoted by
  \(\mathscr{H}_Z^0({\mathcal{F}})\).
\item
  Let \(U=X-Z\), and let \(j: U \rightarrow X\) be the inclusion. Show
  there is an exact sequence of sheaves on \(X\) Furthermore, if
  \({\mathcal{F}}\) is flasque, the map
  \({\mathcal{F}}\rightarrow i_*({ \left.{{{\mathcal{F}}}} \right|_{{U}} })\)
  is surjective.
\end{enumerate}

\hypertarget{ii.1.21.-some-examples-of-sheaves-on-varieties.}{%
\subsubsection{II.1.21. Some Examples of Sheaves on
Varieties.}\label{ii.1.21.-some-examples-of-sheaves-on-varieties.}}

Let \(X\) be a variety over an algebraically closed field \(k\), as in
Ch. I. Let \({\mathcal{O}}_X\) be the sheaf of regular functions on
\(X\) (See (1.0 .1)).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(Y\) be a closed subset of \(X\). For each open set
  \(U \subseteq X\), let \(\mathscr{I}_Y(U)\) be the ideal in the ring
  \({\mathcal{O}}_X(U)\) consisting of those regular functions which
  vanish at all points of \(Y \cap U\). Show that the presheaf
  \(U \mapsto \mathscr{I}_Y(U)\) is a sheaf. It is called the sheaf of
  ideals \({\mathcal{I}}_Y\) of \(Y\), and it is a subsheaf of the sheaf
  of rings \({\mathcal{O}}_X\).
\item
  If \(Y\) is a subvariety, then the quotient sheaf
  \(C_1 \mathcal{T}_{,}\)is isomorphic to \(i_*\left(C_1\right)\), where
  \(i: Y \rightarrow X\) is the inclusion, and \({\mathcal{O}}_Y\) is
  the sheaf of regular functions on \(Y\).
\item
  Now let \(X=\mathbf{P}^1\), and let \(Y\) be the union of two distinct
  points \(P, Q \in X\). Then there is an exact sequence of sheaves on
  \(X\) where
  \({\mathcal{F}}= i_* {\mathcal{O}}_P \oplus i_* {\mathcal{O}}_Q\):
  Show however that the induced map on global sections
  \(\Gamma(X; {\mathcal{O}}_X) \to \Gamma(X; {\mathcal{F}})\) is not
  surjective. This shows that the global section functor
  \(\Gamma(X, \cdot)\) is not exact (cf.~(Ex. 1.8) which shows that it
  is left exact).
\item
  Again let \(X=\mathbf{P}^1\), and let \({\mathcal{O}}\) be the sheaf
  of regular functions. Let \(\mathscr{K}\) be the constant sheaf on
  \(X\) associated to the function field \(K\) of \(X\). Show that there
  is a natural injection \({\mathcal{O}}\rightarrow \mathscr{K}\). Show
  that the quotient sheaf \(\mathscr K / {\mathcal{O}}\) is isomorphic
  to the direct sum of sheaves \(\sum_{P \in X} i_P\left(I_P\right)\),
  where \(I_P\) is the group \(K / {\mathcal{O}}_P\), and
  \(i_P\left(I_P\right)\) denotes the skyscraper sheaf (Ex. 1.17) given
  by \(I_P\) at the point \(P\).
\item
  Finally show that in the case of (d) the sequence is exact.\footnote{This
    is an analogue of what is called the \emph{first Cousin problem} in
    several complex variables. See Gunning and Rossi (1, p.~248)}
\end{enumerate}

\hypertarget{ii.1.22.-glueing-sheaves.}{%
\subsubsection{II.1.22. Glueing
Sheaves.}\label{ii.1.22.-glueing-sheaves.}}

Let \(X\) be a topological space, let
\(\mathfrak{U} = \left\{{U_i}\right\}\) be an open cover of \(X\), and
suppose we are given for each \(i\) a sheaf \({{\mathcal{F}}}_i\) on
\(U_i\), and for each \(i, j\) an isomorphism such that

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\tightlist
\item
  for each \(i, \varphi_{ii}=\mathrm{id}\), and
\item
  for each \(i, j, k\), \(\phi_{ik} = \phi_{jk} \circ \phi_{ij}\) on
  \(U_i \cap U_j \cap U_k\).
\end{enumerate}

Then there exists a unique sheaf \({\mathcal{F}}\) on \(X\), together
with isomorphisms
\(\psi_i:\left.{\mathcal{F}}\right|_{U_i} { \, \xrightarrow{\sim}\, }{\mathcal{F}}_i\),
such that for each \(i, j, \psi_j= \varphi_{i j} \circ \psi_i\) on
\(U_i \cap U_j\). We say loosely that \({\mathcal{F}}\) is obtained by
\textbf{glueing} the sheaves \({\mathcal{F}}_i\) via the isomorphisms
\(\varphi_i\).

\hypertarget{ii.2-schemes}{%
\subsection{II.2: Schemes}\label{ii.2-schemes}}

\hypertarget{ii.2.1}{%
\subsubsection{II.2.1}\label{ii.2.1}}

Let \(A\) be a ring, let \(X=\operatorname{Spec} A\), let \(f \in A\)
and let \(D(f) \subseteq X\) be the open complement of \(V(f)\). Show
that the locally ringed space
\(\left(D(f),{ \left.{{{\mathcal{O}}_X}} \right|_{{D(f)}} } \right)\) is
isomorphic to \(\operatorname{spec} A_f\).

\hypertarget{ii.2.2}{%
\subsubsection{II.2.2}\label{ii.2.2}}

Let \(\left(X, \mathcal{O}_X\right)\) be a scheme, and let
\(U \subseteq X\) be any open subset. Show that
\(\left(U,\left.\mathcal{O}_X\right|_U\right)\) is a scheme. We call
this the \textbf{induced scheme structure} on the open set \(U\), and we
refer to \(\left(U,\left.\mathcal{O}_X\right|_U\right)\) as an
\textbf{open subscheme} of \(X\).

\hypertarget{ii.2.3-reduced-schemes}{%
\subsubsection{II.2.3 Reduced Schemes}\label{ii.2.3-reduced-schemes}}

A scheme \(\left(X, \mathcal{O}_X\right)\) is \textbf{reduced} if for
every open set \(U \subseteq X\), the ring \(\mathcal{O}_X(U)\) has no
nilpotent elements.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(\left(X, \mathcal{O}_X\right)\) is reduced if and only if
  for every \(P \in X\), the local ring \(\mathcal{O}_{X, P}\) has no
  nilpotent elements.
\item
  Let \(\left(X, \mathcal{O}_X\right)\) be a scheme. Let
  \(\left(\mathcal{O}_X\right)_{\text {red }}\) be the sheaf associated
  to the presheaf \(U \mapsto \mathcal{O}_X(U)_{\text {red }}\), where
  for any ring \(A\), we denote by \(A_{\text {red }}\) the quotient of
  \(A\) by its ideal of nilpotent elements. Show that
  \(\left(X,\left({\mathcal{O}}_X\right)_{\text {red }}\right)\) is a
  scheme. We call it the \textbf{reduced scheme} associated to \(X\),
  and denote it by \(X_{\text {red }}\). Show that there is a morphism
  of schemes \(X_{\text {red }} \rightarrow X\), which is a
  homeomorphism on the underlying topological spaces.
\item
  Let \(f: X \rightarrow Y\) be a morphism of schemes, and assume that
  \(X\) is reduced. Show that there is a unique morphism
  \(g: X \rightarrow Y_{\mathrm{red}}\) such that \(f\) is obtained by
  composing \(g\) with the natural map
  \(Y_{\text {red }} \rightarrow Y\).
\end{enumerate}

\hypertarget{ii.2.4}{%
\subsubsection{II.2.4}\label{ii.2.4}}

Let \(A\) be a ring and let \(\left(X, \mathcal{O}_X\right)\) be a
scheme. Given a morphism \(f: X \rightarrow \operatorname{Spec} A\), we
have an associated map on sheaves
\(f^{\sharp}: \mathcal{U}_{\operatorname{Spec}(A)} \rightarrow f_* \mathcal{O}_X\).
Taking global sections we obtain a homomorphism
\(A \rightarrow \Gamma\left(X, {\mathcal{O}}_X\right)\). Thus there is a
natural map Show that \(\alpha\) is bijective (cf.~(I, 3.5) for an
analogous statement about varieties).

\hypertarget{ii.2.5}{%
\subsubsection{II.2.5}\label{ii.2.5}}

Describe Spec \(\mathbf{Z}\), and show that it is a final object for the
category of schemes, i.e., each scheme \(X\) admits a unique morphism to
Spec \(\mathbf{Z}\).

\hypertarget{ii.2.6}{%
\subsubsection{II.2.6}\label{ii.2.6}}

Describe the spectrum of the zero ring, and show that it is an initial
object for the category of schemes. (According to our conventions, all
ring homomorphisms must take 1 to 1 . Since \(0=1\) in the zero ring, we
see that each ring \(R\) admits a unique homomorphism to the zero ring,
but that there is no homomorphism from the zero ring to \(R\) unless
\(0=1\) in \(R\).)

\hypertarget{ii.2.7}{%
\subsubsection{II.2.7}\label{ii.2.7}}

Let \(X\) be a scheme. For any \(x \in X\), let \(\mathcal{O}_x\) be the
local ring at \(x\), and \(\mathfrak{m}_x\) its maximal ideal. We define
the residue field of \(x\) on \(X\) to be the field
\(k(x)=\mathcal{O}_x / \mathfrak{m}_x\). Now let \(K\) be any field.
Show that to give a morphism of \(\operatorname{Spec} K\) to \(X\) it is
equivalent to give a point \(x \in X\) and an inclusion map
\(k(x) \rightarrow K\).

\hypertarget{ii.2.8}{%
\subsubsection{II.2.8}\label{ii.2.8}}

Let \(X\) be a scheme. For any point \(x \in X\), we define the
\textbf{Zariski tangent space} \(T_x\) to \(X\) at \(x\) to be the dual
of the \(k(x)\)-vector space \({\mathfrak{m}}_x / {\mathfrak{m}}_x^2\).
Now assume that \(X\) is a scheme over a field \(k\), and let
\(k[\varepsilon] / \varepsilon^2\) be the ring of dual numbers over
\(k\). Show that to give a \(k\)-morphism of
\(\operatorname{Spec} k[\varepsilon] / \varepsilon^2\) to \(X\) is
equivalent to giving a point \(x \in X\), rational over \(k\) (i.e.,
such that \(k(x)=k\) ), and an element of \(T_x\).

\hypertarget{ii.2.9}{%
\subsubsection{II.2.9}\label{ii.2.9}}

If \(X\) is a topological space, and \(Z\) an irreducible closed subset
of \(X\), a \textbf{generic point} for \(Z\) is a point \(\zeta\) such
that \(Z=\{\zeta\}^{-}\). If \(X\) is a scheme, show that every
(nonempty) irreducible closed subset has a unique generic point.

\hypertarget{ii.2.10}{%
\subsubsection{II.2.10}\label{ii.2.10}}

Describe \(\operatorname{Spec} \mathbf{R}[x]\). How does its topological
space compare to the set \(\mathbf{R}\) ? To \(\mathbf{C}\) ?

\hypertarget{ii.2.11}{%
\subsubsection{II.2.11}\label{ii.2.11}}

Let \(k=\mathbf{F}_p\) be the finite field with \(p\) elements. Describe
\(\operatorname{Spec} k[x]\). What are the residue fields of its points?
How many points are there with a given residue field?

\hypertarget{ii.2.12-glueing-lemma}{%
\subsubsection{II.2.12 Glueing Lemma}\label{ii.2.12-glueing-lemma}}

Generalize the glueing procedure described in the text (2.3.5) as
follows. Let \(\left\{X_i\right\}\) be a family of schemes (possible
infinite). For each \(i \neq j\), suppose given an open subset
\(U_{i j} \subseteq X_i\), and let it have the induced scheme structure
(Ex. 2.2). Suppose also given for each \(i \neq j\) an isomorphism of
schemes \(\varphi_{i j}: U_{i j} \rightarrow U_{j i}\) such that

\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\tightlist
\item
  for each \(i, j, \varphi_{j i}=\varphi_{i j}^{-1}\), and
\item
  for each \(i, j, k\),
  \(\varphi_{i j}\left(U_{i j} \cap U_{i k}\right)=U_{j i} \cap U_{j k}\),
  and \(\varphi_{i k}=\varphi_{j k} \circ \varphi_{i j}\) on
  \(U_{i j} \cap U_{i k}\).
\end{enumerate}

Then show that there is a scheme \(X\), together with morphisms
\(\psi_i: X_i \rightarrow X\) for each \(i\), such that

\begin{enumerate}
\def\labelenumi{(\arabic{enumi})}
\tightlist
\item
  \(\psi_i\) is an isomorphism of \(X_i\) onto an open subscheme of
  \(X\)
\item
  the \(\psi_i\left(X_i\right)\) cover \(X\),
\item
  \(\psi_i\left(U_{i j}\right)=\psi_i\left(X_i\right) \cap \psi_j\left(X_j\right)\)
  and (4) \(\psi_i=\psi_j \circ \varphi_{i j}\) on \(U_{i j}\).
\end{enumerate}

We say that \(X\) is obtained by \textbf{glueing} the schemes \(X_i\)
along the isomorphisms \(\varphi_{i j}\). An interesting special case is
when the family \(X_i\) is arbitrary, but the \(U_{i j}\) and
\(\varphi_{i j}\) are all empty. Then the scheme \(X\) is called the
\textbf{disjoint union} of the \(X_i\), and is denoted \(\coprod X_i\).

\hypertarget{ii.2.13}{%
\subsubsection{II.2.13}\label{ii.2.13}}

A topological space is \textbf{quasi-compact} if every open cover has a
finite subcover.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that a topological space is noetherian (I.1) if and only if every
  open subset is quasi-compact.
\item
  If \(X\) is an affine scheme, show that \(\operatorname{sp}(X)\) is
  quasi-compact, but not in general noetherian. We say a scheme \(X\) is
  quasi-compact if \(\operatorname{sp}(X)\) is.
\item
  If \(A\) is a noetherian ring, show that
  \(\operatorname{sp}(\operatorname{Spec} A)\) is a noetherian
  topological space.
\item
  Give an example to show that
  \(\operatorname{sp}(\operatorname{Spec} A)\) can be noetherian even
  when \(A\) is not.
\end{enumerate}

\hypertarget{ii.2.14}{%
\subsubsection{II.2.14}\label{ii.2.14}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(S\) be a graded ring. Show that
  \(\mathop{\mathrm{Proj}}S=\varnothing\) if and only if every element
  of \(S_{+}\)is nilpotent.
\item
  Let \(\varphi: S \rightarrow T\) be a graded homomorphism of graded
  rings (preserving degrees). Let
  \(U=\left\{\mathfrak{p} \in\right.\left.T \mathrel{\Big|}\mathfrak{p} \nsupseteq \varphi\left(S_{+}\right)\right\}\).
  Show that \(U\) is an open subset of \(\mathop{\mathrm{Proj}}T\), and
  show that \(\varphi\) determines a natural morphism
  \(f: U \rightarrow \operatorname{Proj} S\).
\item
  The morphism \(f\) can be an isomorphism even when \(\varphi\) is not.
  For example, suppose that \(\varphi_d: S_d \rightarrow T_d\) is an
  isomorphism for all \(d \geqslant d_0\), where \(d_0\) is an integer.
  Then show that \(U=\operatorname{Proj} T\) and the morphism
  \(f: \operatorname{Proj} T \rightarrow \operatorname{Proj} S\) is an
  isomorphism.
\item
  Let \(V\) be a projective variety with homogeneous coordinate ring
  \(S\) (See I.2). Show that \(t(V) \cong \operatorname{Proj} S\).
\end{enumerate}

\hypertarget{ii.2.15}{%
\subsubsection{II.2.15}\label{ii.2.15}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(V\) be a variety over the algebraically closed field \(k\). Show
  that a point \(P \in t(V)\) is a closed point if and only if its
  residue field is \(k\).
\item
  If \(f: X \rightarrow Y\) is a morphism of schemes over \(k\), and if
  \(P \in X\) is a point with residue field \(k\), then \(f(P) \in Y\)
  also has residue field \(k\).
\item
  Now show that if \(V, W\) are any two varieties over \(k\), then the
  natural map is bijective. (Injectivity is easy. The hard part is to
  show it is surjective.)
\end{enumerate}

\hypertarget{ii.2.16}{%
\subsubsection{II.2.16}\label{ii.2.16}}

Let \(X\) be a scheme, let
\(f \in \Gamma\left(X, \mathcal{O}_X\right)\), and define \(X_f\) to be
the subset of points \(x \in X\) such that the stalk \(f_x\) of \(f\) at
\(x\) is not contained in the maximal ideal \(\mathfrak{m}_x\) of the
local ring \(\mathcal{O}_x\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(U=\operatorname{Spec} B\) is an open affine subscheme of \(X\),
  and if
  \(\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu \in B=\Gamma\left(U,\left.\mathcal{O}_X\right|_U\right)\)
  is the restriction of \(f\), show that
  \(U \cap X_f=D(\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu)\).
  Conclude that \(X_f\) is an open subset of \(X\).
\item
  Assume that \(X\) is quasi-compact. Let
  \(A=\Gamma\left(X, \mathcal{O}_X\right)\), and let \(a \in A\) be an
  element whose restriction to \(X_f\) is 0 . Show that for some
  \(n>0, f^n a=0\).
\end{enumerate}

\begin{quote}
Hint: Use an open affine cover of \(X\).
\end{quote}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{2}
\item
  Now assume that \(X\) has a finite cover by open affines \(U_i\) such
  that each intersection \(U_i \cap U_j\) is quasi-compact. (This
  hypothesis is satisfied, for example, if \(\operatorname{sp}(X)\) is
  noetherian.) Let \(b \in \Gamma\left(X_f, \mathcal{O}_{X_f}\right)\).
  Show that for some \(n>0, f^n b\) is the restriction of an element of
  \(A\).
\item
  With the hypothesis of (c), conclude that
  \(\Gamma\left(X_f, \mathcal{O}_{X_f}\right) \cong A_f\).
\end{enumerate}

\hypertarget{ii.2.17-a-criterion-for-affineness}{%
\subsubsection{II.2.17 A Criterion for
Affineness}\label{ii.2.17-a-criterion-for-affineness}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(f: X \rightarrow Y\) be a morphism of schemes, and suppose that
  \(Y\) can be covered by open subsets \(U_i\), such that for each
  \(i\), the induced map \(f^{-1}\left(U_i\right) \rightarrow U_i\) is
  an isomorphism. Then \(f\) is an isomorphism.
\item
  A scheme \(X\) is affine if and only if there is a finite set of
  elements \(f_1, \ldots, f_r \in\)
  \(A=\Gamma\left(X, \mathcal{O}_X\right)\), such that the open subsets
  \(X_{f_i}\) are affine, and \(f_1, \ldots, f_r\) generate the unit
  ideal in A.\footnote{Hint: Use (Ex. 2.4) and (Ex. 2.16d) above.}
\end{enumerate}

\hypertarget{ii.2.18}{%
\subsubsection{II.2.18}\label{ii.2.18}}

In this exercise, we compare some properties of a ring homomorphism to
the induced morphism of the spectra of the rings.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(A\) be a ring, \(X=\operatorname{Spec} A\), and \(f \in A\).
  Show that \(f\) is nilpotent if and only if \(D(f)\) is empty.
\item
  Let \(\varphi: A \rightarrow B\) be a homomorphism of rings, and let
  \(f: Y=\operatorname{Spec} B \rightarrow X = \operatorname{Spec}A\) be
  the induced morphism of affine schemes. Show that \(\varphi\) is
  injective if and only if the map of sheaves
  \(f^{\sharp}: \mathcal{O}_X \rightarrow f_* \mathcal{O}_Y\) is
  injective. Show furthermore in that case \(f\) is dominant, i.e.,
  \(f(Y)\) is dense in \(X\).
\item
  With the same notation, show that if \(\varphi\) is surjective, then
  \(f\) is a homeomorphism of \(Y\) onto a closed subset of \(X\), and
  \(f^{\sharp}: \mathcal{O}_X \rightarrow f_* \mathcal{O}_Y\) is
  surjective.
\item
  Prove the converse to (c), namely, if \(f: Y \rightarrow X\) is a
  homeomorphism onto a closed subset, and
  \(f^{\sharp}: \mathcal{O}_X \rightarrow f_* \mathcal{O}_Y\) is
  surjective, then \(\varphi\) is surjective.\footnote{Hint: Consider
    \(X^{\prime}=\operatorname{Spec}(A / \operatorname{ker} \varphi)\)
    and use (b) and (c).}
\end{enumerate}

\hypertarget{ii.2.19}{%
\subsubsection{II.2.19}\label{ii.2.19}}

Let \(A\) be a ring. Show that the following conditions are equivalent:

\begin{enumerate}
\def\labelenumi{(\roman{enumi})}
\item
  \(\operatorname{Spec} A\) is disconnected;
\item
  there exist nonzero elements \(e_1, e_2 \in A\) such that
  \(e_1 e_2=0, e_1^2=e_1, e_2^2=e_2\), \(e_1+e_2=1\) (these elements are
  called \textbf{orthogonal idempotents});
\item
  \(A\) is isomorphic to a direct product \(A_1 \times A_2\) of two
  nonzero rings.
\end{enumerate}

\hypertarget{ii.3-first-properties-of-schemes}{%
\subsection{II.3: First Properties of
Schemes}\label{ii.3-first-properties-of-schemes}}

\hypertarget{ii.3.1}{%
\subsubsection{II.3.1}\label{ii.3.1}}

Show that a morphism \(f: X \rightarrow Y\) is locally of finite type if
and only if for every open affine subset \(V=\operatorname{Spec} B\) of
\(Y, f^{-1}(V)\) can be covered by open affine subsets
\(U_j=\operatorname{Spec} A_j\), where each \(A_j\) is a finitely
generated \(B\)-algebra.

\hypertarget{ii.3.2}{%
\subsubsection{II.3.2}\label{ii.3.2}}

A morphism \(f: X \rightarrow Y\) of schemes is \textbf{quasi-compact}
if there is a cover of \(Y\) by open affines \(V_i\) such that
\(f^{-1}\left(V_i\right)\) is quasi-compact for each \(i\).

Show that \(f\) is quasicompact if and only if for every open affine
subset \(V \subseteq Y, f^{-1}(V)\) is quasi-compact.

\hypertarget{ii.3.3}{%
\subsubsection{II.3.3}\label{ii.3.3}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that a morphism \(f: X \rightarrow Y\) is of finite type if and
  only if it is locally of finite type and quasi-compact.
\item
  Conclude from this that \(f\) is of finite type if and only if for
  every open affine subset \(V=\operatorname{Spec} B\) of
  \(Y, f^{-1}(V)\) can be covered by a finite number of open affines
  \(U_j=\operatorname{Spec} A_j\), where each \(A_j\) is a finitely
  generated \(B\)-algebra.
\item
  Show also if \(f\) is of finite type, then for every open affine
  subset \(V=\operatorname{Spec} B \subseteq\) \(Y\), and for every open
  affine subset \(U=\operatorname{Spec} A \subseteq f^{-1}(V), A\) is a
  finitely generated \(B\)-algebra.
\end{enumerate}

\hypertarget{ii.3.4}{%
\subsubsection{II.3.4}\label{ii.3.4}}

Show that a morphism \(f: X \rightarrow Y\) is finite if and only if for
every open affine subset \(V=\operatorname{Spec} B\) of \(Y, f^{-1}(V)\)
is affine, equal to \(\operatorname{Spec} A\), where \(A\) is a finite
\(B\)-module.

\hypertarget{ii.3.5}{%
\subsubsection{II.3.5}\label{ii.3.5}}

A morphism \(f: X \rightarrow Y\) is \textbf{quasi-finite} if for every
point \(y \in Y, f^{-1}(y)\) is a finite set.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that a finite morphism is quasi-finite.
\item
  Show that a finite morphism is closed, i.e., the image of any closed
  subset is closed.
\item
  Show by example that a surjective, finite-type, quasi-finite morphism
  need not be finite.
\end{enumerate}

\hypertarget{ii.3.6}{%
\subsubsection{II.3.6}\label{ii.3.6}}

Let \(X\) be an integral scheme. Show that the local ring
\(\mathcal{O}_{\xi}\) of the generic point \(\xi\) of \(X\) is a field.
It is called the \textbf{function field} of \(X\), and is denoted by
\(K(X)\).

Show also that if \(U=\operatorname{Spec} A\) is any open affine subset
of \(X\), then \(K(X)\) is isomorphic to the quotient field of \(A\).

\hypertarget{ii.3.7}{%
\subsubsection{II.3.7}\label{ii.3.7}}

A morphism \(f: X \rightarrow Y\), with \(Y\) irreducible, is
\textbf{generically finite} if \(f^{-1}(\eta)\) is a finite set, where
\(\eta\) is the generic point of \(Y\). A morphism
\(f: X \rightarrow Y\) is \textbf{dominant} if \(f(X)\) is dense in
\(Y\).

Now let \(f: X \rightarrow Y\) be a dominant, generically finite
morphism of finite type of integral schemes. Show that there is an open
dense subset \(U \subseteq Y\) such that the induced morphism
\(f^{-1}(U) \rightarrow U\) is finite.\footnote{Hint: First show that
  the function field of \(X\) is a finite field extension of the
  function field of \(Y\).}

\hypertarget{ii.3.8.-normalization.}{%
\subsubsection{II.3.8. Normalization.}\label{ii.3.8.-normalization.}}

A scheme is \textbf{normal} if all of its local rings are integrally
closed domains. Let \(X\) be an integral scheme. For each open affine
subset \(U=\operatorname{Spec} A\) of \(X\), let \(\tilde{A}\) be the
integral closure of \(A\) in its quotient field, and let
\(\tilde{U}=\operatorname{Spec} \tilde{A}\).

Show that one can glue the schemes \(\tilde{U}\) to obtain a normal
integral scheme \(\tilde{X}\), called the \textbf{normalization} of
\(X\).

Show also that there is a morphism \(\tilde{X} \rightarrow X\), having
the following universal property: for every normal integral scheme
\(Z\), and for every dominant morphism \(f: Z \rightarrow X, f\) factors
uniquely through \(\tilde{X}\). If \(X\) is of finite type over a field
\(k\), then the morphism \(\tilde{X} \rightarrow X\) is a finite
morphism. This generalizes (I, Ex. 3.17).

\hypertarget{ii.3.9.-the-topological-space-of-a-product.}{%
\subsubsection{II.3.9. The Topological Space of a
Product.}\label{ii.3.9.-the-topological-space-of-a-product.}}

Recall that in the category of varieties, the Zariski topology on the
product of two varieties is not equal to the product topology (I, Ex.
1.4). Now we see that in the category of schemes, the underlying point
set of a product of schemes is not even the product set.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(k\) be a field, and let
  \(\mathbf{A}_k^1=\operatorname{Spec} k[x]\) be the affine line over
  \(k\). Show that
  \(\mathbf{A}_k^1 \underset{\scriptscriptstyle {\operatorname{Spec}k} }{\times} \mathbf{A}_k^1 \cong \mathbf{A}_k^2\),
  and show that the underlying point set of the product is not the
  product of the underlying point sets of the factors (even if \(k\) is
  algebraically closed).
\item
  Let \(k\) be a field, let \(s\) and \(t\) be indeterminates over
  \(k\). Then \(\operatorname{Spec}k(s)\), \(\operatorname{Spec}k(t)\),
  and \(\operatorname{Spec}k\) are all one-point spaces. Describe the
  product scheme Spec
  \(k(s) \underset{\scriptscriptstyle {\operatorname{Spec}k} }{\times} \operatorname{Spec} k(t)\).
\end{enumerate}

\hypertarget{ii.3.10.-fibres-of-a-morphism.}{%
\subsubsection{II.3.10. Fibres of a
Morphism.}\label{ii.3.10.-fibres-of-a-morphism.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(f: X \rightarrow Y\) is a morphism, and \(y \in Y\) a point, show
  that \(\operatorname{sp}\left(X_y\right)\) is homeomorphic to
  \(f^{-1}(y)\) with the induced topology.
\item
  Let \(X=\operatorname{Spec} k[s, t] /\left(s-t^2\right)\), let
  \(Y=\operatorname{Spec} k[s]\), and let \(f: X \rightarrow Y\) be the
  morphism defined by sending \(s \rightarrow s\).

  \begin{itemize}
  \tightlist
  \item
    If \(y \in Y\) is the point \(a \in k\) with \(a \neq 0\), show that
    the fibre \(X_y\) consists of two points, with residue field \(k\).
  \item
    If \(y \in Y\) corresponds to \(0 \in k\), show that the fibre
    \(X_y\) is a nonreduced one-point scheme.
  \item
    If \(\eta\) is the generic point of \(Y\), show that \(X_\eta\) is a
    one-point scheme, whose residue field is an extension of degree two
    of the residue field of \(\eta\). (Assume \(k\) algebraically
    closed.)
  \end{itemize}
\end{enumerate}

\hypertarget{ii.3.11.-closed-subschemes.}{%
\subsubsection{II.3.11. Closed
Subschemes.}\label{ii.3.11.-closed-subschemes.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  Closed immersions are stable under base extension: if
  \(f: Y \rightarrow X\) is a closed immersion, and if
  \(X^{\prime} \rightarrow X\) is any morphism, then
  \(f^{\prime}: Y \times_X X^{\prime} \rightarrow X^{\prime}\) is also a
  closed immersion.
\end{enumerate}

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\setcounter{enumi}{1}
\tightlist
\item
  * If \(Y\) is a closed subscheme of an affine scheme
  \(X=\operatorname{Spec} A\), then \(Y\) is also affine, and in fact
  \(Y\) is the closed subscheme determined by a suitable ideal
  \(\mathfrak{a} \subseteq A\) as the image of the closed immersion
  \(\operatorname{Spec} A / \mathfrak{a} \rightarrow \operatorname{Spec} A\).\footnote{Hints:
    First show that \(Y\) can be covered by a finite number of open
    affine subsets of the form \(D\left(f_i\right) \cap Y\), with
    \(f_i \in A\). By adding some more \(f_i\) with
    \(D\left(f_i\right) \cap Y=\varnothing\), if necessary, show that we
    may assume that the \(D\left(f_i\right)\) cover \(X\). Next show
    that \(f_1, \ldots, f_r\) generate the unit ideal of \(A\). Then use
    (Ex. 2.17b) to show that \(Y\) is affine, and (Ex. 2.18d) to show
    that \(Y\) comes from an ideal \(\mathfrak{a} \subseteq A\).{]}}\footnote{Note:
    We will give another proof of this result using sheaves of ideals
    later (V.10).}
\end{enumerate}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{2}
\item
  Let \(Y\) be a closed subset of a scheme \(X\), and give \(Y\) the
  reduced induced subscheme structure. If \(Y^{\prime}\) is any other
  closed subscheme of \(X\) with the same underlying topological space,
  show that the closed immersion \(Y \rightarrow X\) factors through
  \(Y^{\prime}\). We express this property by saying that \textbf{the
  reduced induced structure is the smallest subscheme structure on a
  closed subset}.
\item
  Let \(f: Z \rightarrow X\) be a morphism. Then there is a unique
  closed subscheme \(Y\) of \(X\) with the following property: the
  morphism \(f\) factors through \(Y\), and if \(Y^{\prime}\) is any
  other closed subscheme of \(X\) through which \(f\) factors, then
  \(Y \rightarrow X\) factors through \(Y^{\prime}\) also. We call \(Y\)
  the \textbf{scheme-theoretic image} of \(f\). If \(Z\) is a reduced
  scheme, then \(Y\) is just the reduced induced structure on the
  closure of the image \(f(Z)\).
\end{enumerate}

\hypertarget{ii.3.12.-closed-subschemes-of-proj-s.}{%
\subsubsection{\texorpdfstring{II.3.12. Closed Subschemes of Proj
\(S\).}{II.3.12. Closed Subschemes of Proj S.}}\label{ii.3.12.-closed-subschemes-of-proj-s.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(\varphi: S \rightarrow T\) be a surjective homomorphism of
  graded rings, preserving degrees. Show that the open set \(U\) of (Ex.
  2.14) is equal to \(\mathop{\mathrm{Proj}}T\), and the morphism
  \(f: \mathop{\mathrm{Proj}}T\to \mathop{\mathrm{Proj}}S\) is a closed
  immersion.
\item
  If \(I \subseteq S\) is a homogeneous ideal, take \(T=S / I\) and let
  \(Y\) be the closed subscheme of \(X=\operatorname{Proj} S\) defined
  as image of the closed immersion
  \(\operatorname{Proj} S / I \rightarrow X\). Show that different
  homogeneous ideals can give rise to the same closed subscheme. For
  example, let \(d_0\) be an integer, and let
  \(I^{\prime}=\bigoplus_{d \geqslant d_0} I_d\). Show that \(I\) and
  \(I^{\prime}\) determine the same closed subscheme.\footnote{We will
    see later (5.16) that every closed subscheme of \(X\) comes from a
    homogeneous ideal \(I\) of \(S\) (at least in the case where \(S\)
    is a polynomial ring over \(S_0\) ).}
\end{enumerate}

\hypertarget{ii.3.13.-properties-of-morphisms-of-finite-type.}{%
\subsubsection{II.3.13. Properties of Morphisms of Finite
Type.}\label{ii.3.13.-properties-of-morphisms-of-finite-type.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  A closed immersion is a morphism of finite type.
\item
  A quasi-compact open immersion (Ex. 3.2) is of finite type.
\item
  A composition of two morphisms of finite type is of finite type.
\item
  Morphisms of finite type are stable under base extension.
\item
  If \(X\) and \(Y\) are schemes of finite type over \(S\), then
  \(X \times{ }_S Y\) is of finite type over \(S\).
\item
  If \(X \stackrel{f}{\rightarrow} Y \stackrel{g}{\rightarrow} Z\) are
  two morphisms, and if \(f\) is quasi-compact, and \(g \circ f\) is of
  finite type, then \(f\) is of finite type.
\item
  If \(f: X \rightarrow Y\) is a morphism of finite type, and if \(Y\)
  is noetherian, then \(X\) is noetherian.
\end{enumerate}

\hypertarget{ii.3.14}{%
\subsubsection{II.3.14}\label{ii.3.14}}

If \(X\) is a scheme of finite type over a field, show that the closed
points of \(X\) are dense. Give an example to show that this is not true
for arbitrary schemes.

\hypertarget{ii.3.15}{%
\subsubsection{II.3.15}\label{ii.3.15}}

Let \(X\) be a scheme of finite type over a field \(k\) (not necessarily
algebraically closed).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that the following three conditions are equivalent (in which case
  we say that \(X\) is \textbf{geometrically irreducible}):

  \begin{itemize}
  \tightlist
  \item
    i:
    \(X \underset{\scriptscriptstyle {k} }{\times} \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)
    is irreducible, where
    \(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)
    denotes the algebraic closure of \(k\).\footnote{By abuse of
      notation, we write
      \(X \underset{\scriptscriptstyle {k} }{\times} \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)
      to denote
      \(X \underset{\scriptscriptstyle {\operatorname{Spec}k} }{\times} \operatorname{Spec}\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\).}
  \item
    ii: \(X \underset{\scriptscriptstyle {k} }{\times} k_s\) is
    irreducible, where \(k_s\) denotes the separable closure of \(k\).
  \item
    iii: \(X \underset{\scriptscriptstyle {k} }{\times} K\) is
    irreducible for every extension field \(K\) of \(k\).
  \end{itemize}
\item
  Show that the following three conditions are equivalent (in which case
  we say \(X\) is \textbf{geometrically reduced}):

  \begin{itemize}
  \tightlist
  \item
    i:
    \(X \underset{\scriptscriptstyle {k} }{\times} \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)
    is reduced.
  \item
    ii: \(X \underset{\scriptscriptstyle {k} }{\times} k_p\) is reduced,
    where \(k_p\) denotes the perfect closure of \(k\).
  \item
    iii: \(X \underset{\scriptscriptstyle {k} }{\times} K\) is reduced
    for all extension fields \(K\) of \(k\).
  \end{itemize}
\item
  We say that \(X\) is \textbf{geometrically integral} if
  \(X \underset{\scriptscriptstyle {k} }{\times} \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\)
  is integral. Give examples of integral schemes which are neither
  geometrically irreducible nor geometrically reduced.
\end{enumerate}

\hypertarget{ii.3.16.-noetherian-induction.}{%
\subsubsection{II.3.16. Noetherian
Induction.}\label{ii.3.16.-noetherian-induction.}}

Let \(X\) be a noetherian topological space, and let \(\mathscr{P}\) be
a property of closed subsets of \(X\). Assume that for any closed subset
\(Y\) of \(X\), if \(\mathscr{P}\) holds for every proper closed subset
of \(Y\), then \(\mathscr{P}\) holds for \(Y\). (In particular,
\(\mathscr{P}\) must hold for the empty set.) Then \(\mathscr{P}\) holds
for \(X\).

\hypertarget{ii.3.17.-zariski-spaces.}{%
\subsubsection{II.3.17. Zariski
Spaces.}\label{ii.3.17.-zariski-spaces.}}

A topological space \(X\) is a \textbf{Zariski space} if it is
noetherian and every (nonempty) closed irreducible subset has a unique
generic point (Ex. 2.9).

For example, let \(R\) be a discrete valuation ring, and let
\(T=\operatorname{sp}(\operatorname{Spec} R)\). Then \(T\) consists of
two points \(t_0=\) the maximal ideal, \(t_1=\) the zero ideal. The open
subsets are \(\varnothing,\left\{t_1\right\}\), and \(T\). This is an
irreducible Zariski space with generic point \(t_1\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that if \(X\) is a noetherian scheme, then
  \(\operatorname{sp}(X)\) is a Zariski space.
\item
  Show that any minimal nonempty closed subset of a Zariski space
  consists of one point. We call these closed points.
\item
  Show that a Zariski space \(X\) satisfies the axiom \(T_0\) :given any
  two distinct points of \(X\), there is an open set containing one but
  not the other.
\item
  If \(X\) is an irreducible Zariski space, then its generic point is
  contained in every nonempty open subset of \(X\) of \(x_1\), or that
  \(x_1\) is a generization of \(x_0\). Now let \(X\) be a Zariski
  space.

  \begin{itemize}
  \item
    Show that the minimal points, for the partial ordering determined by
    \(x_1>x_0\) if \(x_1 \leftrightarrow\) \(x_0\), are the closed
    points, and the maximal points are the generic points of the
    irreducible components of \(X\).
  \item
    Show also that a closed subset contains every specialization of any
    of its points. (We say closed subsets are \textbf{stable under
    specialization}.) Similarly, open subsets are stable under
    generization.
  \end{itemize}
\item
  Let \(t\) be the functor on topological spaces introduced in the proof
  of (2.6). If \(X\) is a noetherian topological space, show that
  \(t(X)\) is a Zariski space. Furthermore \(X\) itself is a Zariski
  space if and only if the map \(\alpha: X \rightarrow t(X)\) is a
  homeomorphism.
\end{enumerate}

\hypertarget{ii.3.18-constructible-sets.}{%
\subsubsection{II.3.18 Constructible
Sets.}\label{ii.3.18-constructible-sets.}}

Let \(X\) be a Zariski topological space. A constructible subset of
\(X\) is a subset which belongs to the smallest family \(\mathscr{F}\)
of subsets such that (1) every open subset is in \(\mathscr{F},(2)\) a
finite intersection of elements of \(\mathscr{F}\) is in
\(\mathscr{F}\), and (3) the complement of an element of
\(\mathfrak{F}\) is in \(\mathfrak{F}\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  A subset of \(X\) is locally closed if it is the intersection of an
  open subset with a closed subset. Show that a subset of \(X\) is
  constructible if and only if it can be written as a finite disjoint
  union of locally closed subsets.
\item
  Show that a constructible subset of an irreducible Zariski space \(X\)
  is dense if and only if it contains the generic point. Furthermore, in
  that case it contains a nonempty open subset.
\item
  A subset \(S\) of \(X\) is closed if and only if it is constructible
  and stable under specialization. Similarly, a subset \(T\) of \(X\) is
  open if and only if it is constructible and stable under generization.
\item
  If \(f: X \rightarrow Y\) is a continuous map of Zariski spaces, then
  the inverse image of any constructible subset of \(Y\) is a
  constructible subset of \(X\).
\end{enumerate}

\hypertarget{ii.3.19}{%
\subsubsection{II.3.19}\label{ii.3.19}}

Let \(f: X \rightarrow Y\) be a morphism of finite type of noetherian
schemes. Then the image of any constructible subset of \(X\) is a
constructible subset of \(Y\). In particular, \(f(X)\), which need not
be either open or closed, is a constructible subset of \(Y\).\footnote{The
  real importance of the notion of constructible subsets derives from
  the following theorem of Chevalley-see Cartan and Chevalley (1, exposé
  7) and see also Matsumura (2, Ch. 2, 6).}

Prove this theorem in the following steps.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Reduce to showing that \(f(X)\) itself is constructible, in the case
  where \(X\) and \(Y\) are affine, integral noetherian schemes, and
  \(f\) is a dominant morphism.
\item
  * In that case, show that \(f(X)\) contains a nonempty open subset of
  \(Y\) by using the following result from commutative algebra: let
  \(A \subseteq B\) be an inclusion of noetherian integral domains, such
  that \(B\) is a finitely generated \(A\)-algebra. Then given a nonzero
  element \(b \in B\), there is a nonzero element \(a \in A\) with the
  following property: if \(\varphi: A \rightarrow K\) is any
  homomorphism of \(A\) to an algebraically closed field \(K\), such
  that \(\varphi(a) \neq 0\), then \(\varphi\) extends to a homomorphism
  \(\varphi^{\prime}\) of \(B\) into \(K\), such that
  \(\varphi^{\prime}(b) \neq 0\).\footnote{Hint: Prove this algebraic
    result by induction on the number of generators of \(B\) over \(A\).
    For the case of one generator, prove the result directly. In the
    application, take \(b=1\).}
\item
  Now use noetherian induction on \(Y\) to complete the proof.
\item
  Give some examples of morphisms \(f: X \rightarrow Y\) of varieties
  over an algebraically closed field \(k\), to show that \(f(X)\) need
  not be either open or closed.
\end{enumerate}

\hypertarget{ii.3.20.-dimension.}{%
\subsubsection{II.3.20. Dimension.}\label{ii.3.20.-dimension.}}

Let \(X\) be an integral scheme of finite type over a field \(k\) (not
necessarily algebraically closed). Use appropriate results from I.1 to
prove the following.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  For any closed point
  \(P \in X, \operatorname{dim} X=\operatorname{dim} \mathcal{O}_P\),
  where for rings, we always mean the Krull dimension.
\item
  Let \(K(X)\) be the function field of \(X\) (Ex. 3.6). Then
\item
  If \(Y\) is a closed subset of \(X\), then
\item
  If \(Y\) is a closed subset of \(X\), then
\item
  If \(U\) is a nonempty open subset of \(X\), then
  \(\operatorname{dim} U=\operatorname{dim} X\).
\item
  If \(k \subseteq k^{\prime}\) is a field extension, then every
  irreducible component of
  \(X^{\prime}=X \underset{\scriptscriptstyle {k} }{\times} k^{\prime}\)
  has dimension \(=\operatorname{dim} X\).
\end{enumerate}

\hypertarget{ii.3.21}{%
\subsubsection{II.3.21}\label{ii.3.21}}

Let \(R\) be a discrete valuation ring containing its residue field
\(k\). Let \(X=\) Spec \(R[t]\) be the affine line over Spec \(R\). Show
that statements (a), (d), (e) of (Ex. 3.20) are false for \(X\).

\hypertarget{dimension-of-the-fibres-of-a-morphism.}{%
\subsubsection{3.22. * Dimension of the Fibres of a
Morphism.}\label{dimension-of-the-fibres-of-a-morphism.}}

Let \(f: X \rightarrow Y\) be a dominant morphism of integral schemes of
finite type over a field \(k\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(Y^{\prime}\) be a closed irreducible subset of \(Y\), whose
  generic point \(\eta^{\prime}\) is contained in \(f(X)\). Let \(Z\) be
  any irreducible component of \(f^{-1}\left(Y^{\prime}\right)\), such
  that \(\eta^{\prime} \in f(Z)\), and show that
  \(\operatorname{codim}(Z, X) \leqslant \operatorname{codim}\left(Y^{\prime}, Y\right)\).
\item
  Let \(e=\operatorname{dim} X-\operatorname{dim} Y\) be the relative
  dimension of \(X\) over \(Y\). For any point \(y \in f(X)\), show that
  every irreducible component of the fibre \(X_y\) has dimension
  \(\geqslant e\).\footnote{Hint: Let \(Y^{\prime}=\{y\}^{-}\), and use
    (a) and (Ex. 3.20b).}
\item
  Show that there is a dense open subset \(U \subseteq X\), such that
  for any \(y \in f(U)\), \(\operatorname{dim} U_y=e\).\footnote{Hint:
    First reduce to the case where \(X\) and \(Y\) are affine, say
    \(X=\operatorname{Spec} A\) and \(Y=\operatorname{Spec} B\). Then
    \(A\) is a finitely generated \(B\)-algebra. Take
    \(t_1, \ldots, t_e \in A\) which form a transcendence base of
    \(K(X)\) over \(K(Y)\), and let
    \(X_1=\operatorname{Spec} B\left[t_1, \ldots, t_e\right]\). Then
    \(X_1\) is isomorphic to affine \(e\)-space over \(Y\), and the
    morphism \(X \rightarrow X_1\) is generically finite. Now use (Ex.
    3.7) above.}
\item
  Going back to our original morphism \(f: X \rightarrow Y\), for any
  integer \(h\), let \(E_h\) be the set of points \(x \in X\) such that,
  letting \(y=f(x)\), there is an irreducible component \(Z\) of the
  fibre \(X_y\), containing \(x\), and having
  \(\operatorname{dim} Z \geqslant h\). Show that

  \begin{itemize}
  \item
    \begin{enumerate}
    \def\labelenumii{\arabic{enumii})}
    \tightlist
    \item
      \(E_e=X\)\footnote{Use (b) above.};
    \end{enumerate}
  \item
    \begin{enumerate}
    \def\labelenumii{\arabic{enumii})}
    \setcounter{enumii}{1}
    \tightlist
    \item
      if \(h>e\), then \(E_h\) is not dense in \(X\)\footnote{Use (c)
        above.}; and
    \end{enumerate}
  \item
    \begin{enumerate}
    \def\labelenumii{\arabic{enumii})}
    \setcounter{enumii}{2}
    \tightlist
    \item
      \(E_h\) is closed, for all \(h\)\footnote{Use induction on
        \(\operatorname{dim} X\).}.
    \end{enumerate}
  \end{itemize}
\item
  Prove the following theorem of Chevalley-see Cartan and Chevalley (1,
  exposé 8): For each integer \(h\), let \(C_h\) be the set of points
  \(y \in Y\) such that dim \(X_y=h\). Then the subsets \(C_h\) are
  constructible, and \(C_e\) contains an open dense subset of \(Y\).
\end{enumerate}

\hypertarget{ii.3.23}{%
\subsubsection{II.3.23}\label{ii.3.23}}

If \(V, W\) are two varieties over an algebraically closed field \(k\),
and if \(V \times W\) is their product, as defined in (I, Ex. 3.15,
3.16), and if \(t\) is the functor of \((2.6)\), then
\(t(V \times W)=t(V) \underset{\scriptscriptstyle {\operatorname{Spec}k} }{\times} t(W)\).

\hypertarget{ii.4-separated-and-proper-morphisms}{%
\subsection{II.4: Separated and Proper
Morphisms}\label{ii.4-separated-and-proper-morphisms}}

\hypertarget{ii.4.1}{%
\subsubsection{II.4.1}\label{ii.4.1}}

Show that a finite morphism is proper.

\hypertarget{ii.4.2}{%
\subsubsection{II.4.2}\label{ii.4.2}}

Let \(S\) be a scheme, let \(X\) be a reduced scheme over \(S\), and let
\(Y\) be a separated scheme over \(S\). Let \(f\) and \(g\) be two
\(S\)-morphisms of \(X\) to \(Y\) which agree on an open dense subset of
\(X\). Show that \(f=g\). Give examples to show that this result fails
if either

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(X\) is nonreduced, or
\item
  \(Y\) is nonseparated.\footnote{Hint: Consider the map
    \(h: X \rightarrow Y \times{ }_s Y\) obtained from \(f\) and \(g\).}
\end{enumerate}

\hypertarget{ii.4.3}{%
\subsubsection{II.4.3}\label{ii.4.3}}

Let \(X\) be a separated scheme over an affine scheme \(S\). Let \(U\)
and \(V\) be open affine subsets of \(X\). Then \(U \cap V\) is also
affine. Give an example to show that this fails if \(X\) is not
separated.

\hypertarget{ii.4.4.-the-image-of-a-proper-scheme-is-proper.}{%
\subsubsection{II.4.4. The image of a proper scheme is
proper.}\label{ii.4.4.-the-image-of-a-proper-scheme-is-proper.}}

Let \(f: X \rightarrow Y\) be a morphism of separated schemes of finite
type over a noetherian scheme \(S\). Let \(Z\) be a closed subscheme of
\(X\) which is proper over \(S\). Show that \(f(Z)\) is closed in \(Y\),
and that \(f(Z)\) with its image subscheme structure (Ex. 3.11d) is
proper over \(S\).\footnote{Hint: Factor \(f\) into the graph morphism
  \(\Gamma_f: X \rightarrow X \times_s Y\) followed by the second
  projection \(p_2\), and show that \(\Gamma_f\) is a closed immersion.}

\hypertarget{ii.4.5}{%
\subsubsection{II.4.5}\label{ii.4.5}}

Let \(X\) be an integral scheme of finite type over a field \(k\),
having function field \(K\). We say that a valuation of \(K / k\) (see
\(\mathrm{I}, \S 6\) ) has \textbf{center} \(x\) on \(X\) if its
valuation ring \(R\) dominates the local ring \(\mathcal{O}_{x, X}\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(X\) is separated over \(k\), then the center of any valuation of
  \(K / k\) on \(X\) (if it exists) is unique.
\item
  If \(X\) is proper over \(k\), then every valuation of \(K / k\) has a
  unique center on \(X\).\footnote{Note: if \(X\) is a variety over
    \(k\), the criterion of (b) is sometimes taken as the definition of
    a complete variety.}
\item
  * Prove the converses of (a) and (b).\footnote{Hint: While parts (a)
    and (b) follow quite easily from (4.3) and (4.7), their converses
    will require some comparison of valuations in different fields.}
\item
  If \(X\) is proper over \(k\), and if \(k\) is algebraically closed,
  show that \(\Gamma\left(X, \mathcal{O}_X\right)=k\). This result
  generalizes (I, 3.4a).\footnote{Hint: Let
    \(a \in \Gamma\left(X, \mathcal{O}_X\right)\), with \(a \notin k\).
    Show that there is a valuation ring \(R\) of \(K / k\) with
    \(a^{-1} \in \mathfrak{m}_R\). Then use (b) to get a contradiction.}
\end{enumerate}

\hypertarget{ii.4.6}{%
\subsubsection{II.4.6}\label{ii.4.6}}

Let \(f: X \rightarrow Y\) be a proper morphism of affine varieties over
\(k\). Then \(f\) is a finite morphism.\footnote{Hint: Use (4.11A).}

\hypertarget{ii.4.7.-schemes-over-mathbfr.-to_work}{%
\subsubsection{\texorpdfstring{II.4.7. Schemes Over
\(\mathbf{R}\).}{II.4.7. Schemes Over \textbackslash mathbf\{R\}.}}\label{ii.4.7.-schemes-over-mathbfr.-to_work}}

For any scheme \(X_0\) over \(\mathbf{R}\), let
\(X=X_0 \times_{\mathbf{R}} \mathbf{C}\). Let
\(\alpha: \mathbf{C} \rightarrow \mathbf{C}\) be complex conjugation,
and let \(\sigma: X \rightarrow X\) be the automorphism obtained by
keeping \(X_0\) fixed and applying \(\alpha\) to \(\mathbf{C}\). Then
\(X\) is a scheme over \(\mathbf{C}\), and \(\sigma\) is a semi-linear
automorphism, in the sense that we have a commutative diagram:

\begin{tikzcd}
    X && X \\
    \\
    {\operatorname{Spec}{\mathbf{C}}} && {\operatorname{Spec}{\mathbf{C}}}
    \arrow["\sigma", from=1-1, to=1-3]
    \arrow["\alpha", from=3-1, to=3-3]
    \arrow[from=1-1, to=3-1]
    \arrow[from=1-3, to=3-3]
\end{tikzcd}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJYIl0sWzIsMCwiWCJdLFswLDIsIlxcc3BlYyBcXENDIl0sWzIsMiwiXFxzcGVjIFxcQ0MiXSxbMCwxLCJcXHNpZ21hIl0sWzIsMywiXFxhbHBoYSJdLFswLDJdLFsxLDNdXQ==}{Link
to Diagram}
\end{quote}

Since \(\sigma^2=\mathrm{id}\), we call \(\sigma\) an involution.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Now let \(X\) be a separated scheme of finite type over
  \(\mathbf{C}\), let \(\sigma\) be a semilinear involution on \(X\),
  and assume that for any two points \(x_1, x_2 \in X\), there is an
  open affine subset containing both of them. (This last condition is
  satisfied for example if \(X\) is quasi-projective.) Show that there
  is a unique separated scheme \(X_0\) of finite type over
  \(\mathbf{R}\), such that
  \(X_0 \times_{\mathbf{R}} \mathbf{C} \cong X\), and such that this
  isomorphism identifies the given involution of \(X\) with the one on
  \(X_0 \times_{\mathbf{R}} \mathbf{C}\) described above.

  For the following statements, \(X_0\) will denote a separated scheme
  of finite type over \(\mathbf{R}\), and \(X, \sigma\) will denote the
  corresponding scheme with involution over \(\mathbf{C}\).
\item
  Show that \(X_0\) is affine if and only if \(X\) is.
\item
  If \(X_0, Y_0\) are two such schemes over \(\mathbf{R}\), then to give
  a morphism \(f_0: X_0 \rightarrow Y_0\) is equivalent to giving a
  morphism \(f: X \rightarrow Y\) which commutes with the involutions,
  i.e., \(f \circ \sigma_X=\sigma_Y \circ f\).
\item
  If \(X \cong \mathbf{A}_{\mathbf{C}}^1\), then
  \(X_0 \cong \mathbf{A}_{\mathbf{R}}^1\).
\item
  If \(X \cong \mathbf{P}_{\mathbf{C}}^1\), then either
  \(X_0 \cong \mathbf{P}_{\mathbf{R}}^1\), or \(X_0\) is isomorphic to
  the conic in \(\mathbf{P}_{\mathbf{R}}^2\) given by the homogeneous
  equation \(x_0^2+x_1^2+x_2^2=0\).
\end{enumerate}

\hypertarget{ii.4.8}{%
\subsubsection{II.4.8}\label{ii.4.8}}

Let \(\mathscr{P}\) be a property of morphisms of schemes such that:

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  a closed immersion has \(\mathscr{P}\);
\item
  a composition of two morphisms having \(\mathscr{P}\) has
  \(\mathscr{P}\);
\item
  \(\mathscr{P}\) is stable under base extension.

  Then show that:
\item
  a product of morphisms having \(\mathscr{P}\) has \(\mathscr{P}\);
\item
  if \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) are two
  morphisms, and if \(g \circ f\) has \(\mathscr{P}\) and \(g\) is
  separated, then \(f\) has \(\mathscr{P}\);\footnote{Hint: For (e),
    consider the graph morphism
    \(\Gamma_f: X \rightarrow X \times{ }_Z Y\) and note that it is
    obtained by base extension from the diagonal morphism
    \(\Delta: Y \rightarrow Y \times_Z Y\).}
\item
  If \(f: X \rightarrow Y\) has \(\mathscr{P}\), then
  \(f_{\text {red }}: X_{\text {red }} \rightarrow Y_{\text {red }}\)
  has \(\mathscr{P}\).
\end{enumerate}

\hypertarget{ii.4.9}{%
\subsubsection{II.4.9}\label{ii.4.9}}

Show that a composition of projective morphisms is
projective.\footnote{Hint: Use the Segre embedding defined in (I, Ex.
  2.14) and show that it gives a closed immersion
  \(\left.\mathbf{P}^r \times \mathbf{P}^s \rightarrow \mathbf{P}^{r+r+s}.\right.\).}
Conclude that projective morphisms have properties (a)-(f) of (Ex. 4.8)
above.

\hypertarget{ii.4.10-chows-lemma.}{%
\subsubsection{II.4.10 * Chow's Lemma.}\label{ii.4.10-chows-lemma.}}

This result says that proper morphisms are fairly close to projective
morphisms.

Let \(X\) be proper over a noetherian scheme \(S\). Then there is a
scheme \(X^{\prime}\) and a morphism \(g: X^{\prime} \rightarrow X\)
such that \(X^{\prime}\) is projective over \(S\), and there is an open
dense subset \(U \subseteq X\) such that \(g\) induces an isomorphism of
\(g^{-1}(U)\) to \(U\). Prove this result in the following steps.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Reduce to the case \(X\) irreducible.
\item
  Show that \(X\) can be covered by a finite number of open subsets
  \(U_i, i=1, \ldots, n\), each of which is quasi-projective over \(S\).
  Let \(U_i \rightarrow P_i\) be an open immersion of \(U_i\) into a
  scheme \(P_i\) which is projective over \(S\).
\item
  Let \(U=\bigcap U_i\), and consider the map deduced from the given
  maps \(U \rightarrow X\) and \(U \rightarrow P_i\). Let \(X^{\prime}\)
  be the closed image subscheme structure\footnote{See Ex. 3.11d.}
  \(f(U)^{-}\). Let \(g: X^{\prime} \rightarrow X\) be the projection
  onto the first factor, and let be the projection onto the product of
  the remaining factors. Show that \(h\) is a closed immersion, hence
  \(X^{\prime}\) is projective over \(S\).
\item
  Show that \(g^{-1}(U) \rightarrow U\) is an isomorphism, thus
  completing the proof.
\end{enumerate}

\hypertarget{ii.4.11}{%
\subsubsection{II.4.11}\label{ii.4.11}}

If you are willing to do some harder commutative algebra, and stick to
noetherian schemes, then we can express the valuative criteria of
separatedness and properness using only \textbf{discrete} valuation
rings.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\mathcal{O}, {\mathfrak{m}}\) is a noetherian local domain with
  quotient field \(K\), and if \(L\) is a finitely generated field
  extension of \(K\), then there exists a discrete valuation ring \(R\)
  of \(L\) dominating \(\mathcal{O}\).

  Prove this in the following steps.

  \begin{itemize}
  \tightlist
  \item
    By taking a polynomial ring over \(\mathcal{O}\), reduce to the case
    where \(L\) is a finite extension field of \(K\).
  \item
    Then show that for a suitable choice of generators
    \(x_1, \ldots, x_n\) of \(m\), the ideal
    \(\mathfrak{a}=\left(x_1\right)\) in
    \(\mathcal{O}^{\prime}=\mathbb{C}\left[x_2 / x_1, \ldots, x_n / x_1\right]\)
    is not equal to the unit ideal.
  \item
    Then let \(\mathfrak{p}\) be a minimal prime ideal of \(a\), and let
    \(\mathcal{O}_p^{\prime}\) be the localization of
    \(\mathcal{O}^{\prime}\) at \(\mathfrak{p}\). This is a noetherian
    local domain of dimension 1 dominating \(\mathcal{O}\).
  \item
    Let \(\tilde{\mathscr{O}}_p^{\prime}\) be the integral closure of
    \(\mathcal{O}_p^{\prime}\) in \(L\). Use the theorem of
    Krull-Akizuki\footnote{See Nagata 7, p.~115.} to show that
    \(\tilde{U}_{\mathrm{p}}^{\prime}\) is noetherian of dimension 1.
  \item
    Finally, take \(R\) to be a localization of
    \(\tilde{{\mathcal{O}}}_p^{\prime}\) at one of its maximal ideals.
  \end{itemize}
\item
  Let \(f: X \rightarrow Y\) be a morphism of finite type of noetherian
  schemes. Show that \(f\) is separated (respectively, proper) if and
  only if the criterion of \((4.3)\) (respectively, (4.7)) holds for all
  discrete valuation rings.
\end{enumerate}

\hypertarget{ii.4.12-examples-of-valuation-rings.}{%
\subsubsection{II.4.12 Examples of Valuation
Rings.}\label{ii.4.12-examples-of-valuation-rings.}}

Let \(k\) be an algebraically closed field.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(K\) is a function field of dimension 1 over \(k\) then every
  valuation ring of \(K / k\) (except for \(K\) itself) is discrete.
  Thus the set of all of them is just the abstract nonsingular curve
  \(C_K\) of \((I, \S 6)\).
\item
  If \(K / k\) is a function field of dimension two, there are several
  different kinds of valuations. Suppose that \(X\) is a complete
  nonsingular surface with function field \(K\).
\end{enumerate}

\begin{itemize}
\item
  If \(Y\) is an irreducible curve on \(X\), with generic point \(x_1\),
  then the local ring \(R=\mathcal{O}_{x_1, X}\) is a discrete valuation
  ring of \(K / k\) with center at the (nonclosed) point \(x_1\) on
  \(X\).
\item
  If \(f: X^{\prime} \rightarrow X\) is a birational morphism, and if
  \(Y^{\prime}\) is an irreducible curve in \(X^{\prime}\) whose image
  in \(X\) is a single closed point \(x_0\), then the local ring \(R\)
  of the generic point of \(Y^{\prime}\) on \(X^{\prime}\) is a discrete
  valuation ring of \(K / k\) with center at the closed point \(x_0\) on
  \(X\).
\item
  Let \(x_0 \in X\) be a closed point. Let \(f: X_1 \rightarrow X\) be
  the blowing-up of \(x_0\) (I, §4) and let
  \(E_1=f^{-1}\left(x_0\right)\) be the exceptional curve. Choose a
  closed point \(x_1 \in E_1\), let \(f_2: X_2 \rightarrow X_1\) be the
  blowing-up of \(x_1\), and let \(E_2=\) \(f_2^{-1}\left(x_1\right)\)
  be the exceptional curve. Repeat.

  In this manner we obtain a sequence of varieties \(X_i\) with closed
  points \(x_i\) chosen on them, and for each \(i\), the local ring
  \(\mathcal{O}_{x_{i+1}, X_{i+1}}\) dominates
  \(\mathcal{O}_{x_i, X_i}\). Let
  \(R_0=\bigcup_{i=0}^{\infty} \mathcal{O}_{x_i, X_i}\). Then \(R_0\) is
  a local ring, so it is dominated by some valuation ring \(R\) of
  \(K / k\) by (I, 6.1A).

  Show that \(R\) is a valuation ring of \(K / k\), and that it has
  center \(x_0\) on \(X\). When is \(R\) a discrete valuation
  ring?\footnote{Note. We will see later (V, Ex. 5.6) that in fact the
    \(R_0\) of \((3)\) is already a valuation ring itself, so \(R_0=R\).
    Furthermore, every valuation ring of \(K / k\) (except for \(K\)
    itself) is one of the three kinds just described.}
\end{itemize}

\hypertarget{ii.5-sheaves-of-modules}{%
\subsection{II.5: Sheaves of Modules}\label{ii.5-sheaves-of-modules}}

\hypertarget{ii.5.1.}{%
\subsubsection{II.5.1.}\label{ii.5.1.}}

Let \(\left(X, {\mathcal{O}}_X\right)\) be a ringed space, and let
\(\mathcal{E}\) be a locally free \({\mathcal{O}}_X\)-module of finite
rank. We define the dual of \(\mathcal{E}\), denoted
\({\mathcal{E}} {}^{ \vee }\), to be the sheaf
\(\mathop{\mathcal{H}\! \mathit{om}}_{{\mathcal{O}}_X}({\mathcal{E}}, {\mathcal{O}}_X)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that
  \(({\mathcal{E}} {}^{ \vee }) {}^{ \vee }\cong \mathcal{E}\).
\item
  For any \({\mathcal{O}}_X\)-module \(\mathcal{F}\),
\item
  (Projection Formula). If
  \(f:\left(X, {\mathcal{O}}_X\right) \rightarrow\left(Y, {\mathcal{O}}_Y\right)\)
  is a morphism of ringed spaces, if \(\mathcal{F}\) is an
  \({\mathcal{O}}_X\)-module, and if \(\mathcal{E}\) is a locally free
  \({\mathcal{O}}_Y\)-module of finite rank, then there is a natural
  isomorphism
\end{enumerate}

\hypertarget{ii.5.2.}{%
\subsubsection{II.5.2.}\label{ii.5.2.}}

Let \(R\) be a discrete valuation ring with quotient field \(K\), and
let \(X=\operatorname{Spec} R\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  To give an \({\mathcal{O}}_X\)-module is equivalent to giving an
  \(R\)-module \(M\), a \(K\)-vector space \(L\), and a homomorphism
  \(\rho: M \otimes_R K \rightarrow L\).
\item
  That \({\mathcal{O}}_X\)-module is quasi-coherent if and only if
  \(\rho\) is an isomorphism.
\end{enumerate}

\hypertarget{ii.5.3.}{%
\subsubsection{II.5.3.}\label{ii.5.3.}}

Let \(X=\operatorname{Spec} A\) be an affine scheme. Show that the
functors \(\sim\) and \(\Gamma\) are adjoint, in the following sense:
for any \(A\)-module \(M\), and for any sheaf of
\({\mathcal{O}}_X\)-modules \(\mathcal{F}\), there is a natural
isomorphism

\hypertarget{ii.5.4.}{%
\subsubsection{II.5.4.}\label{ii.5.4.}}

Show that a sheaf of \({\mathcal{O}}_X\)-modules \(\mathcal{F}\) on a
scheme \(X\) is quasi-coherent if and only if every point of \(X\) has a
neighborhood \(U\), such that \(\left.\mathcal{F}\right|_U\) is
isomorphic to a cokernel of a morphism of free sheaves on \(U\). If
\(X\) is noetherian, then \(\mathcal{F}\) is coherent if and only if it
is locally a cokernel of a morphism of free sheaves of finite rank.
(These properties were originally the definition of quasi-coherent and
coherent sheaves.)

\hypertarget{ii.5.5.}{%
\subsubsection{II.5.5.}\label{ii.5.5.}}

Let \(f: X \rightarrow Y\) be a morphism of schemes.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show by example that if \(\mathcal{F}\) is coherent on \(X\), then
  \(f_* \mathcal{F}\) need not be coherent on \(Y\), even if \(X\) and
  \(Y\) are varieties over a field \(k\).
\item
  Show that a closed immersion is a finite morphism (\(\S 3\)).
\item
  If \(f\) is a finite morphism of noetherian schemes, and if
  \(\mathcal{F}\) is coherent on \(X\), then \(f_* \mathcal{F}\) is
  coherent on \(Y\).
\end{enumerate}

\hypertarget{ii.5.6.-support.}{%
\subsubsection{II.5.6. Support.}\label{ii.5.6.-support.}}

Recall the notions of support of a section of a sheaf, support of a
sheaf, and subsheaf with supports from (Ex. 1.14) and (Ex. 1.20).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(A\) be a ring, let \(M\) be an \(A\)-module, let
  \(X=\operatorname{Spec} A\), and let \(\mathcal{F}=\tilde{M}\). For
  any \(m \in M=\Gamma(X, \mathcal{F})\), show that
  \(\mathop{\mathrm{supp}}m = V(\operatorname{Ann}m )\), where
  \(\operatorname{Ann}m\) is the annihilator of
  \(m=\{a \in A \mathrel{\Big|}a m=0\}\).
\item
  Now suppose that \(A\) is noetherian, and \(M\) finitely generated.
  Show that
  \(\mathop{\mathrm{supp}}\mathcal{F}=V( \operatorname{Ann}M)\).
\item
  The support of a coherent sheaf on a noetherian scheme is closed.
\item
  For any ideal \(\mathfrak{a} \subseteq A\), we define a submodule
  \(\Gamma_{{\mathfrak{a}}}(M)\) of \(M\) by Assume that \(A\) is
  noetherian, and \(M\) any \(A\)-module. Show that
  \(\Gamma_{\mathfrak{a}}(M)^{\sim} \cong \mathcal{H}_Z^0(\mathcal{F})\),
  where \(Z=V(\mathfrak{a})\) and \(\mathcal{F}=\tilde{M}\).\footnote{Hint:
    Use (Ex. 1.20) and (5.8) to show a priori that
    \(\mathcal{H}_Z^0(\mathcal{F})\) is quasi-coherent. Then show that
    \(\Gamma_{\mathrm{a}}(M) \cong \Gamma_Z(\mathcal{F})\).}
\end{enumerate}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{4}
\tightlist
\item
  Let \(X\) be a noetherian scheme, and let \(Z\) be a closed subset. If
  \(\mathcal{F}\) is a quasicoherent (respectively, coherent)
  \({\mathcal{O}}_X\)-module, then \(\mathcal{H}_Z^0(\mathcal{F})\) is
  also quasicoherent (respectively, coherent).
\end{enumerate}

\hypertarget{ii.5.7.}{%
\subsubsection{II.5.7.}\label{ii.5.7.}}

Let \(X\) be a noetherian scheme, and let \(\mathcal{F}\) be a coherent
sheaf.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If the stalk \(\mathcal{F}_x\) is a free \({\mathcal{O}}_x\)-module
  for some point \(x \in X\), then there is a neighborhood \(U\) of
  \(x\) such that \(\left.\mathcal{F}\right|_U\) is free.
\item
  \(\mathcal{F}\) is locally free if and only if its stalks
  \(\mathcal{F}_x\) are free \({\mathcal{O}}_x\)-modules for all
  \(x \in X\).
\item
  \(\mathcal{F}\) is invertible (i.e., locally free of rank 1) if and
  only if there is a coherent sheaf \(\mathcal{G}\) such that
  \(\mathcal{F} \otimes \mathcal{G} \cong {\mathcal{O}}_X\).\footnote{This
    justifies the terminology invertible: it means that \(\mathcal{F}\)
    is an invertible element of the monoid of coherent sheaves under the
    operation \(\otimes\).}
\end{enumerate}

\hypertarget{ii.5.8.}{%
\subsubsection{II.5.8.}\label{ii.5.8.}}

Again let \(X\) be a noetherian scheme, and \(\mathcal{F}\) a coherent
sheaf on \(X\). We will consider the function where
\(k(x)={\mathcal{O}}_x / m_x\) is the residue field at the point \(x\).
Use Nakayama's lemma to prove the following results.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  The function \(\varphi\) is upper semi-continuous, i.e., for any
  \(n \in \mathbf{Z}\), the set
  \(\{x \in X \mathrel{\Big|}\varphi(x) \geqslant n\}\) is closed.
\item
  If \(\mathcal{F}\) is locally free, and \(X\) is connected, then
  \(\varphi\) is a constant function.
\item
  Conversely, if \(X\) is reduced, and \(\varphi\) is constant, then
  \(\mathcal{F}\) is locally free.
\end{enumerate}

\hypertarget{ii.5.9.}{%
\subsubsection{II.5.9.}\label{ii.5.9.}}

Let \(S\) be a graded ring, generated by \(S_1\) as an \(S_0\)-algebra,
let \(M\) be a graded \(S\) module, and let \(X=\) Proj S.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that there is a natural homomorphism
  \(\alpha: M \rightarrow \Gamma_*(\tilde{M})\).
\item
  Assume now that \(S_0=A\) is a finitely generated \(k\)-algebra for
  some field \(k\), that \(S_1\) is a finitely generated \(A\)-module,
  and that \(M\) is a finitely generated \(S\)-module. Show that the map
  \(\alpha\) is an isomorphism in all large enough degrees, i.e., there
  is a \(d_0 \in \mathbf{Z}\) such that for all
  \(d \geqslant d_0, \alpha_d: M_d \rightarrow \Gamma(X, \tilde{M}(d))\)
  is an isomorphism.\footnote{Hint: Use the methods of the proof of
    (5.19).}
\item
  With the same hypotheses, we define an equivalence relation
  \(\approx\) on graded \(S\)-modules by saying \(M \approx M^{\prime}\)
  if there is an integer \(d\) such that
  \(M_{\geqslant d} \cong M_{\geqslant d}^{\prime}\). Here
  \(M_{\geqslant d}=\bigoplus_{n \geqslant d} M_n\). We will say that a
  graded \(S\)-module \(M\) is \textbf{quasifinitely generated} if it is
  equivalent to a finitely generated module.

  Now show that the functors \(\sim\) and \(\Gamma_*\) induce an
  equivalence of categories between the category of quasi-finitely
  generated graded \(S\)-modules modulo the equivalence relation
  \(\approx\), and the category of coherent \({\mathcal{O}}_X\)-modules.
\end{enumerate}

\hypertarget{ii.5.10.}{%
\subsubsection{II.5.10.}\label{ii.5.10.}}

Let \(A\) be a ring, let \(S=A\left[x_0, \ldots, x_r\right]\) and let
\(X=\operatorname{Proj} S\). We have seen that a homogeneous ideal \(I\)
in \(S\) defines a closed subscheme of \(X\) (Ex. 3.12), and that
conversely every closed subscheme of \(X\) arises in this way (5.16).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  For any homogeneous ideal \(I \subseteq S\), we define the saturation
  \(\mkern 1.5mu\overline{\mkern-1.5muI\mkern-1.5mu}\mkern 1.5mu\) of
  \(I\) to be \(\left\{s \in S \mathrel{\Big|}\right.\) for each
  \(i=0, \ldots, r\), there is an \(n\) such that
  \(\left.x_i^n s \in I\right\}\). We say that \(I\) is saturated if
  \(I=\mkern 1.5mu\overline{\mkern-1.5muI\mkern-1.5mu}\mkern 1.5mu\).
  Show that
  \(\mkern 1.5mu\overline{\mkern-1.5muI\mkern-1.5mu}\mkern 1.5mu\) is a
  homogeneous ideal of \(S\).
\item
  Two homogeneous ideals \(I_1\) and \(I_2\) of \(S\) define the same
  closed subscheme of \(X\) if and only if they have the same
  saturation.
\item
  If \(Y\) is any closed subscheme of \(X\), then the ideal
  \(\Gamma_*\left(\mathcal{I}_Y\right)\) is saturated. Hence it is the
  largest homogeneous ideal defining the subscheme \(Y\).
\item
  There is a 1-1 correspondence between saturated ideals of \(S\) and
  closed subschemes of \(X\).
\end{enumerate}

\hypertarget{ii.5.11.}{%
\subsubsection{II.5.11.}\label{ii.5.11.}}

Let \(S\) and \(T\) be two graded rings with \(S_0=T_0=A\). We define
the \textbf{Cartesian product}
\(S \underset{\scriptscriptstyle {A} }{\times} T\) to be the graded ring
\(\bigoplus_{d \geqslant 0} S_d \otimes_A T_d\). If
\(X=\operatorname{Proj} S\) and \(Y=\operatorname{Proj} T\), show that
\(\operatorname{Proj}\left(S \times{ }_A T\right) \cong X \times{ }_A Y\),
and show that the sheaf \({\mathcal{O}}(1)\) on
\(\operatorname{Proj}\left(S \times{ }_A T\right)\) is isomorphic to the
sheaf
\(p_1^*\left({\mathcal{O}}_X(1)\right) \otimes p_2^*\left({\mathcal{O}}_Y(1)\right)\)
on \(X \times Y\).

The Cartesian product of rings is related to the \textbf{Segre
embedding} of projective spaces (I, Ex. 2.14) in the following way. If
\(x_0, \ldots, x_r\) is a set of generators for \(S_1\) over \(A\),
corresponding to a projective embedding
\(X \hookrightarrow \mathbf{P}_A^r\), and if \(y_0, \ldots, y_s\) is a
set of generators for \(T_1\), corresponding to a projective embedding
\(Y \hookrightarrow P_A^s\), then \(\left\{x_i \otimes y_j\right\}\) is
a set of generators for \(\left(S \times{ }_A T\right)_1\), and hence
defines a projective embedding
\(\mathop{\mathrm{Proj}}(S \underset{\scriptscriptstyle {A} }{\times} T) \hookrightarrow{\mathbf{P}}^N_A\),
with \(N=r s+r+s\). This is just the image of
\(X \times Y \subseteq \mathbf{P}^r \times \mathbf{P}^s\) in its Segre
embedding.

\hypertarget{ii.5.12.}{%
\subsubsection{II.5.12.}\label{ii.5.12.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(X\) be a scheme over a scheme \(Y\), and let
  \(\mathcal{L}, \mathcal{M}\) be two very ample invertible sheaves on
  \(X\). Show that \(\mathcal{L} \otimes \mathcal{M}\) is also very
  ample.\footnote{Hint: Use a Segre embedding.}
\item
  Let \(f: X \rightarrow Y\) and \(g: Y \rightarrow Z\) be two morphisms
  of schemes. Let \(\mathcal{L}\) be a very ample invertible sheaf on
  \(X\) relative to \(Y\), and let \(\mathcal{M}\) be a very ample
  invertible sheaf on \(Y\) relative to \(Z\). Show that
  \(\mathcal{L} \otimes f^* \mathcal{M}\) is a very ample invertible
  sheaf on \(X\) relative to \(Z\).
\end{enumerate}

\hypertarget{ii.5.13.}{%
\subsubsection{II.5.13.}\label{ii.5.13.}}

Let \(S\) be a graded ring, generated by \(S_1\) as an \(S_0\)-algebra.
For any integer \(d>0\), let \(S^{(d)}\) be the graded ring
\(\bigoplus_{n \geqslant 0} S_n^{(d)}\) where \(S_n^{(d)}=S_{n d}\). Let
\(X=\) Proj \(S\). Show that Proj \(S^{(d)} \cong X\), and that the
sheaf \({\mathcal{O}}(1)\) on Proj \(S^{(d)}\) corresponds via this
isomorphism to \({\mathcal{O}}_X(d)\).

This construction is related to the \(d\)-uple embedding (I, Ex. 2.12)
in the following way. If \(x_0, \ldots, x_r\) is a set of generators for
\(S_1\), corresponding to an embedding
\(X \hookrightarrow \mathbf{P}_A^r\), then the set of monomials of
degree \(d\) in the \(x_i\) is a set of generators for
\(S_1^{(d)}=S_d\). These define a projective embedding of Proj
\(S^{(d)}\) which is none other than the image of \(X\) under the
\(d\)-uple embedding of \(\mathbf{P}_A^r\).

\hypertarget{ii.5.14.}{%
\subsubsection{II.5.14.}\label{ii.5.14.}}

Let \(A\) be a ring, and let \(X\) be a closed subscheme of
\({\mathbf{P}}_A^r\). We define the \textbf{homogeneous coordinate ring}
\(S(X)\) of \(X\) for the given embedding to be
\(A\left[x_0, \ldots, x_r\right] / I\), where \(I\) is the ideal
\(\Gamma_*\left(\mathcal{I}_X\right)\) constructed in the proof of
\((5.16)\). Of course if \(A\) is a field and \(X\) a variety, this
coincides with the definition given in (I, §2)! Recall that a scheme
\(X\) is \textbf{normal} if its local rings are integrally closed
domains.

A closed subscheme \(X \subseteq \mathbf{P}_A^r\) is
\textbf{projectively normal} for the given embedding, if its homogeneous
coordinate ring \(S(X)\) is an integrally closed domain (cf.~(I, Ex.
3.18)).

Now assume that \(k\) is an algebraically closed field, and that \(X\)
is a connected, normal closed subscheme of \(\mathbf{P}_k^r\). Show that
for some \(d>0\), the \(d\)-uple embedding of \(X\) is projectively
normal, as follows.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(S\) be the homogeneous coordinate ring of \(X\), and let
  \(S^{\prime}=\bigoplus_{n \geqslant 0} \Gamma\left(X, {\mathcal{O}}_X(n)\right)\).
  Show that \(S\) is a domain, and that \(S^{\prime}\) is its integral
  closure.\footnote{Hint: First show that \(X\) is integral. Then regard
    \(S^{\prime}\) as the global sections of the sheaf of rings
    \(\mathcal{S}=\bigoplus_{n \geq 0} {\mathcal{O}}_X(n)\) on \(X\),
    and show that \(\mathcal{S}\) is a sheaf of integrally closed
    domains.}
\item
  Use (Ex. 5.9) to show that \(S_d=S_d^{\prime}\) for all sufficiently
  large \(d\).
\item
  Show that \(S^{(d)}\) is integrally closed for sufficiently large
  \(d\), and hence conclude that the \(d\)-uple embedding of \(X\) is
  projectively normal.
\item
  As a corollary of (a), show that a closed subscheme
  \(X \subseteq \mathbf{P}_A^r\) is projectively normal if and only if
  it is normal, and for every \(n \geqslant 0\) the natural map
  \(\Gamma\left(\mathbf{P}^r, {\mathcal{O}}_{\mathbf{P}^r}(n)\right) \rightarrow \Gamma\left(X, {\mathcal{O}}_X(n)\right)\)
  is surjective.
\end{enumerate}

\hypertarget{ii.5.15.-extension-of-coherent-sheaves.}{%
\subsubsection{II.5.15. Extension of Coherent
Sheaves.}\label{ii.5.15.-extension-of-coherent-sheaves.}}

We will prove the following theorem in several steps: Let \(X\) be a
noetherian scheme, let \(U\) be an open subset, and let \(\mathcal{F}\)
be a coherent sheaf on \(U\). Then there is a coherent sheaf
\(\mathcal{F}^{\prime}\) on \(X\) such that
\(\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  On a noetherian affine scheme, every quasi-coherent sheaf is the union
  of its coherent subsheaves. We say a sheaf \(\mathcal{F}\) is the
  union of its subsheaves \(\mathcal{F}\) if for every open set \(U\),
  the group \(\mathcal{F}(U)\) is the union of the subgroups
  \(\mathcal{F}_\alpha(U)\).
\item
  Let \(X\) be an affine noetherian scheme, \(U\) an open subset, and
  \(\mathcal{F}\) coherent on \(U\). Then there exists a coherent sheaf
  \(\mathcal{F}^{\prime}\) on \(X\) with
  \(\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}\).\footnote{Hint:
    Let \(i: U \rightarrow X\) be the inclusion map. Show that
    \(i_* \mathcal{F}\) is quasi-coherent, then use (a).}
\item
  With \(X, U, \mathcal{F}\) as in (b), suppose furthermore we are given
  a quasi-coherent sheaf \(\mathcal{G}\) on \(X\) such that
  \(\left.\mathcal{F} \subseteq \mathcal{G}\right|_U\). Show that we can
  find \(\mathcal{F}^{\prime}\) a coherent subsheaf of \(\mathcal{G}\),
  with
  \(\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}\).\footnote{Hint:
    Use the same method, but replace \(i_* \mathcal{F}\) by
    \(\rho^{-1}\left(i_* \mathcal{F}\right)\), where \(\rho\) is the
    natural map
    \(\mathcal{G} \rightarrow i_*\left(\left.\mathcal{G}\right|_U\right)\).}
\item
  Now let \(X\) be any noetherian scheme, \(U\) an open subset,
  \(\mathcal{F}\) a coherent sheaf on \(U\), and \(\mathcal{G}\) a
  quasi-coherent sheaf on \(X\) such that
  \(\left.\mathcal{F} \subseteq \mathcal{G}\right|_U\). Show that there
  is a coherent subsheaf \(\mathcal{F}^{\prime} \subseteq \mathcal{G}\)
  on \(X\) with
  \(\left.\mathcal{F}^{\prime}\right|_U \cong \mathcal{F}\). Taking
  \(\mathcal{G}=i_* \mathcal{F}\) proves the result announced at the
  beginning.\footnote{Hint: Cover \(X\) with open affines, and extend
    over one of them at a time.}
\item
  As an extra corollary, show that on a noetherian scheme, any
  quasi-coherent sheaf \(\mathcal{F}\) is the union of its coherent
  subsheaves.\footnote{Hint: If \(s\) is a section of \(\mathcal{F}\)
    over an open set \(U\), apply (d) to the subsheaf of
    \(\left.\mathcal{F}\right|_U\) generated by \(s\).}
\end{enumerate}

\hypertarget{ii.5.16.-tensor-operations-on-sheaves.}{%
\subsubsection{II.5.16. Tensor Operations on
Sheaves.}\label{ii.5.16.-tensor-operations-on-sheaves.}}

First we recall the definitions of various tensor operations on a
module. Let \(A\) be a ring, and let \(M\) be an \(A\)-module.

\begin{itemize}
\item
  Let \(T^n(M)\) be the tensor product \(M \otimes \ldots \otimes M\) of
  \(M\) with itself \(n\) times, for \(n \geqslant 1\). For \(n=0\) we
  put \(T^0(M)=A\). Then \(T(M)=\bigoplus_{n \geqslant 0} T^n(M)\) is a
  (noncommutative) \(A\)-algebra, which we call the \textbf{tensor
  algebra} of \(M\).
\item
  We define the \textbf{symmetric algebra}
  \(S(M)=\bigoplus_{n \geqslant 0} S^n(M)\) of \(M\) to be the quotient
  of \(T(M)\) by the two-sided ideal generated by all expressions
  \(x \otimes y-y \otimes x\), for all \(x, y \in M\). Then \(S(M)\) is
  a commutative \(A\)-algebra. Its component \(S^n(M)\) in degree \(n\)
  is called the \(n\)th \textbf{symmetric product} of \(M\). We denote
  the image of \(x \otimes y\) in \(S(M)\) by \(x y\), for any
  \(x, y \in M\). As an example, note that if \(M\) is a free
  \(A\)-module of rank \(r\), then \(S(M) \cong\)
  \(A\left[x_1, \ldots, x_r\right]\)
\item
  We define the \textbf{exterior algebra}
  \(\bigwedge(M)=\bigoplus_{n \geqslant 0} \bigwedge^n(M)\) of \(M\) to
  be the quotient of \(T(M)\) by the two-sided ideal generated by all
  expressions \(x \otimes x\) for \(x \in M\). Note that this ideal
  contains all expressions of the form \(x \otimes y+y \otimes x\), so
  that \(\bigwedge(M)\) is a skew commutative graded \(A\)-algebra. This
  means that if \(u \in\) \(\bigwedge^r(M)\) and
  \(v \in \bigwedge^s(M)\), then \(u \wedge v=(-1)^{r s} v \wedge u\)
  (here we denote by \(\wedge\) the multiplication in this algebra; so
  the image of \(x \otimes y\) in \(\bigwedge^2(M)\) is denoted by
  \(x \wedge y\) ). The \(n\)th component \(\bigwedge^n(M)\) is called
  the \(n\)th \textbf{exterior power} of \(M\).
\end{itemize}

Now let \(\left(X, {\mathcal{O}}_X\right)\) be a ringed space, and let
\(\mathcal{F}\) be a sheaf of \({\mathcal{O}}_X\)-modules. We define the
tensor algebra, symmetric algebra, and exterior algebra of
\(\mathcal{F}\) by taking the sheaves associated to the presheaf, which
to each open 'set \(U\) assigns the corresponding tensor operation
applied to \(\mathcal{F}(U)\) as an \({\mathcal{O}}_X(U)\)-module. The
results are \({\mathcal{O}}_x\)-algebras, and their components in each
degree are \({\mathcal{O}}_x\)-modules.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Suppose that \(\mathcal{F}\) is locally free of rank \(n\). Then
  \(T^r(\mathcal{F}), S^r(\mathcal{F})\), and
  \(\bigwedge^r(\mathcal{F})\) are also locally free, of ranks
  \(n^r,\left(\begin{array}{c}n+r-1 \\ n-1\end{array}\right)\), and
  \(\left(\begin{array}{c}r \\ r\end{array}\right)\) respectively.
\item
  Again let \(\mathcal{F}\) be locally free of rank \(n\). Then the
  multiplication map \(\bigwedge^r \mathcal{F} \otimes\)
  \(\bigwedge^{n-r} \mathcal{F} \rightarrow \bigwedge^n \cdot \mathcal{F}\)
  is a perfect pairing for any \(r\), i.c., it induces an isomorphism of
  \(\bigwedge^r \mathcal{F}\) with
  \(\left(\bigwedge^{n-r} \mathcal{F}\right)^{\ulcorner} \otimes \bigwedge^n \mathcal{F}\).
  As a special case, note if \(\mathcal{F}\) has rank 2 , then
  \(\mathcal{F} \cong \mathcal{F}^{\top} \otimes \bigwedge^2 \mathcal{F}\).
\item
  Let
  \(0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \mathcal{F}^{\prime \prime} \rightarrow 0\)
  be an exact sequence of locally free sheaves. Then for any \(r\) there
  is a finite filtration of \(S^r(\mathcal{F})\), with quotients for
  each \(p\).
\item
  Same statement as (c), with exterior powers instead of symmetric
  powers. In particular, if
  \(\mathcal{F}^{\prime}, \mathcal{F}, \mathcal{F}^{\prime \prime}\)
  have ranks \(n^{\prime}, n, n^{\prime \prime}\) respectively, there is
  an isomorphism
\item
  Let \(f: X \rightarrow Y\) be a morphism of ringed spaces, and let
  \(\mathcal{F}\) be an \({\mathcal{O}}_Y\)-module. Then \(f^*\)
  commutes with all the tensor operations on \(\mathcal{F}\), i.e.,
  \(f^*\left(S^n(\mathcal{F})\right)=\)
  \(S^n\left(f^* \mathcal{F}\right)\) etc.
\end{enumerate}

\hypertarget{ii.5.17.-affine-morphisms.}{%
\subsubsection{II.5.17. Affine
Morphisms.}\label{ii.5.17.-affine-morphisms.}}

A morphism \(f: X \rightarrow Y\) of schemes is \textbf{affine} if there
is an open affine cover \(\left\{V_i\right\}\) of \(Y\) such that
\(f^{-1}\left(V_i\right)\) is affine for each \(i\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(f: X \rightarrow Y\) is an affine morphism if and only if
  for every open affine \(V \subseteq Y, f^{-1}(V)\) is
  affine\footnote{Hint: Reduce to the case \(Y\) affine, and use (Ex.
    2.17).}
\item
  An affine morphism is quasi-compact and separated. Any finite morphism
  is affine.
\item
  Let \(Y\) be a scheme, and let \(\mathcal{A}\) be a quasi-coherent
  sheaf of \({\mathcal{O}}_Y\)-algebras (i.e., a sheaf of rings which is
  at the same time a quasi-coherent sheaf of
  \({\mathcal{O}}_Y\)-modules). Show that there is a unique scheme
  \(X\), and a morphism \(f: X \rightarrow Y\), such that for every open
  affine
  \(V \subseteq Y, f^{-1}(V) \cong \operatorname{Spec} \mathcal{A}(V)\),
  and for every inclusion \(U \hookrightarrow V\) of open affines of
  \(Y\), the morphism \(f^{-1}(U) \hookrightarrow f^{-1}(V)\)
  corresponds to the restriction homomorphism
  \(\mathcal{A}(V) \rightarrow \mathcal{A}(U)\). The scheme \(X\) is
  called \(\operatorname{Spec}\mathcal{A}\).\footnote{Hint: Construct
    \(X\) by glueing together the schemes
    \(\operatorname{Spec} \mathcal{A}(V)\), for \(V\) open affine in
    \(Y\).}
\item
  If \(\mathcal{A}\) is a quasi-coherent \({\mathcal{O}}_Y\)-algebra,
  then \(f: X=\) Spec \(\mathcal{A} \rightarrow Y\) is an affine
  morphism, and \(\mathcal{A} \cong f_* {\mathcal{O}}_X\). Conversely,
  if \(f: X \rightarrow Y\) is an affine morphism, then
  \(\mathcal{A}=f_* {\mathcal{O}}_X\) is a quasi-coherent sheaf of
  \({\mathcal{O}}_Y\)-algebras, and
  \(X \cong \operatorname{Spec} \mathcal{A}\).
\item
  Let \(f: X \rightarrow Y\) be an affine morphism, and let
  \(\mathcal{A}=f_* {\mathcal{O}}_X\). Show that \(f_*\) induces an
  equivalence of categories from the category of quasi-coherent
  \({\mathcal{O}}_X\)-modules to the category of quasi-coherent
  \(\mathcal{A}\)-modules (i.e., quasi-coherent
  \({\mathcal{O}}_Y\)-modules having a structure of
  \(\mathcal{A}\)-module).\footnote{Hint: For any quasi-coherent
    \(\mathcal{A}\)-module \(\mathcal{M}\), construct a quasi-coherent
    \({\mathcal{O}}_X\)-module \(\tilde{\mathcal{M}}\), and show that
    the functors \(f_*\) and \(\sim\) are inverse to each other.}
\end{enumerate}

\hypertarget{ii.5.18.-vector-bundles.}{%
\subsubsection{II.5.18. Vector
Bundles.}\label{ii.5.18.-vector-bundles.}}

Let \(Y\) be a scheme. A \textbf{(geometric) vector bundle} of rank
\(n\) over \(Y\) is a scheme \(X\) and a morphism
\(f: X \rightarrow Y\), together with additional data consisting of an
open covering \(\left\{U_i\right\}\) of \(Y\), and isomorphisms
\(\psi_i: f^{-1}\left(U_i\right) \rightarrow \mathbf{A}_{U_i}^n\), such
that for any \(i, j\), and for any open affine subset
\(V=\operatorname{Spec} A \subseteq U_i \cap U_j\), the automorphism
\(\psi=\psi_j \circ \psi_i^{-1}\) of
\(\mathbf{A}_V^n=\operatorname{Spec} A\left[x_1, \ldots, x_n\right]\) is
given by a \emph{linear} automorphism \(\theta\) of
\(A\left[x_1, \ldots, x_n\right]\), i.e., \(\theta(a)=a\) for any
\(a \in A\), and \(\theta\left(x_i\right)=\) \(\sum a_{i j} x_j\) for
suitable \(a_{i j} \in A\).

An \textbf{isomorphism} of one vector bundle of rank \(n\) to another
one is an isomorphism \(g: X \rightarrow X^{\prime}\) of the underlying
schemes, such that \(f=f^{\prime} \circ g\), and such that \(X, f\),
together with the covering of \(Y\) consisting of all the \(U_i\) and
\(U_i^{\prime}\), and the isomorphisms \(\psi_i\) and
\(\psi_i^{\prime} \circ g\), is also a vector bundle structure on \(X\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(\mathcal{E}\) be a locally free sheaf of rank \(n\) on a scheme
  \(Y\). Let \(S(\mathcal{E})\) be the symmetric algebra on
  \(\mathcal{E}\), and let \(X=\operatorname{Spec} S(\mathcal{E})\),
  with projection morphism \(f: X \rightarrow Y\). For each open affine
  subset \(U \subseteq Y\) for which \(\left.\mathcal{E}\right|_U\) is
  free, choose a basis of \(\mathcal{E}\), and let
  \(\psi: f^{-1}(U) \rightarrow \mathbf{A}_U^n\) be the isomorphism
  resulting from the identification of \(S(\mathcal{E}(U))\) with
  \({\mathcal{O}}(U)\left[x_1, \ldots, x_n\right]\).

  Then \((X, f,\{U\},\{\psi\})\) is a vector bundle of rank \(n\) over
  \(Y\), which (up to isomorphism) does not depend on the bases of
  \(\mathcal{E}_U\) chosen. We call it the geometric vector bundle
  associated to \(\mathcal{E}\), and denote it by
  \(\mathbf{V}(\mathcal{E})\).
\item
  For any morphism \(f: X \rightarrow Y\), a section of \(f\) over an
  open set \(U \subseteq Y\) is a morphism \(s: U \rightarrow X\) such
  that \(f \circ s=\mathrm{id}_U\). It is clear how to restrict sections
  to smaller open sets, or how to glue them together, so we see that the
  presheaf \(U \mapsto\{\) set of sections of \(f\) over \(U\}\) is a
  sheaf of sets on \(Y\), which we denote by \(\mathcal{S}(X / Y)\).

  Show that if \(f: X \rightarrow Y\) is a vector bundle of rank \(n\),
  then the sheaf of sections \(\mathcal{S}(X / Y)\) has a natural
  structure of \({\mathcal{O}}_Y\)-module, which makes it a locally free
  \({\mathcal{O}}_Y\)-module of rank \(n\).\footnote{Hint: It is enough
    to define the module structure locally, so we can assume
    \(Y=\operatorname{Spec} A\) is affine, and \(X=\mathbf{A}_Y^n\).
    Then a section \(s: Y \rightarrow X\) comes from an \(A\)-algebra
    homomorphism \(\theta: A\left[x_1, \ldots, x_n\right] \rightarrow\)
    \(A\), which in turn determines an ordered \(n\)-tuple
    \(\left\langle\theta\left(x_1\right), \ldots, \theta\left(x_n\right)\right\rangle\)
    of elements of \(A\). Use this correspondence between sections \(s\)
    and ordered \(n\)-tuples of elements of \(A\) to define the module
    structure.{]}}
\item
  Again let \(\mathcal{E}\) be a locally free sheaf of rank \(n\) on
  \(Y\), let \(X=\mathbf{V}(\mathcal{E})\), and let \(\mathcal{S}=\)
  \(\mathcal{S}(X / Y)\) be the sheaf of sections of \(X\) over \(Y\).
  Show that \(\mathcal{S} \cong \mathcal{E}^2\), as follows. Given a
  section \(s \in \Gamma\left(V, \mathcal{E}^{\curlyvee}\right)\) over
  any open set \(V\), we think of \(s\) as an element of
  \(\operatorname{Hom}\left(\left.\mathcal{E}\right|_V, {\mathcal{O}}_V\right)\).
  So \(s\) determines an \({\mathcal{O}}_V\)-algebra homomorphism
  \(S\left(\left.\mathcal{E}\right|_V\right) \rightarrow {\mathcal{O}}_V\).

  This determines a morphism of spectra
  \(V=\operatorname{Spec} {\mathcal{O}}_V \rightarrow \operatorname{Spec} S\left(\left.\mathcal{E}\right|_V\right)=\)
  \(f^{-1}(V)\), which is a section of \(X / Y\). Show that this
  construction gives an isomorphism of \(\mathcal{E}^2\) to
  \(\mathcal{S}\).
\item
  Summing up, show that we have established a one-to-one correspondence
  between isomorphism classes of locally free sheaves of rank \(n\) on
  \(Y\), and isomorphism classes of vector bundles of rank \(n\) over
  \(Y\). Because of this, we sometimes use the words ``locally free
  sheaf'' and ``vector bundle'' interchangeably, if no confusion seems
  likely to result.
\end{enumerate}

\hypertarget{ii.6-divisors}{%
\subsection{II.6: Divisors}\label{ii.6-divisors}}

In this section we will consider schemes satisfying the following
condition:

\((*) X\) is a noetherian integral separated scheme which is regular in
codimension one.

\hypertarget{ii.6.1.}{%
\subsubsection{II.6.1.}\label{ii.6.1.}}

Let \(X\) be a scheme satisfying \((*)\). Then \(X \times \mathbf{P}^n\)
also satisfies \((*)\), and
\(\operatorname{Cl} \left(X \times \mathbf{P}^n\right) \cong( \operatorname{Cl} X) \times \mathbf{Z}\)

\hypertarget{ii.6.2.-varieties-in-projective-space.}{%
\subsubsection{II.6.2. * Varieties in Projective
Space.}\label{ii.6.2.-varieties-in-projective-space.}}

Let \(k\) be an algebraically closed field, and let \(X\) be a closed
subvariety of \(\mathbf{P}_k^n\) which is nonsingular in codimension one
(hence satisfies \((*)\) ). For any divisor \(D=\sum n_i Y_i\) on \(X\),
we define the \textbf{degree} of \(D\) to be
\(\sum n_i \operatorname{deg} Y_i\), where \(\operatorname{deg} Y_i\) is
the degree of \(Y_i\), considered as a projective variety itself (I,
§7).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(V\) be an irreducible hypersurface in \(\mathbf{P}^n\) which
  does not contain \(X\), and let \(Y_i\) be the irreducible components
  of \(V \cap X\). They all have codimension 1 by (I, Ex. 1.8). For each
  \(i\), let \(f_i\) be a local equation for \(V\) on some open set
  \(U_i\) of \(\mathbf{P}^n\) for which
  \(Y_i \cap U_i \neq \varnothing\), and let
  \(n_i=v_{Y_i}(\mkern 1.5mu\overline{\mkern-1.5muf_i\mkern-1.5mu}\mkern 1.5mu)\),
  where
  \(\mkern 1.5mu\overline{\mkern-1.5muf\mkern-1.5mu}\mkern 1.5mu_i\) is
  the restriction of \(f_i\) to \(U_i \cap X\).

  Then we define the \textbf{divisor} \(V . X\) to be \(\sum n_i Y_i\).
  Extend by linearity, and show that this gives a well-defined
  homomorphism from the subgroup of Div \(\mathbf{P}^n\) consisting of
  divisors, none of whose components contain \(X\), to
  \(\operatorname{Div} X\).
\item
  If \(D\) is a principal divisor on \(\mathbf{P}^n\), for which
  \(D . X\) is defined as in (a), show that \(D . X\) is principal on
  \(X\). Thus we get a homomorphism
  \(\operatorname{Cl} \mathbf{P}^n \rightarrow \operatorname{Cl} X\).
\item
  Show that the integer \(n_i\) defined in (a) is the same as the
  intersection multiplicity \(i\left(X, V ; Y_i\right)\) defined in
  \((\mathrm{I}, \S 7)\). Then use the generalized Bézout theorem
  \((\mathrm{I}, 7.7)\) to show that for any divisor \(D\) on
  \(\mathbf{P}^n\), none of whose components contain \(X\),
\item
  If \(D\) is a principal divisor on \(X\), show that there is a
  rational function \(f\) on \(\mathbf{P}^n\) such that \(D=(f) . X\).
  Conclude that \(\operatorname{deg} D=0\). Thus the degree function
  defines a homomorphism deg:
  \(\operatorname{Cl} X \rightarrow \mathbf{Z}\).\footnote{This gives
    another proof of (6.10), since any complete nonsingular curve is
    projective.} Finally, there is a commutative diagram
\end{enumerate}

\begin{tikzcd}
    { \operatorname{Cl} {\mathbf{P}}^n} && { \operatorname{Cl} X} \\
    \\
    {\mathbf{Z}}&& {\mathbf{Z}}
    \arrow["{\cdot \deg(X)}", from=3-1, to=3-3]
    \arrow[from=1-1, to=1-3]
    \arrow["\deg", from=1-3, to=3-3]
    \arrow["{\deg, \cong}"', from=1-1, to=3-1]
\end{tikzcd}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXENsIFxcUFBebiJdLFsyLDAsIlxcQ2wgWCJdLFswLDIsIlxcWloiXSxbMiwyLCJcXFpaIl0sWzIsMywiXFxjZG90IFxcZGVnKFgpIl0sWzAsMV0sWzEsMywiXFxkZWciXSxbMCwyLCJcXGRlZywgXFxjb25nIiwyXV0=}{Link
to Diagram}
\end{quote}

and in particular, we see that the map
\(\operatorname{Cl} \mathbf{P}^n \rightarrow \operatorname{Cl} X\) is
injective.

\hypertarget{ii.6.3.-cones.}{%
\subsubsection{II.6.3. * Cones.}\label{ii.6.3.-cones.}}

In this exercise we compare the class group of a projective variety
\(V\) to the class group of its cone (I, Ex. 2.10). So let \(V\) be a
projective variety in \(\mathbf{P}^n\), which is of dimension
\(\geqslant 1\) and nonsingular in codimension 1 . Let \(X=C(V)\) be the
affine cone over \(V\) in \(\mathbf{A}^{n+1}\), and let
\(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) be its
projective closure in \(\mathbf{P}^{n+1}\). Let \(P \in X\) be the
vertex of the cone.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let
  \(\pi: \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu-P \rightarrow V\)
  be the projection map. Show that \(V\) can be covered by open subsets
  \(U_i\) such that
  \(\pi^{-1}\left(U_i\right) \cong U_i \times \mathbf{A}^1\) for each
  \(i\), and then show as in (6.6) that
  \(\pi^*: \operatorname{Cl} V \rightarrow \operatorname{Cl} (\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu-P)\)
  is an isomorphism. Since
  \(\operatorname{Cl} \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu \cong\)
  \(\operatorname{Cl} (\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu-P)\),
  we have also
  \(\operatorname{Cl} V \cong \operatorname{Cl} \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\).
\item
  We have
  \(V \subseteq \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\)
  as the hyperplane section at infinity. Show that the class of the
  divisor \(V\) in
  \(\operatorname{Cl} \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\)
  is equal to \(\pi^*\) (class of \(V . H\) ) where \(H\) is any
  hyperplane of \(\mathbf{P}^n\) not containing \(V\). Thus conclude
  using \((6.5)\) that there is an exact sequence where the first arrow
  sends \(1 \mapsto V . H\), and the second is \(\pi^*\) followed by the
  restriction to \(X-P\) and inclusion in \(X\). (The injectivity of the
  first arrow follows from the previous exercise.)
\item
  Let \(S(V)\) be the homogeneous coordinate ring of \(V\) (which is
  also the affine coordinate ring of \(X\) ). Show that \(S(V)\) is a
  unique factorization domain if and only if

  \begin{itemize}
  \tightlist
  \item
    \(V\) is projectively normal (Ex. 5.14), and
  \item
    \(\operatorname{Cl} V \cong \mathrm{Z}\) and is generated by the
    class of \(V . H\).
  \end{itemize}
\item
  Let \({\mathcal{O}}_P\) be the local ring of \(P\) on \(X\). Show that
  the natural restriction map induces an isomorphism
  \(\operatorname{Cl} X \rightarrow \operatorname{Cl} \left(\operatorname{Spec}{\mathcal{O}}_P\right)\).
\end{enumerate}

\hypertarget{ii.6.4.}{%
\subsubsection{II.6.4.}\label{ii.6.4.}}

Let \(k\) be a field of characteristic \(\neq 2\). Let
\(f \in k\left[x_1, \ldots, x_n\right]\) be a square-free nonconstant
polynomial, i.e., in the unique factorization of \(f\) into irreducible
polynomials, there are no repeated factors. Let
\(A=k\left[x_1, \ldots, x_n, z\right] /\left(z^2-f\right)\). Show that
\(A\) is an integrally closed ring.\footnote{Hint: The quotient field
  \(K\) of \(A\) is just
  \(k\left(x_1, \ldots, x_n\right)[z] /\left(z^2-f\right)\). It is a
  Galois extension of \(k\left(x_1, \ldots, x_n\right)\) with Galois
  group \(\mathbf{Z} / 2 \mathbf{Z}\) generated by \(z \mapsto-z\). If
  \(\alpha=g+h z \in K\), where
  \(g, h \in k\left(x_1, \ldots, x_n\right)\), then the minimal
  polynomial of \(\alpha\) is Now show that \(\alpha\) is integral over
  \(k\left[x_1, \ldots, x_n\right]\) if and only if
  \(g, h \in k\left[x_1, \ldots, x_n\right]\).}

Conclude that \(A\) is the integral closure of
\(k\left[x_1, \ldots, x_n\right]\) in \(K\).

\hypertarget{ii.6.5.-quadric-hypersurfaces.}{%
\subsubsection{II.6.5. * Quadric
Hypersurfaces.}\label{ii.6.5.-quadric-hypersurfaces.}}

Let char \(k \neq 2\), and let \(X\) be the affine quadric
hypersurface\footnote{cf.~(I, Ex. 5.12).}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(X\) is normal if \(r \geqslant 2\) (use (Ex. 6.4)).
\item
  Show by a suitable linear change of coordinates that the equation of
  \(X\) could be written as \(x_0 x_1=x_2^2+\ldots+x_r^2\). Now imitate
  the method of \((6.5 .2)\) to show that:

  \begin{enumerate}
  \def\labelenumii{(\arabic{enumii})}
  \tightlist
  \item
    If \(r=2\), then
    \(\operatorname{Cl} X \cong \mathbf{Z} / 2 \mathbf{Z}\);
  \item
    If \(r=3\), then \(\operatorname{Cl} X \cong \mathbf{Z}\) (use
    (6.6.1) and (Ex. 6.3) above);
  \item
    If \(r \geqslant 4\) then \(\operatorname{Cl} X=0\).
  \end{enumerate}
\item
  Now let \(Q\) be the projective quadric hypersurface in
  \(\mathbf{P}^n\) defined by the same equation. Show that:

  \begin{enumerate}
  \def\labelenumii{(\arabic{enumii})}
  \tightlist
  \item
    If \(r=2, \operatorname{Cl} Q \cong \mathbf{Z}\), and the class of a
    hyperplane section \(Q . H\) is twice the generator;
  \item
    If \(r=3, \operatorname{Cl} Q \cong \mathbf{Z} \oplus \mathbf{Z}\);
  \item
    If \(r \geqslant 4, \operatorname{Cl} Q \cong \mathbf{Z}\),
    generated by \(Q . H\).
  \end{enumerate}
\item
  Prove Klein's theorem, which says that if \(r \geqslant 4\), and if
  \(Y\) is an irreducible subvariety of codimension 1 on \(Q\), then
  there is an irreducible hypersurface \(V \subseteq \mathbf{P}^n\) such
  that \(V \cap Q=Y\), with multiplicity one. In other words, \(Y\) is a
  complete intersection.

  (First show that for \(r \geqslant 4\), the homogeneous coordinate
  ring
  \(S(Q)=k\left[x_0, \ldots, x_n\right] /\left(x_0^2+\ldots+x_r^2\right)\)
  is a UFD.)
\end{enumerate}

\hypertarget{ii.6.6.}{%
\subsubsection{II.6.6.}\label{ii.6.6.}}

Let \(X\) be the nonsingular plane cubic curve \(y^2 z=x^3-x z^2\) of
\((6.10 .2)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that three points \(P, Q, R\) of \(X\) are collinear if and only
  if \(P+Q+R=0\) in the group law on \(X\).

  Note that the point \(P_0=(0,1,0)\) is the zero element in the group
  structure on \(X\).
\item
  A point \(P \in X\) has order 2 in the group law on \(X\) if and only
  if the tangent line at \(P\) passes through \(P_0\).
\item
  A point \(P \in X\) has order 3 in the group law on \(X\) if and only
  if \(P\) is an inflection point.\footnote{An inflection point of a
    plane curve is a nonsingular point \(P\) of the curve, whose tangent
    line (I, Ex. 7.3) has intersection multiplicity \(\geqslant 3\) with
    the curve at \(P\).}
\item
  Let \(k=\mathbf{C}\). Show that the points of \(X\) with coordinates
  in \(\mathbf{Q}\) form a subgroup of the group \(X\). Can you
  determine the structure of this subgroup explicitly?
\end{enumerate}

\hypertarget{ii.6.7.}{%
\subsubsection{II.6.7. *}\label{ii.6.7.}}

Let \(X\) be the nodal cubic curve \(y^2 z=x^3+x^2 z\) in
\(\mathbf{P}^2\). Imitate (6.11.4) and show that the group of Cartier
divisors of degree \(0, \mathrm{CaCl}^{\circ} X\), is naturally
isomorphic to the multiplicative group \(\mathbf{G}_m\).

\hypertarget{ii.6.8.}{%
\subsubsection{II.6.8.}\label{ii.6.8.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(f: X \rightarrow Y\) be a morphism of schemes. Show that
  \(\mathcal{L} \mapsto f^* \mathcal{L}\) induces a homomorphism of
  Picard groups,
  \(f^*:\operatorname{Pic}Y \rightarrow \operatorname{Pic}X\).
\item
  If \(f\) is a finite morphism of nonsingular curves, show that this
  homomorphism corresponds to the homomorphism
  \(f^*: \operatorname{Cl} Y \rightarrow \operatorname{Cl} X\) defined
  in the text, via the isomorphisms of (6.16).
\item
  If \(X\) is a locally factorial integral closed subscheme of
  \(\mathbf{P}_k^n\), and if \(f: X \rightarrow \mathbf{P}^n\) is the
  inclusion map, then \(f^*\) on Pic agrees with the homomorphism on
  divisor class groups defined in (Ex. 6.2) via the isomorphisms of
  (6.16).
\end{enumerate}

\hypertarget{ii.6.9.-singular-curves.}{%
\subsubsection{II.6.9. * Singular
Curves.}\label{ii.6.9.-singular-curves.}}

Here we give another method of calculating the Picard group of a
singular curve. Let \(X\) be a projective curve over \(k\), let
\(\tilde{X}\) be its normalization, and let
\(\pi: \tilde{X} \rightarrow X\) be the projection map (Ex. 3.8). For
each point \(P \in X\), let \({\mathcal{O}}_P\) be its local ring, and
let \(\tilde{{\mathcal{O}}}_P\) be the integral closure of
\({\mathcal{O}}_P\). We use a \(*\) to denote the group of units in a
ring.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  Show there is an exact sequence\footnote{Hint: Represent
    \(\operatorname{Pic}X\) and Pic \(\tilde{X}\) as the groups of
    Cartier divisors modulo principal divisors, and use the exact
    sequence of sheaves on \(X\)}
\item
  Use (a) to give another proof of the fact that if \(X\) is a plane
  cuspidal cubic curve, then there is an exact sequence and if \(X\) is
  a plane nodal cubic curve, there is an exact sequence
\end{enumerate}

\hypertarget{ii.6.10.-the-grothendieck-group-kx.-to_work}{%
\subsubsection{\texorpdfstring{II.6.10. The Grothendieck Group
\(K(X)\).}{II.6.10. The Grothendieck Group K(X).}}\label{ii.6.10.-the-grothendieck-group-kx.-to_work}}

Let \(X\) be a noetherian scheme. We define \(K(X)\) to be the quotient
of the free abelian group generated by all the coherent sheaves on
\(X\), by the subgroup generated by all expressions
\({\mathcal{F}}-{\mathcal{F}}^{\prime}-{\mathcal{F}}^{\prime \prime}\),
whenever there is an exact sequence
\(0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow {\mathcal{F}}\rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0\)
of coherent sheaves on \(X\). If \({\mathcal{F}}\) is a coherent sheaf,
we denote by \(\gamma({\mathcal{F}})\) its image in \(K(X)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(X=\mathbf{A}_k^1\), then \(K(X) \cong \mathbf{Z}\).
\item
  If \(X\) is any integral scheme, and \({\mathcal{F}}\) a coherent
  sheaf, we define the \textbf{rank} of \({\mathcal{F}}\) to be
  \(\operatorname{dim}_K {\mathcal{F}}_{\xi}\), where \(\xi\) is the
  generic point of \(X\), and \(K={\mathcal{O}}_{\xi}\) is the function
  field of \(X\).

  Show that the rank function defines a surjective homomorphism rank:
  \(K(X) \rightarrow \mathbf{Z}\).
\item
  If \(Y\) is a closed subscheme of \(X\), there is an exact sequence
  where the first map is extension by zero, and the second map is
  restriction.\footnote{Hint: For exactness in the middle, show that if
    \({\mathcal{F}}\) is a coherent sheaf on \(X\), whose support is
    contained in \(Y\), then there is a finite filtration
    \({\mathcal{F}}={\mathcal{F}}_0 \supseteq\)
    \({\mathcal{F}}_1 \supseteq \ldots \supseteq {\mathcal{F}}_n=0\),
    such that each \({\mathcal{F}}_i / {\mathcal{F}}_{i+1}\) is an
    \({\mathcal{O}}_Y\)-module. To show surjectivity on the right, use
    (Ex. 5.15).\\
    For further information about \(K(X)\), and its applications to the
    generalized Riemann-Roch theorem, see Borel-Serre \([1]\), Manin
    \([1]\), and Appendix A.}
\end{enumerate}

\hypertarget{ii.6.11.-the-grothendieck-group-of-a-nonsingular-curve.}{%
\subsubsection{II.6.11. *The Grothendieck Group of a Nonsingular
Curve.}\label{ii.6.11.-the-grothendieck-group-of-a-nonsingular-curve.}}

Let \(X\) be a nonsingular curve over an algebraically closed field
\(k\). We will show that
\(K(X) \cong \operatorname{Pic}X \oplus \mathbf{Z}\), in several steps.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  For any divisor \(D=\sum n_i P_i\) on \(X\), let
  \(\psi(D)=\sum n_i \gamma\left(k\left(P_i\right)\right) \in K(X)\),
  where \(k\left(P_i\right)\) is the skyscraper sheaf \(k\) at \(P_i\)
  and 0 elsewhere. If \(D\) is an effective divisor, let
  \({\mathcal{O}}_D\) be the structure sheaf of the associated subscheme
  of codimension 1, and show that
  \(\psi(D)=\gamma\left({\mathcal{O}}_D\right)\). Then use (6.18) to
  show that for any \(D, \psi(D)\) depends only on the linear
  equivalence class of \(D\), so \(\psi\) defines a homomorphism
  \(\psi: \operatorname{Cl} X \rightarrow K(X)\)
\item
  For any coherent sheaf \({\mathcal{F}}\) on \(X\), show that there
  exist locally free sheaves \(\mathcal{E}_0\) and \(\mathcal{E}_1\) and
  an exact sequence
  \(0 \rightarrow \mathcal{E}_1 \rightarrow \mathcal{E}_0 \rightarrow {\mathcal{F}}\rightarrow 0\).
  Let \(r_0=\) rank \(\mathcal{E}_0\),
  \(r_1=\operatorname{rank} \mathcal{E}_1\), and define Here
  \(\bigwedge\) denotes the exterior power (Ex. 5.16). Show that
  \(\operatorname{det}{\mathcal{F}}\) is independent of the resolution
  chosen, and that it gives a homomorphism det:
  \(K(X) \rightarrow \operatorname{Pic}X\). Finally show that if \(D\)
  is a divisor, then \(\operatorname{det}(\psi(D))=\mathcal{L}(D)\).
\item
  If \({\mathcal{F}}\) is any coherent sheaf of rank \(r\), show that
  there is a divisor \(D\) on \(X\) and an exact sequence where
  \(\mathcal{T}\) is a torsion sheaf. Conclude that if \({\mathcal{F}}\)
  is a sheaf of rank \(r\), then
\item
  Using the maps \(\psi, \operatorname{det}, \mathrm{rank}\), and
  \(1 \mapsto \gamma\left({\mathcal{O}}_X\right)\) from
  \(\mathbf{Z} \rightarrow K(X)\), show that
  \(K(X) \cong \operatorname{Pic}X \oplus \mathbf{Z}\).
\end{enumerate}

\hypertarget{ii.6.12.}{%
\subsubsection{II.6.12.}\label{ii.6.12.}}

Let \(X\) be a complete nonsingular curve. Show that there is a unique
way to define the degree of any coherent sheaf on \(X\), deg
\({\mathcal{F}}\in \mathbf{Z}\), such that:

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  If \(D\) is a divisor,
  \(\operatorname{deg} \mathcal{L}(D)=\operatorname{deg} D\);
\item
  If \({\mathcal{F}}\) is a torsion sheaf(meaning a sheaf whose stalk at
  the generic point is zero), then
  \(\operatorname{deg} {\mathcal{F}}=\sum_{P \in X}\) length
  \(\left({\mathcal{F}}_P\right)\); and
\item
  If
  \(0 \rightarrow {\mathcal{F}}^{\prime} \rightarrow \mathcal{\mathcal { F }} \rightarrow {\mathcal{F}}^{\prime \prime} \rightarrow 0\)
  is an exact sequence, then
  \(\operatorname{deg} {\mathcal{F}}=\operatorname{deg} {\mathcal{F}}^{\prime}+\)
  \(\operatorname{deg} {\mathcal{F}}^{\prime \prime}\).
\end{enumerate}

\hypertarget{ii.7-projective-morphisms}{%
\subsection{II.7: Projective
Morphisms}\label{ii.7-projective-morphisms}}

\hypertarget{ii.7.1.}{%
\subsubsection{II.7.1.}\label{ii.7.1.}}

Let \(\left(X, {\mathcal{O}}_X\right)\) be a locally ringed space, and
let \(f: \mathcal{L} \rightarrow \mathcal{M}\) be a surjective map of
invertible sheaves on \(X\). Show that \(f\) is an
isomorphism.\footnote{Hint: Reduce to a question of modules over a local
  ring by looking at the stalks.}

\hypertarget{ii.7.2.}{%
\subsubsection{II.7.2.}\label{ii.7.2.}}

Let \(X\) be a scheme over a field \(k\). Let \(\mathcal{L}\) be an
invertible sheaf on \(X\), and let \(\left\{s_0, \ldots, s_n\right\}\)
and \(\left\{t_0, \ldots, t_m\right\}\) be two sets of sections of
\(\mathcal{L}\), which generate the same subspace
\(V \subseteq \Gamma(X, \mathcal{L})\), and which generate the sheaf
\(\mathcal{L}\) at every point. Suppose \(n \leqslant m\).

Show that the corresponding morphisms
\(\varphi: X \rightarrow \mathbf{P}_k^n\) and \(\psi: X \rightarrow\)
\(\mathbf{P}_k^m\) differ by a suitable linear projection
\(\mathbf{P}^m-L \rightarrow \mathbf{P}^n\) and an automorphism of
\(\mathbf{P}^n\), where \(L\) is a linear subspace of \(\mathbf{P}^m\)
of dimension \(m-n-1\).

\hypertarget{ii.7.3.}{%
\subsubsection{II.7.3.}\label{ii.7.3.}}

Let \(\varphi: \mathbf{P}_k^n \rightarrow \mathbf{P}_k^m\) be a
morphism. Then:

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Either \(\varphi\left(\mathbf{P}^n\right) = {\operatorname{pt}}\) or
  \(m \geqslant n\) and
  \(\operatorname{dim} \varphi\left(\mathbf{P}^n\right)=n\);
\item
  In the second case, \(\varphi\) can be obtained as the composition of

  \begin{enumerate}
  \def\labelenumii{(\arabic{enumii})}
  \tightlist
  \item
    a \(d\)-uple embedding \(\mathbf{P}^n \rightarrow \mathbf{P}^N\) for
    a uniquely determined \(d \geqslant 1,\)
  \item
    a linear projection
    \(\mathbf{P}^N-\mathbf{L} \rightarrow \mathbf{P}^m\), and
  \item
    an automorphism of \(\mathbf{P}^m\).
  \end{enumerate}

  Also, \(\varphi\) has finite fibres.
\end{enumerate}

\hypertarget{ii.7.4.}{%
\subsubsection{II.7.4.}\label{ii.7.4.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Use (7.6) to show that if \(X\) is a scheme of finite type over a
  noetherian \(\operatorname{ring} A\), and if \(X\) admits an ample
  invertible sheaf, then \(X\) is separated.
\item
  Let \(X\) be the affine line over a field \(k\) with the origin
  doubled (4.0.1). Calculate \(\operatorname{Pic}X\), determine which
  invertible sheaves are generated by global sections, and then show
  directly (without using (a)) that there is no ample invertible sheaf
  on \(X\).
\end{enumerate}

\hypertarget{ii.7.5.}{%
\subsubsection{II.7.5.}\label{ii.7.5.}}

Establish the following properties of ample and very ample invertible
sheaves on a noetherian scheme \(X\). \(\mathcal{L}, \mathcal{M}\) will
denote invertible sheaves, and for (d), (e) we assume furthermore that
\(X\) is of finite type over a noetherian ring \(A\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\mathcal{L}\) is ample and \(\mathcal{M}\) is generated by global
  sections, then \(\mathcal{L} \otimes \mathcal{M}\) is ample.
\item
  If \(\mathcal{L}\) is ample and \(\mathcal{M}\) is arbitrary, then
  \(\mathcal{M} \otimes \mathcal{L}^n\) is ample for sufficiently large
  \(n\).
\item
  If \(\mathcal{L}, \mathcal{M}\) are both ample, so is
  \(\mathcal{L} \otimes \mathcal{M}\).
\item
  If \(\mathcal{L}\) is very ample and \(\mathcal{M}\) is generated by
  global sections, then \(\mathcal{L} \otimes \mathcal{M}\) is very
  ample.
\item
  If \(\mathcal{L}\) is ample, then there is an \(n_0>0\) such that
  \(\mathcal{L}^n\) is very ample for all \(n \geqslant n_0\).
\end{enumerate}

\hypertarget{ii.7.6.-the-riemann-roch-problem.}{%
\subsubsection{II.7.6. The Riemann-Roch
Problem.}\label{ii.7.6.-the-riemann-roch-problem.}}

Let \(X\) be a nonsingular projective variety over an algebraically
closed field, and let \(D\) be a divisor on \(X\). For any \(n>0\) we
consider the complete linear system \(|n D|\). Then the Riemann-Roch
problem is to determine \(\operatorname{dim}|n D|\) as a function of
\(n\), and, in particular, its behavior for large \(n\).

If \(\mathcal{L}\) is the corresponding invertible sheaf, then
\(\operatorname{dim}|n D|=\operatorname{dim} \Gamma\left(X, \mathcal{L}^n\right)-1\),
so an equivalent problem is to determine
\(\operatorname{dim} \Gamma\left(X, \mathcal{L}^n\right)\) as a function
of \(n\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that if \(D\) is very ample, and if
  \(X \hookrightarrow \mathbf{P}_k^n\) is the corresponding embedding in
  projective space, then for all \(n\) sufficiently large,
  \(\operatorname{dim}|n D|=P_X(n)-1\), where \(P_X\) is the Hilbert
  polynomial of \(X(\mathbf{I}, \S 7)\). Thus in this case
  \(\operatorname{dim}|n D|\) is a polynomial function of \(n\), for
  \(n\) large.
\item
  If \(D\) corresponds to a torsion element of \(\operatorname{Pic}X\),
  of order \(r\), then \(\operatorname{dim}|n D|=0\) if
  \(r \mathrel{\Big|}n\) and \(-1\) otherwise. In this case the function
  is periodic of period \(r\). It follows from the general Riemann-Roch
  theorem that \(\operatorname{dim}|n D|\) is a polynomial function for
  \(n\) large, whenever \(D\) is an ample divisor.\footnote{See (IV,
    1.3.2), \((\mathrm{V}, 1.6)\), and Appendix A.}

  In the case of algebraic surfaces, Zariski \([7]\) has shown for any
  effective divisor \(D\), that there is a finite set of polynomials
  \(P_1, \ldots, P_r\), such that for all \(n\) sufficiently large,
  \(\operatorname{dim}|n D|=P_{i(n)}(n)\), where
  \(i(n) \in\{1,2, \ldots, r\}\) is a function of \(n\).
\end{enumerate}

\hypertarget{ii.7.7.-some-rational-surfaces.}{%
\subsubsection{II.7.7. Some Rational
Surfaces.}\label{ii.7.7.-some-rational-surfaces.}}

Let \(X=\mathbf{P}_k^2\), and let \(|D|\) be the complete linear system
of all divisors of degree 2 on \(X\) (conics). \(D\) corresponds to the
invertible sheaf \({\mathcal{O}}(2)\), whose space of global sections
has a basis \(x^2, y^2, z^2, x y, x z, y z\), where \(x, y, z\) are the
homogeneous coordinates of \(X\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  The complete linear system \(|D|\) gives an embedding of
  \(\mathbf{P}^2\) in \(\mathbf{P}^5\), whose image is the Veronese
  surface.\footnote{I, Ex. 2.13.}
\item
  Show that the subsystem defined by \(x^2, y^2, z^2, y(x-z),(x-y) z\)
  gives a closed immersion of \(X\) into \(\mathbf{P}^4\). The image is
  called the Veronese surface in \(\mathbf{P}^4\). Cf. (IV, Ex. 3.11).
\item
  Let \(\nu \subseteq|D|\) be the linear system of all conics passing
  through a fixed point \(P\). Then \(\nu\) gives an immersion of
  \(U=X-P\) into \(\mathbf{P}^4\). Furthermore, if we blow up \(P\), to
  get a surface \(\tilde{X}\), then this map extends to give a closed
  immersion of \(\tilde{X}\) in \(\mathbf{P}^4\).

  Show that \(\tilde{X}\) is a surface of degree 3 in \(\mathbf{P}^4\),
  and that the lines in \(X\) through \(P\) are transformed into
  straight lines in \(\tilde{X}\) which do not meet. \(\tilde{X}\) is
  the union of all these lines, so we say \(\tilde{X}\) is a
  \textbf{ruled surface} \((\mathrm{V}, 2.19 .1)\).
\end{enumerate}

\hypertarget{ii.7.8.}{%
\subsubsection{II.7.8.}\label{ii.7.8.}}

Let \(X\) be a noetherian scheme, let \(\mathcal{E}\) be a coherent
locally free sheaf on \(X\), and let
\(\pi: \mathbf{P}(\mathcal{E}) \rightarrow X\) be the corresponding
projective space bundle. Show that there is a natural \(1-1\)
correspondence between sections of \(\pi\) (i.e., morphisms
\(\sigma: X \rightarrow\) \(\mathbf{P}(\mathcal{E})\) such that
\(\left.\pi \circ \sigma=\mathrm{id}_X\right)\) and quotient invertible
sheaves \(\mathcal{E} \rightarrow \mathcal{L} \rightarrow 0\) of
\(\mathcal{E}\).

\hypertarget{ii.7.9.}{%
\subsubsection{II.7.9.}\label{ii.7.9.}}

Let \(X\) be a regular noetherian scheme, and \(\mathcal{E}\) a locally
free coherent sheaf of rank \(\geqslant 2\) on \(X\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that
  \(\operatorname{Pic}\mathbf{P}(\mathcal{E}) \cong \operatorname{Pic}X \times \mathbf{Z}\).
\item
  If \(\mathcal{E}^{\prime}\) is another locally free coherent sheaf on
  \(X\), show that
  \(\mathbf{P}(\mathcal{E}) \cong \mathbf{P}\left(\mathcal{E}^{\prime}\right)\)
  (over \(X\) ) if and only if there is an invertible sheaf
  \(\mathcal{L}\) on \(X\) such that
  \(\mathcal{E}^{\prime} \cong \mathcal{E} \otimes \mathcal{L}\).
\end{enumerate}

\hypertarget{ii.7.10.-mathbfpn-bundles-over-a-scheme.-to_work}{%
\subsubsection{\texorpdfstring{II.7.10. \(\mathbf{P}^n\)-Bundles Over a
Scheme.}{II.7.10. \textbackslash mathbf\{P\}\^{}n-Bundles Over a Scheme.}}\label{ii.7.10.-mathbfpn-bundles-over-a-scheme.-to_work}}

Let \(X\) be a noetherian scheme.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  By analogy with the definition of a vector bundle (Ex. 5.18), define
  the notion of a projective \(n\)-space bundle over \(X\), as a scheme
  \(P\) with a morphism \(\pi: P \rightarrow X\) such that \(P\) is
  locally isomorphic to \(U \times \mathbf{P}^n, U \subseteq X\) open,
  and the transition automorphisms on
  \(\operatorname{Spec} A \times \mathbf{P}^n\) are given by
  \(A\)-linear automorphisms of the homogeneous coordinate ring
  \(A\left[x_0, \ldots, x_n\right]\)

  E.g., \(x_i^{\prime}=\sum a_{i j} x_j, a_{i j} \in A\).
\item
  If \(\mathcal{E}\) is a locally free sheaf of rankof rank \(n+1\) on
  \(X\) then \(\mathbf{P}(\mathcal E)\) is a
  \(\mathbf P^n{\hbox{-}}\)bundle over \(X\).
\item
  * Assume that \(X\) is regular, and show that every
  \(\mathbf{P}^n\)-bundle \(P\) over \(X\) is isomorphic to
  \(\mathbf{P}(\mathcal{E})\) for some locally free sheaf
  \({\mathcal{E}}\) on \(X\).\footnote{Hint: Let \(U \subseteq X\) be an
    open set such that \(\pi^{-1}(U) \cong U \times \mathbf{P}^n\), and
    let \(\mathcal{L}_0\) be the invertible sheaf \(\mathcal{C}(1)\) on
    \(U \times \mathbf{P}^n\). Show that \(\mathcal{L}_0\) extends to an
    invertible sheaf \(\mathcal{L}\) on \(P\). Then show that
    \(\pi_* \mathcal{L}=\mathcal{E}\) is a locally free sheaf on \(X\)
    and that \(P \cong \mathbf{P}(\mathcal{E})\).{]} Can you weaken the
    hypothesis " \(X\) regular"?}
\item
  Conclude (in the case \(X\) regular) that we have a 1-1 correspondence
  between \(\mathbf{P}^n\)-bundles over \(X\), and equivalence classes
  of locally free sheaves \(\mathcal{E}\) of rank \(n+1\) under the
  equivalence relation \({\mathcal{E}}\sim {\mathcal{E}}'\) if and only
  if \({\mathcal{E}}' \cong {\mathcal{E}}\otimes{\mathcal{M}}\) for some
  invertible sheaf \({\mathcal{M}}\) on \(X\).
\end{enumerate}

\hypertarget{ii.7.11.}{%
\subsubsection{II.7.11.}\label{ii.7.11.}}

On a noetherian scheme \(X\), different sheaves of ideals can give rise
to isomorphic blown up schemes.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\mathcal{I}\) is any coherent sheaf of ideals on \(X\), show that
  blowing up \(\mathcal{I}^d\) for any \(d \geqslant 1\) gives a scheme
  isomorphic to the blowing up of \(\mathcal{I}\) (cf.~Ex. 5.13).
\item
  If \(\mathcal{I}\) is any coherent sheaf of ideals, and if
  \(\mathcal{J}\) is an invertible sheaf of ideals, then \(\mathcal{I}\)
  and \(\mathcal{I} \cdot \mathcal{J}\) give isomorphic blowings-up.
\item
  If \(X\) is regular, show that (7.17) can be strengthened as follows.
  Let \(U \subseteq X\) be the largest open set such that
  \(f: f^{-1} U \rightarrow U\) is an isomorphism. Then \(\mathcal{I}\)
  can be chosen such that the corresponding closed subscheme \(Y\) has
  support equal to \(X-U\)
\end{enumerate}

\hypertarget{ii.7.12.}{%
\subsubsection{II.7.12.}\label{ii.7.12.}}

Let \(X\) be a noetherian scheme, and let \(Y, Z\) be two closed
subschemes, neither one containing the other. Let \(\tilde{X}\) be
obtained by blowing up \(Y \cap Z\) (defined by the ideal sheaf
\(\mathcal{I}_Y+\mathcal{I}_Z\) ). Show that the strict transforms
\(\tilde{Y}\) and \(\tilde{Z}\) of \(Y\) and \(Z\) in \(\tilde{X}\) do
not meet.

\hypertarget{ii.7.13.-a-complete-nonprojective-variety.}{%
\subsubsection{II.7.13. * A Complete Nonprojective
Variety.}\label{ii.7.13.-a-complete-nonprojective-variety.}}

Let \(k\) be an algebraically closed field of char \(\neq 2\). Let
\(C \subseteq \mathbf{P}_k^2\) be the nodal cubic curve If
\(P_0=(0,0,1)\) is the singular point, then \(C-P_0\) is isomorphic to
the multiplicative group
\(\mathbf{G}_m=\operatorname{Spec} k\left[t, t^{-1}\right]\) (Ex. 6.7).
For each \(a \in k, a \neq 0\), consider the translation of
\(\mathbf{G}_m\) given by \(t \mapsto a t\). This induces an
automorphism of \(C\) which we denote by \(\varphi_a\). Now consider
\(C \times\left(\mathbf{P}^1-\{0\}\right)\) and
\(C \times\left(\mathbf{P}^1-\{\infty\}\right)\). We glue their open
subsets \(C \times\left(\mathbf{P}^1-\{0, \infty\}\right)\) by the
isomorphism Thus we obtain a scheme \(X\), which is our example. The
projections to the second factor are compatible with \(\varphi\), so
there is a natural morphism \(\pi: X \rightarrow \mathbf{P}^1\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(\pi\) is a proper morphism, and hence that \(X\) is a
  complete variety over \(k\).
\item
  Use the method of (Ex. 6.9) to show that\footnote{Hint: If \(A\) is a
    domain and if \(*\) denotes the group of units, then
    \((A[u])^* \cong A^*\) and
    \(\left(A\left[u, u^{-1}\right]\right)^* \cong A^* \times \mathbf{Z}\).}
\item
  Now show that the restriction map is of the form
  \(\langle t, n\rangle \mapsto\langle t, 0, n\rangle\), and that the
  automorphism \(\varphi\) of
  \(C \times\left(\mathbf{A}^1-\{0\}\right)\) induces a map of the form
  \(\langle t, d, n\rangle \mapsto\langle t, d+n, n\rangle\) on its
  Picard group.
\item
  Conclude that the image of the restriction map consists entirely of
  divisors of degree 0 on \(C\). Hence \(X\) is not projective over
  \(k\) and \(\pi\) is not a projective morphism.
\end{enumerate}

\hypertarget{ii.7.14.}{%
\subsubsection{II.7.14.}\label{ii.7.14.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Give an example of a noetherian scheme \(X\) and a locally free
  coherent sheaf \(\mathcal{E}\), such that the invertible sheaf
  \({\mathcal{O}}(1)\) on \(\mathbf{P}(\mathcal{E})\) is not very ample
  relative to \(X\).
\item
  Let \(f: X \rightarrow Y\) be a morphism of finite type, let
  \(\mathcal{L}\) be an ample invertible sheaf on \(X\), and let
  \(\mathcal{S}\) be a sheaf of graded \({\mathcal{O}}_X\)-algebras
  satisfying (†). Let \(P=\operatorname{Proj} \mathcal{S}\), let
  \(\pi: P \rightarrow X\) be the projection, and let
  \({\mathcal{O}}_P(1)\) be the associated invertible sheaf. Show that
  for all \(n \gg 0\), the sheaf
  \({\mathcal{O}}_P(1) \otimes \pi^* \mathcal{L}^n\) is very ample on
  \(P\) relative to \(Y\).\footnote{Hint: Use (7.10) and (Ex. 5.12)}
\end{enumerate}

\hypertarget{ii.8-differentials}{%
\subsection{II.8: Differentials}\label{ii.8-differentials}}

\hypertarget{ii.8.1}{%
\subsubsection{II.8.1}\label{ii.8.1}}

Here we will strengthen the results of the text to include information
about the sheaf of differentials at a not necessarily closed point of a
scheme \(X\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Generalize (8.7) as follows. Let \(B\) be a local ring containing a
  field \(k\), and assume that the residue field
  \(k(B)=B / \mathfrak{m}\) of \(B\) is a separably generated extension
  of \(k\). Then the exact sequence of (8.4A), is exact on the left
  also.\footnote{Hint: In copying the proof of (8.7), first pass to
    \(B / \mathrm{m}^2\), which is a complete local ring, and then use
    (8.25A) to choose a field of representatives for
    \(B / \mathrm{m}^2\).}
\item
  Generalize (8.8) as follows. With \(B, k\) as above, assume
  furthermore that \(k\) is perfect, and that \(B\) is a localization of
  an algebra of finite type over \(k\). Then show that \(B\) is a
  regular local ring if and only if \(\Omega_{B / k}\) is free of rank
  \(=\operatorname{dim} B+\) tr.d. \(k(B) / k\).
\item
  Strengthen (8.15) as follows. Let \(X\) be an irreducible scheme of
  finite type over a perfect field \(k\), and let
  \(\operatorname{dim} X=n\). For any point \(x \in X\), not necessarily
  closed, show that the local ring \(\mathcal{O}_{x, X}\) is a regular
  local ring if and only if the stalk \(\left(\Omega_{X / k}\right)_x\)
  of the sheaf of differentials at \(x\) is free of rank \(n\).
\item
  Strengthen (8.16) as follows. If \(X\) is a variety over an
  algebraically closed field \(k\), then
  \(U=\left\{x \in X \mathrel{\Big|}\mathcal{O}_x\right.\) is a regular
  local ring \(\}\) is an open dense subset of \(X\).
\end{enumerate}

\hypertarget{ii.8.2.}{%
\subsubsection{II.8.2.}\label{ii.8.2.}}

Let \(X\) be a variety of dimension \(n\) over \(k\). Let
\(\mathcal{E}\) be a locally free sheaf of rank \(>n\) on \(X\), and let
\(V \subseteq \Gamma(X, \mathcal{E})\) be a vector space of global
sections which generate \(\mathcal{E}\). Then show that there is an
element \(s \in V\), such that for each \(x \in X\), we have
\(s_x \notin \mathfrak{m}_x \mathcal{E}_x\). Conclude that there is a
morphism \(\mathcal{O}_X \rightarrow \mathcal{E}\) giving rise to an
exact sequence where \(\mathcal{E}'\) is also locally free.\footnote{Hint:
  Use a method similar to the proof of Bertini's theorem (8.18).{]}}

\hypertarget{ii.8.3.-product-schemes.}{%
\subsubsection{II.8.3. Product
Schemes.}\label{ii.8.3.-product-schemes.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(X\) and \(Y\) be schemes over another scheme \(S\). Use (8.10)
  and (8.11) to show that
\item
  If \(X\) and \(Y\) are nonsingular varieties over a field \(k\), show
  that
\item
  Let \(Y\) be a nonsingular plane cubic curve, and let \(X\) be the
  surface \(Y \times Y\). Show that \(p_g(X)=1\) but \(p_a(X)=-1\) (I,
  Ex. 7.2). This shows that the arithmetic genus and the geometric genus
  of a nonsingular projective variety may be different.
\end{enumerate}

\hypertarget{ii.8.4.-complete-intersections-in-mathbfpn.-to_work}{%
\subsubsection{\texorpdfstring{II.8.4. Complete Intersections in
\(\mathbf{P}^n\).}{II.8.4. Complete Intersections in \textbackslash mathbf\{P\}\^{}n.}}\label{ii.8.4.-complete-intersections-in-mathbfpn.-to_work}}

A closed subscheme \(Y\) of \(\mathbf{P}_k^n\) is called a
\textbf{(strict, global) complete intersection} if the homogeneous ideal
\(I\) of \(Y\) in \(S=k\left[x_0, \ldots, x_n\right]\) can be generated
by \(r=\operatorname{codim}\left(Y, \mathbf{P}^n\right)\) elements (I,
Ex. 2.17).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(Y\) be a closed subscheme of codimension \(r\) in
  \(\mathbf{P}^n\). Then \(Y\) is a complete intersection if and only if
  there are hypersurfaces (i.e., locally principal subschemes of
  codimension 1) \(H_1, \ldots, H_r\), such that
  \(Y=H_1 \cap \ldots \cap H_r\) as schemes, i.e.,
  \(\mathcal{I}_Y=\mathcal{I}_{H_1}+\ldots+\mathcal{I}_{H_r}\).\footnote{Hint:
    Use the fact that the uniqueness theorem holds in \(S\) (Matsumura
    \([2, p. 107]\)).}
\item
  If \(Y\) is a complete intersection of dimension \(\geqslant 1\) in
  \(\mathbf{P}^n\), and if \(Y\) is normal, then \(Y\) is projectively
  normal (Ex. 5.14).\footnote{Hint: Apply (8.23) to the affine cone over
    \(Y\).}
\item
  With the same hypotheses as (b), conclude that for all
  \(l \geqslant 0\), the natural map
  \(\Gamma\left(\mathbf{P}^n, \mathcal{O}_{\mathbf{P}^n}(l)\right) \rightarrow \Gamma\left(Y, \mathcal{O}_Y(l)\right)\)
  is surjective. In particular, taking \(l=0\), show that \(Y\) is
  connected.
\item
  Now suppose given integers \(d_1, \ldots, d_r \geqslant 1\), with
  \(r<n\). Use Bertini's theorem (8.18) to show that there exist
  nonsingular hypersurfaces \(H_1, \ldots, H_r\) in \(\mathbf{P}^n\),
  with deg \(H_i=d_i\), such that the scheme
  \(Y=H_1 \cap \ldots \cap H_r\) is irreducible and nonsingular of
  codimension \(r\) in \(\mathbf{P}^n\).
\item
  If \(Y\) is a nonsingular complete intersection as in (d), show that
\item
  If \(Y\) is a nonsingular hypersurface of degree \(d\) in
  \(\mathbf{P}^n\), use (c) and (e) above to show that
  \(p_g(Y)={d-1\choose n}\). Thus \(p_g(Y)=p_a(Y)\) (I, Ex. 7.2). In
  particular, if \(Y\) is a nonsingular plane curve of degree \(d\),
  then
\item
  If \(Y\) is a nonsingular curve in \(\mathbf{P}^3\), which is a
  complete intersection of nonsingular surfaces of degrees \(d, e\),
  then Again the geometric genus is the same as the arithmetic genus (I,
  Ex. 7.2).
\end{enumerate}

\hypertarget{ii.8.5.-blowing-up-a-nonsingular-subvariety.}{%
\subsubsection{II.8.5. Blowing up a Nonsingular
Subvariety.}\label{ii.8.5.-blowing-up-a-nonsingular-subvariety.}}

As in (8.24), let \(X\) be a nonsingular variety, let \(Y\) be a
nonsingular subvariety of codimension \(r \geqslant 2\), let
\(\pi: \widetilde{X} \rightarrow X\) be the blowing-up of \(X\) along
\(Y\), and let \(Y^{\prime}=\pi^{-1}(Y)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that the maps
  \(\pi^*:\operatorname{Pic}X \rightarrow\operatorname{Pic}\tilde{X}\),
  and \(\mathbf{Z} \rightarrow \operatorname{Pic}X\) defined by
  \(n \mapsto\) class of \(n Y^{\prime}\), give rise to an isomorphism
  \(\operatorname{Pic}\tilde{X} \cong \operatorname{Pic} X \oplus \mathbf{Z}\).
\item
  Show that\footnote{Hint: By (a) we can write in any case for some
    invertible sheaf \(\mathcal{M}\) on \(X\), and some integer q. By
    restricting to \(\tilde{X}-Y^{\prime} \cong X-Y\), show that
    \(\mathcal{M} \cong \omega_X\). To determine \(q\), proceed as
    follows.

    \begin{itemize}
    \tightlist
    \item
      First show that
      \(\omega_{Y^{\prime}} \cong f^* \omega_X \otimes \mathcal{O}_{Y^{\prime}}(-q-1)\).
    \item
      Then take a closed point \(y \in Y\) and let \(Z\) be the fibre of
      \(Y^{\prime}\) over \(y\).
    \item
      Then show that \(\omega_Z \cong\) \(\mathcal{O}_Z(-q-1)\). But
      since \(Z \cong \mathbf{P}^{r-1}\), we have
      \(\omega_Z \cong \mathcal{O}_Z(-r)\), so \(q=r-1\).
    \end{itemize}}
\end{enumerate}

\hypertarget{ii.8.6.-the-infinitesimal-lifting-property.}{%
\subsubsection{II.8.6. The Infinitesimal Lifting
Property.}\label{ii.8.6.-the-infinitesimal-lifting-property.}}

The following result is very important in studying deformations of
nonsingular varieties. Let \(k\) be an algebraically closed field, let
\(A\) be a finitely generated \(k\)-algebra such that
\(\operatorname{Spec} A\) is a nonsingular variety over \(k\). Let be an
exact sequence, where \(B^{\prime}\) is a \(k\)-algebra, and \(I\) is an
ideal with \(I^2=0\). Finally suppose given a \(k\)-algebra homomorphism
\(f: A \rightarrow B\). Then there exists a \(k\)-algebra homomorphism
\(g: A \rightarrow B^{\prime}\) making a commutative diagram

\begin{tikzcd}
    && {} \\
    \\
    && I \\
    && {B'} \\
    A && B
    \arrow["f", from=5-1, to=5-3]
    \arrow["g", dashed, from=5-1, to=4-3]
    \arrow[hook, from=3-3, to=4-3]
    \arrow[two heads, from=4-3, to=5-3]
\end{tikzcd}

\begin{quote}
\href{https://q.uiver.app/?q=WzAsNSxbMCw0LCJBIl0sWzIsNCwiQiJdLFsyLDIsIkkiXSxbMiwwXSxbMiwzLCJCJyJdLFswLDEsImYiXSxbMCw0LCJnIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzIsNCwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbNCwxLCIiLDIseyJzdHlsZSI6eyJoZWFkIjp7Im5hbWUiOiJlcGkifX19XV0=}{Link
to Diagram}
\end{quote}

We call this result the \textbf{infinitesimal lifting property for
\(A\)}. We prove this result in several steps.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  First suppose that \(g: A \rightarrow B^{\prime}\) is a given
  homomorphism lifting \(f\). If
  \(g^{\prime}: A \rightarrow B^{\prime}\) is another such homomorphism,
  show that \(\theta=g-g^{\prime}\) is a \(k\)-derivation of \(A\) into
  \(I\), which we can consider as an element of
  \(\operatorname{Hom}_A\left(\Omega_{A / k}, I\right)\). Note that
  since \(I^2=0, I\) has a natural structure of \(B\)-module and hence
  also of \(A\)-module. Conversely, for any
  \(\theta \in \operatorname{Hom}_A\left(\Omega_{A / k}, I\right), g^{\prime}=g+\theta\)
  is another homomorphism lifting \(f\). (For this step, you do not need
  the hypothesis about Spec \(A\) being nonsingular.)
\item
  Now let \(P=k\left[x_1, \ldots, x_n\right]\) be a polynomial ring over
  \(k\) of which \(A\) is a quotient, and let \(J\) be the kernel. Show
  that there does exist a homomorphism \(h: P \rightarrow B^{\prime}\)
  making a commutative diagram,

  \begin{tikzcd}
   J && I \\
   P && B \\
   A && {B'}
   \arrow["f"', from=3-1, to=3-3]
   \arrow["h"', from=2-1, to=2-3]
   \arrow[hook, from=1-1, to=2-1]
   \arrow[hook, from=1-3, to=2-3]
   \arrow[two heads, from=2-1, to=3-1]
   \arrow[two heads, from=2-3, to=3-3]
  \end{tikzcd}

  \begin{quote}
  \href{https://q.uiver.app/?q=WzAsNixbMCwwLCJKIl0sWzAsMSwiUCJdLFswLDIsIkEiXSxbMiwwLCJJIl0sWzIsMSwiQiJdLFsyLDIsIkInIl0sWzIsNSwiZiIsMl0sWzEsNCwiaCIsMl0sWzAsMSwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMyw0LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFsxLDIsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dLFs0LDUsIiIsMCx7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dXQ==}{Link
  to Diagram}
  \end{quote}

  and show that \(h\) induces an \(A\)-linear map
  \(\mkern 1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu: J / J^2 \rightarrow I\).
\item
  Now use the hypothesis Spec \(A\) nonsingular and (8.17) to obtain an
  exact sequence Show furthermore that applying the functor
  \(\operatorname{Hom}_A(\cdot, I)\) gives an exact sequence Let
  \(\theta \in \operatorname{Hom}_P\left(\Omega_{P / k}, I\right)\) be
  an element whose image gives
  \(\mkern 1.5mu\overline{\mkern-1.5muh\mkern-1.5mu}\mkern 1.5mu\in \operatorname{Hom}_A\left(J / J^2, I\right)\).
  Consider \(\theta\) as a derivation of \(P\) to \(B^{\prime}\). Then
  let \(h^{\prime}=h-\theta\), and show that \(h^{\prime}\) is a
  homomorphism of \(P \rightarrow B^{\prime}\) such that
  \(h^{\prime}(J)=0\). Thus \(h^{\prime}\) induces the desired
  homomorphism \(g: A \rightarrow B^{\prime}\).
\end{enumerate}

\hypertarget{ii.8.7.}{%
\subsubsection{II.8.7.}\label{ii.8.7.}}

As an application of the infinitesimal lifting property, we consider the
following general problem. Let \(X\) be a scheme of finite type over
\(k\), and let \(\mathcal{F}\) be a coherent sheaf on \(X\). We seek to
classify schemes \(X^{\prime}\) over \(k\), which have a sheaf of ideals
\(\mathcal{I}\) such that \(\mathcal{I}^2=0\) and
\(\left(X^{\prime}, \mathcal{O}_X / \mathcal{I}\right) \cong\left(X, \mathcal{O}_X\right)\),
and such that \(\mathcal{I}\) with its resulting structure of
\(\mathcal{O}_X\)-module is isomorphic to the given sheaf
\(\mathcal{F}\). Such a pair \(X^{\prime}, \mathcal{I}\) we call an
\textbf{infinitesimal extension of the scheme \(X\) by the sheaf
\(\mathcal{F}\)}.

One such extension, the trivial one, is obtained as follows. Take
\(\mathcal{O}_{X^{\prime}}=\mathcal{O}_X \oplus \mathcal{F}\) as sheaves
of abelian groups, and define multiplication by Then the topological
space \(X\) with the sheaf of rings \(\mathcal{O}_{X^{\prime}}\) is an
infinitesimal extension of \(X\) by \(\mathcal{F}\).

The general problem of classifying extensions of \(X\) by
\(\mathcal{F}\) can be quite complicated. So for now, just prove the
following special case: if \(X\) is affine and nonsingular, then any
extension of \(X\) by a coherent sheaf \(\mathcal{F}\) is isomorphic to
the trivial one. See (III, Ex. 4.10) for another case.

\hypertarget{ii.8.8.-plurigenus-and-some-hodge-numbers-are-birational-invariants.}{%
\subsubsection{II.8.8. Plurigenus and (some) Hodge numbers are
birational
invariants.}\label{ii.8.8.-plurigenus-and-some-hodge-numbers-are-birational-invariants.}}

Let \(X\) be a projective nonsingular variety over \(k\). For any
\(n>0\) we define the \textbf{\(n\)th plurigenus of \(X\)} to be Thus in
particular \(P_1=p_g\). Also, for any
\(q, 0 \leqslant q \leqslant \operatorname{dim} X\) we define an integer
is the sheaf of regular \(q\)-forms on \(X\). In particular, for
\(q=\operatorname{dim} X\), we recover the geometric genus again. The
integers \(h^{q, 0}\) are called \textbf{Hodge numbers}.

Using the method of \((8.19)\), show that \(P_n\) and \(h^{q, 0}\) are
birational invariants of \(X\), i.e., if \(X\) and \(X^{\prime}\) are
birationally equivalent nonsingular projective varieties, then
\(P_n(X)=P_n\left(X^{\prime}\right)\) and
\(h^{q, 0}(X)=h^{q, 0}\left(X^{\prime}\right)\).

\hypertarget{ii.9-formal-schemes}{%
\subsection{II.9: Formal Schemes}\label{ii.9-formal-schemes}}

\hypertarget{ii.9.1.}{%
\subsubsection{II.9.1.}\label{ii.9.1.}}

Let \(X\) be a noetherian scheme, \(Y\) a closed subscheme, and
\(\widehat{X}\) the completion of \(X\) along \(Y\). We call the ring
\(\Gamma\left(\widehat{X}, \mathcal{O}_{\widehat{X}}\right)\) the ring
of \textbf{formal-regular} functions on \(X\) along \(Y\). In this
exercise we show that if \(Y\) is a connected, nonsingular, positive
dimensional subvariety of \(X=\mathbf{P}_k^n\) over an algebraically
closed field \(k\), then
\(\Gamma\left(\widehat{X}, \mathcal{O}_{\widehat{X}}\right)=k\)

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(\mathcal{I}\) be the ideal sheaf of \(Y\). Use (8.13) and (8.17)
  to show that there is an inclusion of sheaves on
  \(Y, \mathcal{I} / \mathcal{I}^2 \hookrightarrow \mathcal{O}_Y(-1)^{n+1}\).
\item
  Show that for any
  \(r \geqslant 1, \Gamma\left(Y, \mathcal{I}^r / \mathcal{I}^{r+1}\right)=0\).
\item
  Use the exact sequences and induction on \(r\) to show that
  \(\Gamma\left(Y, \mathcal{O}_X / \mathcal{I}^r\right)=k\) for all
  \(r \geqslant 1\).\footnote{Use 8.21Ae.}
\end{enumerate}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{3}
\tightlist
\item
  Conclude that
  \(\Gamma\left(\widehat{X}, \mathcal{O}_{\widehat{X}}\right)=k\).\footnote{Actually,
    the same result holds without the hypothesis \(Y\) nonsingular, but
    the proof is more difficult-see Hartshorne \([3,(7.3)]\).}
\end{enumerate}

\hypertarget{ii.9.2.}{%
\subsubsection{II.9.2.}\label{ii.9.2.}}

Use the result of (Ex. 9.1) to prove the following geometric result. Let
\(Y \subseteq X=\) \(\mathbf{P}_k^n\) be as above, and let
\(f: X \rightarrow Z\) be a morphism of \(k\)-varieties. Suppose that
\(f(Y)\) is a single closed point \(P \in Z\). Then \(f(X)=P\) also.

\hypertarget{ii.9.3.}{%
\subsubsection{II.9.3.}\label{ii.9.3.}}

Prove the analogue of \((5.6)\) for formal schemes, which says, if
\(\mathrm{X}\) is an affine formal scheme, and if is an exact sequence
of \(\mathcal{O}_x\)-modules, and if \(\mathrm{F}^{\prime}\) is
coherent, then the sequence of global sections is exact. For the proof,
proceed in the following steps.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(\mathrm{I}\) be an ideal of definition for \(\mathrm{X}\), and
  for each \(n>0\) consider the exact sequence Use (5.6), slightly
  modified, to show that for every open affine subset
  \(\mathrm{U} \subseteq \mathrm{X}\), the sequence is exact.
\item
  Now pass to the limit, using (9.1), (9.2), and (9.6). Conclude that
  \(\mathrm{F} \cong \lim \mathrm{F} / \mathrm{J}^n \mathrm{F}^{\prime}\)
  and that the sequence of global sections above is exact.
\end{enumerate}

\hypertarget{ii.9.4.}{%
\subsubsection{II.9.4.}\label{ii.9.4.}}

Use (Ex. 9.3) to prove that if is an exact sequence of
\(\mathcal{O}_x\)-modules on a noetherian formal scheme \(\mathrm{X}\),
and if \(\mathrm{F}^{\prime}, \mathrm{F}^{\prime \prime}\) are coherent,
then \(\mathrm{F}\) is coherent also.

\hypertarget{ii.9.5.}{%
\subsubsection{II.9.5.}\label{ii.9.5.}}

If \(\mathrm{F}\) is a coherent sheaf on a noetherian formal scheme
\(\mathrm{X}\), which can be generated by global sections, show in fact
that it can be generated by a finite number of its global sections.

\hypertarget{ii.9.6.}{%
\subsubsection{II.9.6.}\label{ii.9.6.}}

Let \(\mathrm{X}\) be a noetherian formal scheme, let \(\mathrm{I}\) be
an ideal of definition, and for each \(n\), let \(Y_n\) be the scheme
\(\left(\mathrm{X}, \mathcal{O}_x / \mathrm{J}^n\right)\). Assume that
the inverse system of groups
\(\left(\Gamma\left(Y_n, \mathcal{O}_{Y_n}\right)\right)\) satisfies the
Mittag-Leffler condition. Then prove that
\(\operatorname{Pic}\mathrm{X}=\lim \operatorname{Pic}Y_n\).

As in the case of a scheme, we define \(\operatorname{Pic}\mathrm{X}\)
to be the group of locally free \(\mathcal{O}_x\)-modules of rank 1
under the operation \(\otimes\). Proceed in the following steps.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Use the fact that
  \(\operatorname{ker}\left(\Gamma\left(Y_{n+1}, \mathcal{O}_{Y_{n+1}}\right) \rightarrow \Gamma\left(Y_n, \mathcal{O}_{Y_n}\right)\right)\)
  is a nilpotent ideal to show that the inverse system
  \(\left(\Gamma\left(Y_n, \mathcal{O}_{Y_n}^*\right)\right)\) of units
  in the respective rings also satisfies (ML).
\item
  Let \(\mathrm{F}\) be a coherent sheaf of \(\mathcal{O}_x\)-modules,
  and assume that for each \(n\), there is some isomorphism
  \(\varphi_n: \tilde{F} / \mathrm{J}^n \mathrm{F} \cong \mathcal{O}_{Y_n}\).
  Then show that there is an isomorphism
  \(\tilde{F} \cong \mathcal{O}_x\).\footnote{Be careful, because the
    \(\varphi_n\) may not be compatible with the maps in the two inverse
    systems \(\left(\mathrm{F} / \mathrm{J}^n \mathrm{F}\right)\) and
    \(\left(\mathcal{O}_{Y_n}\right)\)!}

  Conclude that the natural map Pic
  \(\mathrm{X} \rightarrow \cocolim \operatorname{Pic}Y_n\) is
  injective.
\item
  Given an invertible sheaf \(\mathcal{L}_n\) on \(Y_n\) for each \(n\),
  and given isomorphisms \(\mathcal{L}_{n+1} \otimes\)
  \(\mathcal{O}_{Y_n} \cong \mathcal{L}_n\), construct maps
  \(\mathcal{L}_{n^{\prime}} \rightarrow \mathcal{L}_n\) for each
  \(n^{\prime} \geqslant n\) so as to make an inverse system, and show
  that \(\mathrm{L}=\lim \mathcal{L}_n\) is a coherent sheaf on
  \(\mathrm{X}\).

  Then show that \(\mathrm{L}\) is locally free of rank 1, and thus
  conclude that the map
  \(\operatorname{Pic}\mathrm{X} \rightarrow \lim \operatorname{Pic}Y_n\)
  is surjective.\footnote{Again be careful, because even though each
    \(\mathcal{L}_n\) is locally free of rank 1, the open sets needed to
    make them free might get smaller and smaller with \(n\).}
\item
  Show that the hypothesis
  ``\(\left(\Gamma\left(Y_n, \mathcal{O}_{Y_n}\right)\right)\) satisfies
  (ML)'' is satisfied if either \(\mathrm{X}\) is affine, or each
  \(Y_n\) is projective over a field \(k\).\footnote{See (III, Ex.
    11.5-11.7) for further examples and applications.}
\end{enumerate}

\hypertarget{iii-cohomology}{%
\section{III: Cohomology}\label{iii-cohomology}}

\hypertarget{iii.1-derived-functors}{%
\subsection{III.1: Derived Functors}\label{iii.1-derived-functors}}

Amazing! No exercises in this section.

\hypertarget{iii.2-cohomology-of-sheaves}{%
\subsection{III.2: Cohomology of
Sheaves}\label{iii.2-cohomology-of-sheaves}}

\hypertarget{v.2.1.}{%
\subsubsection{V.2.1.}\label{v.2.1.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(X=\mathbf{A}_k^1\) be the affine line over an infinite field
  \(k\). Let \(P, Q\) be distinct closed points of \(X\), and let
  \(U=X-\{P, Q\}\). Show that
  \(H^1\left(X, \mathbf{Z}_U\right) \neq 0\).
\item
  * More generally, let \(Y \subseteq X=\mathbf{A}_k^n\) be the union of
  \(n+1\) hyperplanes in suitably general position, and let \(U=X-Y\).
  Show that \(H^n\left(X, Z_U\right) \neq 0\). Thus the result of
  \((2.7)\) is the best possible.
\end{enumerate}

\hypertarget{v.2.2.}{%
\subsubsection{V.2.2.}\label{v.2.2.}}

Let \(X=\mathbf{P}_k^1\) be the projective line over an algebraically
closed field \(k\). Show that the exact sequence of (II, Ex. 1.21d) is a
flasque resolution of \(\mathcal{O}\). Conclude from (II, Ex. 1.21e)
that \(H^i(X, \mathcal{O})=0\) for all \(i>0\).

\hypertarget{v.2.3.-cohomology-with-supports.}{%
\subsubsection{V.2.3. Cohomology with
Supports.}\label{v.2.3.-cohomology-with-supports.}}

Let \(X\) be a topological space, let \(Y\) be a closed subset, and let
\(\mathcal{F}\) be a sheaf of abelian groups. Let
\(\Gamma_Y(X, \mathcal{F})\) denote the group of sections of
\(\mathcal{F}\) with support in \(Y\) (II, Ex. 1.20).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(\Gamma_Y(X, \cdot)\) is a left exact functor from
  \({\mathsf{Ab}}(X)\) to \({\mathsf{Ab}}\). We denote the right derived
  functors of \(\Gamma_Y(X, \cdot)\) by \(H_Y^i(X, \cdot)\). They are
  the cohomology groups of \(X\) with supports in \(Y\), and
  coefficients in a given sheaf.
\item
  If
  \(0 \rightarrow \mathcal{F}^{\prime} \rightarrow \mathcal{F} \rightarrow \dot{{\mathcal{F}}}^{\prime \prime} \rightarrow 0\)
  is an exact sequence of sheaves, with \(\mathcal{F}^{\prime}\)
  flasque, show that is exact.
\item
  Show that if \(\mathcal{F}\) is flasque, then
  \(H_Y^i(X, \mathcal{F})=0\) for all \(i>0\).
\item
  If \(\mathcal{F}\) is flasque, show that the sequence is exact.
\item
  Let \(U=X-Y\). Show that for any \(\mathcal{F}\), there is a long
  exact sequence of cohomology groups
\item
  \emph{Excision}. Let \(V\) be an open subset of \(X\) containing
  \(Y\). Then there are natural functorial isomorphisms, for all \(i\)
  and \(\mathcal{F}\),
\end{enumerate}

\hypertarget{v.2.4.-mayer-vietoris-sequence.}{%
\subsubsection{V.2.4. Mayer-Vietoris
Sequence.}\label{v.2.4.-mayer-vietoris-sequence.}}

Let \(Y_1, Y_2\) be two closed subsets of \(X\). Then there is a long
exact sequence of cohomology with supports

\hypertarget{v.2.5.}{%
\subsubsection{V.2.5.}\label{v.2.5.}}

Let \(X\) be a Zariski space (II, Ex. 3.17). Let \(P \in X\) be a closed
point, and let \(X_P\) be the subset of \(X\) consisting of all points
\(Q \in X\) such that \(P \in\{Q\}^{-}\). We call \(X_P\) the
\textbf{local space} of \(X\) at \(P\), and give it the induced
topology.

Let \(j: X_P \rightarrow X\) be the inclusion, and for any sheaf
\(\mathcal{F}\) on \(X\), let \(\mathcal{F}_P=j^* \mathcal{F}\). Show
that for all \(i, \mathcal{F}\), we have

\hypertarget{v.2.6.}{%
\subsubsection{V.2.6.}\label{v.2.6.}}

Let \(X\) be a noetherian topological space, and let
\(\left\{\mathcal{I}_\alpha\right\}_{\alpha \in A}\) be a direct system
of injective sheaves of abelian groups on \(X\). Then
\(\colim\, \mathcal{I}_\alpha\) is also injective.\footnote{Hints: First
  show that a sheaf \(\mathcal{I}\) is injective if and only if for
  every open set \(U \subseteq X\), and for every subsheaf
  \(\mathcal{R} \subseteq \mathbf{Z}_U\), and for every map
  \(f: \mathcal{R} \rightarrow \mathcal{I}\), there exists an extension
  of \(f\) to a map of \(\mathbf{Z}_U \rightarrow \mathcal{I}\).
  Secondly, show that any such sheaf \(\mathcal{R}\) is finitely
  generated, so any map
  \(\mathcal{R} \rightarrow \colim\, \mathcal{I}_\alpha\) factors
  through one of the \(\mathcal{I}_\alpha\).}

\hypertarget{v.2.7.}{%
\subsubsection{V.2.7.}\label{v.2.7.}}

Let \(S^1\) be the circle (with its usual topology), and let
\(\mathbf{Z}\) be the constant sheaf \(\mathbf{Z}\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(H^1\left(S^1, \mathbf{Z}\right) \cong \mathbf{Z}\), using
  our definition of cohomology.
\item
  Now let \(\mathcal{R}\) be the sheaf of germs of continuous
  real-valued functions on \(S^1\). Show that
  \(H^1\left(S^1, \mathcal{R}\right)=0\).
\end{enumerate}

\hypertarget{iii.3-cohomology-of-a-noetherian-affine-scheme}{%
\subsection{III.3: Cohomology of a Noetherian Affine
Scheme}\label{iii.3-cohomology-of-a-noetherian-affine-scheme}}

\hypertarget{v.3.1.}{%
\subsubsection{V.3.1.}\label{v.3.1.}}

Let \(X\) be a noetherian scheme. Show that \(X\) is affine if and only
if \(X_{\text {red }}\) (II, Ex. 2.3) is affine.\footnote{Hint: Use
  (3.7), and for any coherent sheaf \(\mathcal{F}\) on \(X\), consider
  the filtration
  \(\mathcal{F} \supseteq \mathcal{N} \cdot \mathcal{F} \supseteq \mathcal{N}^2 \cdot \mathcal{F} \supseteq \ldots\),
  where \(\mathcal{N}\) is the sheaf of nilpotent elements on \(X\).}

\hypertarget{v.3.2.}{%
\subsubsection{V.3.2.}\label{v.3.2.}}

Let \(X\) be a reduced noetherian scheme. Show that \(X\) is affine if
and only if each irreducible component is affine.

\hypertarget{v.3.3.}{%
\subsubsection{V.3.3.}\label{v.3.3.}}

Let \(A\) be a noetherian ring, and let \(\mathfrak{a}\) be an ideal of
\(A\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(\Gamma_{\mathrm{{\mathfrak{a}}}}(\cdot)\) (II, Ex. 5.6) is
  a left-exact functor from the category of \(A\)-modules to itself. We
  denote its right derived functors, calculated in \(\mathsf{Mod}(A)\),
  by \(H_{{\mathfrak{a}}}^i(\cdot)\).
\item
  Now let \(X=\operatorname{Spec} A, Y=V(\mathfrak{a})\). Show that for
  any \(A\)-module \(M\), where \(H_Y^i(X, \cdot)\) denotes cohomology
  with supports in \(Y(\) Ex. 2.3).
\item
  For any \(i\), show that
  \(\Gamma_{{\mathfrak{a}}}\left(H_{\mathfrak{a}}^i(M)\right)=H_{\mathfrak{a}}^i(M)\).
\end{enumerate}

\hypertarget{v.3.4.-cohomological-interpretation-of-depth.}{%
\subsubsection{V.3.4. Cohomological Interpretation of
Depth.}\label{v.3.4.-cohomological-interpretation-of-depth.}}

If \(A\) is a ring, \({\mathfrak{a}}\) an ideal, and \(M\) an \(A\)
module, then \(\operatorname{depth}_{\mathfrak{a}}M\) is the maximum
length of an \(M\)-regular sequence \(x_1, \ldots, x_r\), with all
\(x_i \in \mathfrak{a}\). This generalizes the notion of depth
introduced in \((II, \S 8)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Assume that \(A\) is noetherian. Show that if
  \(\operatorname{depth}_{\mathfrak{a}}M \geqslant 1\), then
  \(\Gamma_{\mathfrak{a}}(M)=0\), and the converse is true if \(M\) is
  finitely generated.\footnote{Hint: When \(M\) is finitely generated,
    both conditions are equivalent to saying that \(\mathfrak{a}\) is
    not contained in any associated prime of \(M\).}
\item
  Show inductively, for \(M\) finitely generated, that for any
  \(n \geqslant 0\), the following conditions are equivalent:

  \begin{enumerate}
  \def\labelenumii{(\roman{enumii})}
  \tightlist
  \item
    \(\operatorname{depth}_{\mathfrak{a}} M \geqslant n\);
  \item
    \(H_{\mathfrak{a}}^i(M)=0\) for all \(i<n\).
  \end{enumerate}
\end{enumerate}

\hypertarget{v.3.5.}{%
\subsubsection{V.3.5.}\label{v.3.5.}}

Let \(X\) be a noetherian scheme, and let \(P\) be a closed point of
\(X\). Show that the following conditions are equivalent:

\begin{enumerate}
\def\labelenumi{(\roman{enumi})}
\tightlist
\item
  \(\operatorname{depth}\mathcal{O}_P \geqslant 2\);
\item
  if \(U\) is any open neighborhood of \(P\), then every section of
  \(\mathcal{O}_X\) over \(U-P\) extends uniquely to a section of
  \(\mathcal{O}_X\) over \(U\).
\end{enumerate}

This generalizes (I, Ex. 3.20), in view of (II, 8.22A).

\hypertarget{v.3.6.}{%
\subsubsection{V.3.6.}\label{v.3.6.}}

Let \(X\) be a noetherian scheme.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that the sheaf \(\mathcal{G}\) constructed in the proof of (3.6)
  is an injective object in the category \({\mathsf{QCoh}}(X)\) of
  quasi-coherent sheaves on \(X\). Thus \({\mathsf{QCoh}}(X)\) has
  enough injectives.
\item
  * Show that any injective object of \({\mathsf{QCoh}}(X)\) is
  flasque.\footnote{Hints: The method of proof of (2.4) will not work,
    because \(\mathcal{O}_U\) is not quasi-coherent on \(X\) in general.
    Instead, use (II, Ex. 5.15) to show that if
    \(\mathcal{I} \in \mathbb{Q} \mathrm{co}(X)\) is injective, and if
    \(U \subseteq X\) is an open subset, then
    \(\left.\mathcal{I}\right|_U\) is an injective object of
    \(\operatorname{Qco}(U)\). Then cover \(X\) with open affines
    \(\cdots\).}
\item
  Conclude that one can compute cohomology as the derived functors of
  \(\Gamma(X, \cdot)\), considered as a functor from
  \({\mathsf{QCoh}}(X)\) to \({\mathsf{Ab}}\).
\end{enumerate}

\hypertarget{v.3.7.}{%
\subsubsection{V.3.7.}\label{v.3.7.}}

Let \(A\) be a noetherian ring, let \(X=\operatorname{Spec} A\), let
\(\mathfrak{a} \subseteq A\) be an ideal, and let \(U \subseteq X\) be
the open set \(X-V(\mathfrak{a})\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  For any \(A\)-module \(M\), establish the following formula of
  Deligne:
\item
  Apply this in the case of an injective \(A\)-module \(I\), to give
  another proof of (3.4).
\end{enumerate}

\hypertarget{v.3.8.}{%
\subsubsection{V.3.8.}\label{v.3.8.}}

Without the noetherian hypothesis, (3.3) and (3.4) are false. Let
\(A=k\left[x_0, x_1, x_2, \ldots\right]\) with the relations
\(x_0^n x_n=0\) for \(n=1,2, \ldots\). Let \(I\) be an injective
\(A\)-module containing \(A\). Show that \(I \rightarrow I_{x_0}\) is
not surjective.

\hypertarget{iii.4-ux10dech-cohomology}{%
\subsection{III.4: Čech Cohomology}\label{iii.4-ux10dech-cohomology}}

\hypertarget{v.4.1.}{%
\subsubsection{V.4.1.}\label{v.4.1.}}

Let \(f: X \rightarrow Y\) be an affine morphism of noetherian separated
schemes (II, Ex. 5.17). Show that for any quasi-coherent sheaf
\(\mathcal{F}\) on \(X\), there are natural isomorphisms for all
\(i \geqslant 0\),\footnote{Hint: Use (II, 5.8).}

\hypertarget{v.4.2.}{%
\subsubsection{V.4.2.}\label{v.4.2.}}

Prove Chevalley's theorem: Let \(f: X \rightarrow Y\) be a finite
surjective morphism of noetherian separated schemes, with \(X\) affine.
Then \(Y\) is affine.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(f: X \rightarrow Y\) be a finite surjective morphism of integral
  noetherian schemes. Show that there is a coherent sheaf
  \(\mathcal{M}\) on \(X\), and a morphism of sheaves
  \(\alpha: \mathcal{O}_Y^r \rightarrow f_* \mathcal{M}\) for some
  \(r>0\), such that \(\alpha\) is an isomorphism at the generic point
  of \(Y\).
\item
  For any coherent sheaf \(\mathcal{F}\) on \(Y\), show that there is a
  coherent sheaf \(\mathcal{G}\) on \(X\), and a morphism
  \(\beta: f_* \mathcal{G} \rightarrow \mathcal{F}^r\) which is an
  isomorphism at the generic point of \(Y\).\footnote{Hint: Apply
    \(\mathop{\mathcal{H}\! \mathit{om}}(\cdot, \mathcal{F})\) to
    \(\alpha\) and use (II, Ex. 5.17e).}
\item
  Now prove Chevalley's theorem. First use (Ex. 3.1) and (Ex. 3.2) to
  reduce to the case \(X\) and \(Y\) integral. Then use (3.7), (Ex.
  4.1), consider \(\operatorname{ker} \beta\) and coker \(\beta\), and
  use noetherian induction on \(Y\).
\end{enumerate}

\hypertarget{v.4.3.}{%
\subsubsection{V.4.3.}\label{v.4.3.}}

Let \(X=\mathbf{A}_k^2=\operatorname{Spec} k[x, y]\), and let
\(U=X-\{(0,0)\}\). Using a suitable cover of \(U\) by open affine
subsets, show that \(H^1\left(U, \mathcal{O}_U\right)\) is isomorphic to
the \(k\)-vector space spanned by
\(\left\{x^i y^j \mathrel{\Big|}i, j<0\right\}\). In particular, it is
infinite-dimensional.\footnote{Using (3.5), this provides another proof
  that \(U\) is not affine-cf.~(I, Ex. 3.6).}

\hypertarget{v.4.4.}{%
\subsubsection{V.4.4.}\label{v.4.4.}}

On an arbitrary topological space \(X\) with an arbitrary abelian sheaf
\(\mathcal{F}\), Čech cohomology may not give the same result as the
derived functor cohomology. But here we show that for \(H^1\), there is
an isomorphism if one takes the limit over all coverings.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(\mathfrak{U}=\left(U_i\right)_{i \in I}\) be an open covering of
  the topological space \(X\). A refinement of \(\mathfrak{U}\) is a
  covering \(\mathfrak{B}=\left(V_j\right)_{j \in J}\), together with a
  map \(\lambda: J \rightarrow I\) of the index sets, such that for each
  \(j \in J, V_j \subseteq U_{\lambda(j)}\). If \(\mathfrak{B}\) is a
  refinement of \(\mathfrak{X}\), show that there is a natural induced
  map on Čech cohomology, for any abelian sheaf \(\mathcal{F}\), and for
  each \(i\), The coverings of \(X\) form a partially ordered set under
  refinement, so we can consider the Ceech cohomology in the limit
\item
  For any abelian sheaf \(\mathcal{F}\) on \(X\), show that the natural
  maps (4.4) for each covering are compatible with the refinement maps
  above.
\item
  Now prove the following theorem. Let \(X\) be a topological space,
  \(\mathcal{F}\) a sheaf of abelian groups. Then the natural map is an
  isomorphism.\footnote{Hint: Embed \(\mathcal{F}\) in a flasque sheaf
    \(\mathcal{G}\), and let \(\mathcal{R}=\mathcal{G}/\mathcal{F}\), so
    that we have an exact sequence Define a complex
    \(D^{\cdot}(\mathfrak{U})\) by Then use the exact cohomology
    sequence of this sequence of complexes, and the natural map of
    complexes and see what happens under refinement.}
\end{enumerate}

\hypertarget{v.4.5.}{%
\subsubsection{V.4.5.}\label{v.4.5.}}

For any ringed space \(\left(X, \mathcal{O}_X\right)\), let
\(\operatorname{Pic}X\) be the group of isomorphism classes of
invertible sheaves (II, §6). Show that
\(\operatorname{Pic}X \cong H^1\left(X, \mathcal{O}_X^*\right)\), where
\(\mathcal{O}_X^*\) denotes the sheaf whose sections over an open set
\(U\) are the units in the ring \(\Gamma\left(U, \mathcal{O}_X\right)\),
with multiplication as the group operation.\footnote{Hint: For any
  invertible sheaf \(\mathcal{L}\) on \(X\), cover \(X\) by open sets
  \(U_i\) on which \(\mathcal{L}\) is free, and fix isomorphisms
  \(\phi_i: {\mathcal{O}}_{U_i} { \, \xrightarrow{\sim}\, }{ \left.{{{\mathcal{L}}}} \right|_{{U_i}} }\).
  Then on \(U_i \cap U_j\), we get an isomorphism
  \(\varphi_i^{-1} \circ \varphi_j\) of \(\mathcal{O}_{U_i \cap U_j}\)
  with itself. These isomorphisms give an element of
  \(\check{H}^1\left(\mathfrak{U}, \mathcal{O}_X^*\right)\). Now use
  (Ex. 4.4).}

\hypertarget{v.4.6.}{%
\subsubsection{V.4.6.}\label{v.4.6.}}

Let \(\left(X, \mathcal{O}_X\right)\) be a ringed space, let
\(\mathcal{I}\) be a sheaf of ideals with \(\mathcal{I}^2=0\), and let
\(X_0\) be the ringed space
\(\left(X, \mathcal{O}_X / \mathcal{I}\right)\). Show that there is an
exact sequence of sheaves of abelian groups on \(X\), where
\(\mathcal{O}_X^*\) (respectively, \(\mathcal{O}_{X_0}^*\) ) denotes the
sheaf of (multiplicative) groups of units in the sheaf of rings
\(\mathcal{O}_X\) (respectively, \(\mathcal{O}_{X_0}\) ) the map
\(\mathcal{I} \rightarrow \mathcal{O}_X^*\) is defined by \(a \mapsto\)
\(1+a\), and \(\mathcal{I}\) has its usual (additive) group structure.
Conclude there is an exact sequence of abelian groups

\hypertarget{v.4.7.}{%
\subsubsection{V.4.7.}\label{v.4.7.}}

Let \(X\) be a subscheme of \(\mathbf{P}_k^2\) defined by a single
homogeneous equation \(f\left(x_0, x_1, x_2\right)=0\) of degree \(d\).
(Do not assume \(f\) is irreducible.) Assume that \((1,0,0)\) is not on
\(X\). Then show that \(X\) can be covered by the two open affine
subsets \(U=X \cap\left\{x_1 \neq 0\right\}\) and
\(V=X \cap\left\{x_2 \neq 0\right\}\). Now calculate the Čech complex
explicitly, and thus show that

\hypertarget{v.4.8.-cohomological-dimension.}{%
\subsubsection{V.4.8. Cohomological
Dimension.}\label{v.4.8.-cohomological-dimension.}}

Let \(X\) be a noetherian separated scheme. We define the
\textbf{cohomological dimension} of \(X\), denoted
\(\operatorname{cd}(X)\), to be the least integer \(n\) such that
\(H^i(X, \mathcal{F})=0\) for all quasi-coherent sheaves \(\mathcal{F}\)
and all \(i>n\).

Thus for example, Serre's theorem (3.7) says that
\(\operatorname{cd}(X)=0\) if and only if \(X\) is affine.
Grothendieck's theorem (2.7) implies that
\(\operatorname{cd}(X) \leqslant \operatorname{dim} X\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  In the definition of \(\operatorname{cd}(X)\), show that it is
  sufficient to consider only coherent sheaves on \(X\). Use (II, Ex.
  5.15) and (2.9).
\item
  If \(X\) is quasi-projective over a field \(k\), then it is even
  sufficient to consider only locally free coherent sheaves on \(X\).
  Use (II, 5.18).
\item
  Suppose \(X\) has a covering by \(r+1\) open affine subsets. Use Čech
  cohomology to show that \(\operatorname{cd}(X) \leqslant r\).
\item
  * If \(X\) is a quasi-projective scheme of dimension \(r\) over a
  field \(k\), then \(X\) can be covered by \(r+1\) open affine subsets.
  Conclude (independently of (2.7)) that
  \(\operatorname{cd}(X) \leqslant \operatorname{dim} X\).
\item
  Let \(Y\) be a set-theoretic complete intersection (I, Ex. 2.17) of
  codimension \(r\) in \(X=\mathbf{P}_k^n\). Show that
  \(\operatorname{cd}(X-Y) \leqslant r-1\).
\end{enumerate}

\hypertarget{v.4.9.}{%
\subsubsection{V.4.9.}\label{v.4.9.}}

Let \(X=\operatorname{Spec} k\left[x_1, x_2, x_3, x_4\right]\) be affine
four-space over a field \(k\). Let \(Y_1\) be the plane \(x_1=x_2=0\)
and let \(Y_2\) be the plane \(x_3=x_4=0\). Show that \(Y=Y_1 \cup Y_2\)
is not a set-theoretic complete intersection in \(X\). Therefore the
projective closure
\(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\) in
\(\mathbf{P}_k^4\) is also not a set-theoretic complete
intersection.\footnote{Hints: Use an affine analogue of (Ex. 4.8e). Then
  show that \(H^2\left(X-Y, \mathcal{O}_X\right) \neq 0\), by using (Ex.
  2.3) and (Ex. 2.4). If \(P=Y_1 \cap Y_2\), imitate (Ex. 4.3) to show
  \(H^3\left(X-P, \mathcal{O}_X\right) \neq 0\).}

\hypertarget{v.4.10.}{%
\subsubsection{V.4.10.}\label{v.4.10.}}

* Let \(X\) be a nonsingular variety over an algebraically closed field
\(k\), and let \(\mathcal{F}\) be a coherent sheaf on \(X\). Show that
there is a one-to-one correspondence between the set of infinitesimal
extensions of \(X\) by \(\mathcal{F}\) (II, Ex. 8.7) up to isomorphism,
and the group \(H^1(X, \mathcal{F} \otimes \mathcal{T})\), where
\(\mathcal{T}\) is the tangent sheaf of \(X\), see
\((II \S 8)\).\footnote{Hint: Use (II, Ex. 8.6) and (4.5).}

\hypertarget{v.4.11.}{%
\subsubsection{V.4.11.}\label{v.4.11.}}

This exercise shows that Cech cohomology will agree with the usual
cohomology whenever the sheaf has no cohomology on any of the open sets.
More precisely, let \(X\) be a topological space, \(\mathcal{F}\) a
sheaf of abelian groups, and \(\mathfrak{U}=\left(U_i\right)\) an open
cover. Assume for any finite intersection
\(V=U_{i_0} \cap \ldots \cap U_{i_p}\) of open sets of the covering, and
for any \(k>0\), that
\(H^k\left(V,\left.\mathcal{F}\right|_V\right)=0\). Then prove that for
all \(p \geqslant 0\), the natural maps of (4.4) are isomorphisms. Show
also that one can recover (4.5) as a corollary of this more general
result.

\hypertarget{iii.5-the-cohomology-of-projective-space}{%
\subsection{III.5: The Cohomology of Projective
Space}\label{iii.5-the-cohomology-of-projective-space}}

\hypertarget{iii.5.1.}{%
\subsubsection{III.5.1.}\label{iii.5.1.}}

Let \(X\) be a projective scheme over a field \(k\), and let
\(\mathcal{F}\) be a coherent sheaf on \(X\). We define the Euler
characteristic of \(\mathcal{F}\) by If is a short exact sequence of
coherent sheaves on \(X\), show that
\(\chi(\mathcal{F})=\chi\left(\mathcal{F}^{\prime}\right)+\)
\(\chi\left(\mathcal{F}^{\prime \prime}\right)\).

\hypertarget{iii.5.2.}{%
\subsubsection{III.5.2.}\label{iii.5.2.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(X\) be a projective scheme over a field \(k\), let
  \(\mathcal{O}_X(1)\) be a very ample invertible sheaf on \(X\) over
  \(k\), and let \(\mathcal{F}\) be a coherent sheaf on \(X\). Show that
  there is a polynomial \(P(z) \in \mathbf{Q}[z]\), such that
  \(\chi(\mathcal{F}(n))=P(n)\) for all \(n \in \mathbf{Z}\). We call
  \(P\) the \textbf{Hilbert polynomial} of \(\mathcal{F}\) with respect
  to the sheaf \(\mathcal{O}_X(1)\).\footnote{Hints: Use induction on
    dim Supp \(\mathcal{F}\), general properties of numerical
    polynomials (I, 7.3), and suitable exact sequences}
\item
  Now let \(X=\mathbf{P}_k^r\), and let \(M=\Gamma_*(\mathcal{F})\),
  considered as a graded \(S=k\left[x_0, \ldots, x_r\right]-\) module.
  Use (5.2) to show that the Hilbert polynomial of \(\mathcal{F}\) just
  defined is the same as the Hilbert polynomial of \(M\) defined in
  \((I, \S 7 )\).
\end{enumerate}

\hypertarget{iii.5.3.-arithmetic-genus.}{%
\subsubsection{III.5.3. Arithmetic
Genus.}\label{iii.5.3.-arithmetic-genus.}}

Let \(X\) be a projective scheme of dimension \(r\) over a field \(k\).
We define the arithmetic genus \(p_a\) of \(X\) by Note that it depends
only on \(X\), not on any projective embedding.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(X\) is integral, and \(k\) algebraically closed, show that
  \(H^0\left(X, \mathcal{O}_X\right) \cong k\), so that In particular,
  if \(X\) is a curve, we have\footnote{Hint: Use (I, 3.4).}
\item
  If \(X\) is a closed subvariety of \(\mathbf{P}_k^r\), show that this
  \(p_a(X)\) coincides with the one defined in (I, Ex. 7.2), which
  apparently depended on the projective embedding.
\item
  If \(X\) is a nonsingular projective curve over an algebraically
  closed field \(k\), show that \(p_a(X)\) is in fact a birational
  invariant. Conclude that a nonsingular plane curve of degree
  \(d \geqslant 3\) is not rational.\footnote{This gives another proof
    of (II, 8.20.3) where we used the geometric genus.}
\end{enumerate}

\hypertarget{iii.5.4.}{%
\subsubsection{III.5.4.}\label{iii.5.4.}}

Recall from (II, Ex. 6.10) the definition of the Grothendieck group
\(K(X)\) of a noetherian scheme \(X\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(X\) be a projective scheme over a field \(k\), and let
  \(\mathcal{O}_X(1)\) be a very ample invertible sheaf on \(X\). Show
  that there is a (unique) additive homomorphism such that for each
  coherent sheaf \(\mathcal{F}\) on \(X, P(\gamma(\mathcal{F}))\) is the
  Hilbert polynomial of \(\mathcal{F}\) (Ex. 5.2).
\item
  Now let \(X=\mathbf{P}_k^r\). For each \(i=0,1, \ldots, r\), let
  \(L_i\) be a linear space of dimension \(i\) in \(X\). Then show that

  \begin{enumerate}
  \def\labelenumii{(\arabic{enumii})}
  \tightlist
  \item
    \(K(X)\) is the free abelian group generated by
    \(\left\{\gamma\left(\mathcal{O}_{L_i}\right) \mathrel{\Big|}i=0, \ldots, r\right\}\),
    and
  \item
    the map \(P: K(X) \rightarrow \mathbf{Q}[z]\) is
    injective.\footnote{Hint: Show that (1) \(\Rightarrow\) (2). Then
      prove (1) and (2) simultaneously, by induction on \(r\), using
      (II, Ex. 6.10c).}
  \end{enumerate}
\end{enumerate}

\hypertarget{iii.5.5.}{%
\subsubsection{III.5.5.}\label{iii.5.5.}}

Let \(k\) be a field, let \(X=\mathbf{P}_k^r\), and let \(Y\) be a
closed subscheme of dimension \(q \geqslant 1\), which is a complete
intersection (II, Ex. 8.4). Then:

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  for all \(n \in \mathbf{Z}\), the natural map is
  surjective.\footnote{This gives a generalization and another proof of
    (II, Ex. 8.4c), where we assumed \(Y\) was normal.}
\item
  \(Y\) is connected;
\item
  \(H^i\left(Y, \mathcal{O}_Y(n)\right)=0\) for \(0<i<q\) and all
  \(n \in \mathbf{Z}\);
\item
  \(p_a(Y)=\operatorname{dim}_k H^q\left(Y, \mathcal{O}_Y\right)\).\footnote{Hint:
    Use exact sequences and induction on the codimension, starting from
    the case \(Y=X\) which is (5.1).}
\end{enumerate}

\hypertarget{iii.5.6.-curves-on-a-nonsingular-quadric-surface.}{%
\subsubsection{III.5.6. Curves on a Nonsingular Quadric
Surface.}\label{iii.5.6.-curves-on-a-nonsingular-quadric-surface.}}

Let \(Q\) be the nonsingular quadric surface \(x y=z w\) in
\(X=\mathbf{P}_k^3\) over a field \(k\). We will consider locally
principal closed subschemes \(Y\) of \(Q\). These correspond to Cartier
divisors on \(Q\) by (II, 6.17.1). On the other hand, we know that
\(\operatorname{Pic}Q \cong \mathbf{Z} \oplus \mathbf{Z}\), so we can
talk about the type \((a, b)\) of \(Y(\) II, 6.16) and (II, 6.6.1).

Let us denote the invertible sheaf \(\mathcal{L}(Y)\) by
\(\mathcal{O}_Q(a, b)\). Thus for any
\(n \in \mathbf{Z}, \mathcal{O}_Q(n)=\mathcal{O}_Q(n, n)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Use the special cases \((q, 0)\) and \((0, q)\), with \(q>0\), when
  \(Y\) is a disjoint union of \(q\) lines \(\mathbf{P}^1\) in \(Q\), to
  show:

  \begin{enumerate}
  \def\labelenumii{(\arabic{enumii})}
  \tightlist
  \item
    if \(|a-b| \leqslant 1\), then
    \(H^1\left(Q, \mathcal{O}_Q(a, b)\right)=0\);
  \item
    if \(a, b<0\), then \(H^1\left(Q \mathcal{O}_Q(a, b)\right)=0\)
  \item
    If \(a \leqslant-2\), then
    \(H^1\left(Q, \mathcal{O}_Q(a, 0)\right) \neq 0\).
  \end{enumerate}
\item
  Now use these results to show:

  \begin{enumerate}
  \def\labelenumii{(\arabic{enumii})}
  \tightlist
  \item
    if \(Y\) is a locally principal closed subscheme of type \((a, b)\),
    with \(a, b>0\), then \(Y\) is connected;
  \item
    now assume \(k\) is algebraically closed. Then for any \(a, b>0\),
    there exists an irreducible nonsingular curve \(Y\) of type (a,b).
    Use (II, 7.6.2) and (II, 8.18).
  \item
    an irreducible nonsingular curve \(Y\) of type \((a, b), a, b>0\) on
    \(Q\) is projectively normal (II, Ex. 5.14) if and only if
    \(|a-b| \leqslant 1\). In particular, this gives lots of examples of
    nonsingular, but not projectively normal curves in \(\mathbf{P}^3\).
    The simplest is the one of type \((1,3)\), which is just the
    rational quartic curve (I, Ex. 3.18).
  \end{enumerate}
\item
  If \(Y\) is a locally principal subscheme of type \((a, b)\) in \(Q\),
  show that\footnote{Hint: Calculate Hilbert polynomials of suitable
    sheaves, and again use the special case \((q, 0)\) which is a
    disjoint union of \(q\) copies of \(\mathbf{P}^1\). See
    \((\mathrm{V}, 1.5 .2)\) for another method.}
\end{enumerate}

\hypertarget{iii.5.7.}{%
\subsubsection{III.5.7.}\label{iii.5.7.}}

Let \(X\) (respectively, \(Y\) ) be proper schemes over a noetherian
ring \(A\). We denote by \(\mathcal{L}\) an invertible sheaf.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\mathcal{L}\) is ample on \(X\), and \(Y\) is any closed
  subscheme of \(X\), then \(i^* \mathcal{L}\) is ample on \(Y\), where
  \(i: Y \rightarrow X\) is the inclusion.
\item
  \(\mathcal{L}\) is ample on \(X\) if and only if
  \(\mathcal{L}_{\text {red }}=\mathcal{L} \otimes \mathcal{O}_{X_{\text {red }}}\)
  is ample on \(X_{\text {red }}\).
\item
  Suppose \(X\) is reduced. Then \(\mathcal{L}\) is ample on \(X\) if
  and only if \(\mathcal{L} \otimes \mathcal{O}_{X_i}\) is ample on
  \(X_i\), for each irreducible component \(X_i\) of \(X\).
\item
  Let \(f: X \rightarrow Y\) be a finite surjective morphism, and let
  \(\mathcal{L}\) be an invertible sheaf on \(Y\). Then \(\mathcal{L}\)
  is ample on \(Y\) if and only if \(f^* \mathcal{L}\) is ample on
  \(X\).\footnote{Hints: Use (5.3) and compare (Ex. 3.1, Ex. 3.2, Ex.
    4.1, Ex. 4.2). See also Hartshorne \([5, Ch. I \S 4]\) for more
    details.}
\end{enumerate}

\hypertarget{iii.5.8.}{%
\subsubsection{III.5.8.}\label{iii.5.8.}}

Prove that every one-dimensional proper scheme \(X\) over an
algebraically closed field \(k\) is projective.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(X\) is irreducible and nonsingular, then \(X\) is projective by
  (II, 6.7).
\item
  If \(X\) is integral, let \(\tilde{X}\) be its normalization (II, Ex.
  3.8). Show that \(\tilde{X}\) is complete and nonsingular, hence
  projective by (a).

  Let \(f: \tilde{X} \rightarrow X\) be the projection. Let
  \(\mathcal{L}\) be a very ample invertible sheaf on \(\tilde{X}\).
  Show there is an effective divisor \(D=\sum P_i\) on \(\tilde{X}\)
  with \(\mathcal{L}(D) \cong \mathcal{L}\), and such that
  \(f\left(P_i\right)\) is a nonsingular point of \(X\), for each \(i\).

  Conclude that there is an invertible sheaf \(\mathcal{L}_0\) on \(X\)
  with \(f^* \mathcal{L}_0 \cong\) \(\mathcal{L}\). Then use (Ex. 5.7d),
  (II, 7.6) and (II, 5.16.1) to show that \(X\) is projective.
\item
  If \(X\) is reduced, but not necessarily irreducible, let
  \(X_1, \ldots, X_r\) be the irreducible components of \(X\). Use (Ex.
  4.5) to show
  \(\operatorname{Pic}X \rightarrow \bigoplus \operatorname{Pic} X_i\)
  is surjective. Then use (Ex. 5.7c) to show \(X\) is projective.
\item
  Finally, if \(X\) is any one-dimensional proper scheme over \(k\), use
  (2.7) and (Ex. 4.6) to show that
  \(\operatorname{Pic}X \rightarrow \operatorname{Pic}X_{\text {red }}\)
  is surjective. Then use (Ex. 5.7b) to show \(X\) is projective.
\end{enumerate}

\hypertarget{iii.5.9.-a-nonprojective-scheme.}{%
\subsubsection{III.5.9. A Nonprojective
Scheme.}\label{iii.5.9.-a-nonprojective-scheme.}}

We show the result of (Ex. 5.8) is false in dimension 2. Let \(k\) be an
algebraically closed field of characteristic 0 , and let
\(X=\mathbf{P}_k^2\). Let \(\omega\) be the sheaf of differential
2-forms (II, §8). Define an infinitesimal extension \(X^{\prime}\) of
\(X\) by \(\omega\) by giving the element
\(\xi \in H^1(X, \omega \otimes \mathcal{T})\) defined as follows (Ex.
4.10).

Let \(x_0, x_1, x_2\) be the homogeneous coordinates of \(X\), let
\(U_0, U_1, U_2\) be the standard open covering, and let
\(\xi_{i j}=\left(x_j / x_i\right) d\left(x_i / x_j\right)\). This gives
a Čech 1-cocycle with values in \(\Omega_X^1\), and since
\(\operatorname{dim} X=2\), we have
\(\omega \otimes \mathcal{T} \cong \Omega^1\) (II, Ex. 5.16b). Now use
the exact sequence of (Ex. 4.6) and show \(\delta\) is injective. We
have \(\omega \cong \mathcal{O}_X(-3)\) by (II, 8.20.1), so
\(H^2(X, \omega) \cong k\). Since char \(k=0\), you need only show that
\(\delta(\mathcal{O}(1)) \neq 0\), which can be done by calculating in
Čech cohomology.

Since \(H^1(X, \omega)=0\), we see that
\(\operatorname{Pic}X^{\prime}=0\). In particular, \(X^{\prime}\) has no
ample invertible sheaves, so it is not projective.\footnote{Note. In
  fact, this result can be generalized to show that for any nonsingular
  projective surface \(X\) over an algebraically closed field \(k\) of
  characteristic 0 , there is an infinitesimal extension \(X^{\prime}\)
  of \(X\) by \(\omega\), such that \(X^{\prime}\) is not projective
  over \(k\).\\
  Indeed, let \(D\) be an ample divisor on \(X\). Then \(D\) determines
  an element \(c_1(D) \in\) \(H^1\left(X, \Omega^1\right)\) which we use
  to define \(X^{\prime}\), as above. Then for any divisor \(E\) on
  \(X\) one can show that \(\delta(\mathcal{L}(E))=(D . E)\), where
  \((D . E)\) is the intersection number (Chapter V), considered as an
  element of \(k\). Hence if \(E\) is ample,
  \(\delta(\mathcal{L}(E)) \neq 0\). Therefore \(X^{\prime}\) has no
  ample divisors.\\
  On the other hand, over a field of characteristic \(p>0\), a proper
  scheme \(X\) is projective if and only if \(X_{\text {red }}\) is!}

\hypertarget{iii.5.10.}{%
\subsubsection{III.5.10.}\label{iii.5.10.}}

Let \(X\) be a projective scheme over a noetherian ring \(A\), and let
\(\mathcal{F}^1 \rightarrow \mathcal{F}^2 \rightarrow \ldots \rightarrow\)
\(\mathcal{F}^r\) be an exact sequence of coherent sheaves on \(X\).
Show that there is an integer \(n_0\), such that for all
\(n \geqslant n_0\), the sequence of global sections is exact.

\hypertarget{iii.6-ext-groups-and-sheaves}{%
\subsection{III.6: Ext Groups and
Sheaves}\label{iii.6-ext-groups-and-sheaves}}

\hypertarget{iii.6.1.}{%
\subsubsection{III.6.1.}\label{iii.6.1.}}

Let \(\left(X, \mathcal{O}_X\right)\) be a ringed space, and let
\(\mathcal{F}^{\prime}, \mathcal{F}^{\prime \prime} \in {\mathsf{Mod}}(X)\).
An \textbf{extension} of \(\mathcal{F}^{\prime \prime}\) by
\(\mathcal{F}^{\prime}\) is a short exact sequence in
\({\mathsf{Mod}}(X)\). Two extensions are isomorphic if there is an
isomorphism of the short exact sequences, inducing the identity maps on
\(\mathcal{F}^{\prime}\) and \(\mathcal{F}^{\prime \prime}\). Given an
extension as above consider the long exact sequence arising from
\(\operatorname{Hom}\left(\mathcal{F}^{\prime \prime}, \cdot\right)\),
in particular the map and let
\(\xi \in \operatorname{Ext}^1\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime}\right)\)
be \(\delta\left(1_{\mathcal{F}^{\prime \prime}}\right)\). Show that
this process gives a one-to-one correspondence between isomorphism
classes of extensions of \(\mathcal{F}^{\prime \prime}\) by
\(\mathcal{F}^{\prime}\), and elements of the group
\(\operatorname{Ext}^1\left(\mathcal{F}^{\prime \prime}, \mathcal{F}^{\prime}\right)\).

\hypertarget{iii.6.2.}{%
\subsubsection{III.6.2.}\label{iii.6.2.}}

Let \(X=\mathbf{P}_k^1\), with \(k\) an infinite field.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  Show that there does not exist a projective object
  \(\mathcal{P} \in {\mathsf{Mod}}(X)\), together with a surjective map
  \(\mathcal{P} \rightarrow \mathcal{O}_X \rightarrow 0\).\footnote{Hint:
    Consider surjections of the form \(\mathcal{O}_V \rightarrow\)
    \(k(x) \rightarrow 0\), where \(x \in X\) is a closed point, \(V\)
    is an open neighborhood of \(x\), and
    \(\mathcal{O}_V=j_{!}\left(\left.\mathcal{O}_X\right|_V\right)\),
    where \(j: V \rightarrow X\) is the inclusion.}
\end{enumerate}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item
  Show that there does not exist a projective object \(\mathcal{P}\) in
  either \({\mathsf{QCoh}}(X)\) or \({\mathsf{Coh}}(X)\) together with a
  surjection
  \(\mathcal{P} \rightarrow \mathcal{O}_X \rightarrow 0\).\footnote{Hint:
    Consider surjections of the form
    \(\mathcal{L} \rightarrow \mathcal{L} \otimes k(x) \rightarrow 0\),
    where \(x \in X\) is a closed point, and \(\mathcal{L}\) is an
    invertible sheaf on \(X\).}
\end{enumerate}

\hypertarget{iii.6.3.}{%
\subsubsection{III.6.3.}\label{iii.6.3.}}

Let \(X\) be a noetherian scheme, and let
\(\mathcal{F}, \mathcal{G} \in {\mathsf{Mod}}(X)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\mathcal{F}, \mathcal{G}\) are both coherent, then
  \(\mathcal{E x t}(\mathcal{F}, \mathcal{G})\) is coherent, for all
  \(i \geqslant 0\).
\item
  If \(\mathcal{F}\) is coherent and \(\mathcal{G}\) is quasi-coherent,
  then \(\mathcal{E} x t^i(\mathcal{F}, \mathcal{G})\) is
  quasi-coherent, for all \(i \geqslant 0\).
\end{enumerate}

\hypertarget{iii.6.4.}{%
\subsubsection{III.6.4.}\label{iii.6.4.}}

Let \(X\) be a noetherian scheme, and suppose that every coherent sheaf
on \(X\) is a quotient of a locally free sheaf. In this case we say
\({\mathsf{Coh}}(X)\) has enough locally frees. Then for any
\(\mathcal{G} \in {\mathsf{Mod}}(X)\), show that the \(\delta\)-functor
\(\left(\mathcal{E} x t^i(\cdot, \mathcal{G})\right)\), from
\({\mathsf{Coh}}(X)\) to \({\mathsf{Mod}}(X)\) is a contravariant
universal \(\delta\)-functor.\footnote{Hint: Show
  \(\mathcal{E} x t^i(\cdot, \mathcal{G})\) is coeffaceable for \(i>0\).}

\hypertarget{iii.6.5.}{%
\subsubsection{III.6.5.}\label{iii.6.5.}}

Let \(X\) be a noetherian scheme, and assume that \({\mathsf{Coh}}(X)\)
has enough locally frees (Ex. 6.4). Then for any coherent sheaf
\(\mathcal{F}\) we define the \textbf{homological dimension} of
\(\mathcal{F}\), denoted \(\mathrm{hd} (\mathcal{F})\), to be the least
length of a locally free resolution of \(\mathcal{F}\) (or \(+\infty\)
if there is no finite one). Show:

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(\mathcal{F}\) is locally free
  \(\Leftrightarrow \mathcal{E} x t^1(\mathcal{F}, \mathcal{G})=0\) for
  all \(\mathcal{G} \in {\mathsf{Mod}}(X)\);
\item
  \(\operatorname{hd}(\mathcal{F}) \leqslant n \Leftrightarrow \mathcal{E x t}(\mathcal{F}, \mathcal{G})=0\)
  for all \(i>n\) and all \(\mathcal{G} \in {\mathsf{Mod}}(X)\);
\item
  \(\operatorname{hd}(\mathcal{F})=\sup _x \operatorname{pd}_{{\mathcal{O}}_x} \mathcal{F}_x\).
\end{enumerate}

\hypertarget{iii.6.6.}{%
\subsubsection{III.6.6.}\label{iii.6.6.}}

Let \(A\) be a regular local ring, and let \(M\) be a finitely generated
\(A\)-module. In this case, strengthen the result \((6.10 \mathrm{~A})\)
as follows.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(M\) is projective if and only if \(\operatorname{Ext}^i(M, A)=0\)
  for all \(i>0\).\footnote{Hint: Use (6.11A) and descending induction
    on \(i\) to show that \(\operatorname{Ext}^i(M, N)=0\) for all
    \(i>0\) and all finitely generated \(A\)-modules \(N\). Then show
    \(M\) is a direct summand of a free \(A\)-module (Matsumura
    \([2, p. 129]\)).}
\item
  Use (a) to show that for any \(n\), pd \(M \leqslant n\) if and only
  if \(\operatorname{Ext}^i(M, A)=0\) for all \(i>n\).
\end{enumerate}

\hypertarget{iii.6.7.}{%
\subsubsection{III.6.7.}\label{iii.6.7.}}

Let \(X=\operatorname{Spec} A\) be an affine noetherian scheme. Let
\(M, N\) be \(A\)-modules, with \(M\) finitely generated. Then and

\hypertarget{iii.6.8.}{%
\subsubsection{III.6.8.}\label{iii.6.8.}}

Prove the following theorem of Kleiman (see Borelli \([1]\)): if \(X\)
is a noetherian, integral, separated, locally factorial scheme, then
every coherent sheaf on \(X\) is a quotient of a locally free sheaf (of
finite rank).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  First show that open sets of the form \(X_s\), for various
  \(s \in \Gamma(X, \mathcal{L})\), and various invertible sheaves
  \(\mathcal{L}\) on \(X\), form a base for the topology of
  \(X\).\footnote{Hint: Given a closed point \(x \in X\) and an open
    neighborhood \(U\) of \(x\), to show there is an \(\mathcal{L}, s\)
    such that \(x \in X_s \subseteq U\), first reduce to the case that
    \(Z=X-U\) is irreducible. Then let \(\zeta\) be the generic point of
    \(Z\). Let \(f \in K(X)\) be a rational function with
    \(f \in \mathcal{O}_x, f \notin \mathcal{O}_\zeta\). Let
    \(D=(f)_{\infty}\), and let
    \(\mathcal{L}=\mathcal{L}(D), s \in \Gamma(X, \mathcal{L}(D))\)
    correspond to \(D\) (II, §6).}
\item
  Now use (II, 5.14) to show that any coherent sheaf is a quotient of a
  direct sum \(\bigoplus \mathcal{L}_i^{n_i}\) for various invertible
  sheaves \(\mathcal{L}_i\) and various integers \(n_i\).
\end{enumerate}

\hypertarget{iii.6.9.}{%
\subsubsection{III.6.9.}\label{iii.6.9.}}

Let \(X\) be a noetherian, integral, separated, regular scheme. (We say
a scheme is regular if all of its local rings are regular local rings.)
Recall the definition of the Grothendieck group \(K(X)\) from (II, Ex.
6.10).

We define similarly another group \(K_1(X)\) using locally free sheaves:
it is the quotient of the free abelian group generated by all locally
free (coherent) sheaves, by the subgroup generated by all expressions of
the form
\(\mathcal{E}-\mathcal{E}^{\prime}-\mathcal{E}^{\prime \prime}\),
whenever
\(0 \rightarrow \mathcal{E}^{\prime} \rightarrow \mathcal{E} \rightarrow \mathcal{E}^{\prime \prime} \rightarrow 0\)
is a short exact sequence of locally free sheaves.

Clearly there is a natural group homomorphism
\(\varepsilon: K_1(X) \rightarrow K(X)\). Show that \(\varepsilon\) is
an isomorphism (Borel and Serre \([1, \S 4])\) as follows.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Given a coherent sheaf \(\mathcal{F}\), use (Ex. 6.8) to show that it
  has a locally free resolution
  \(\mathcal{E} . \rightarrow \mathcal{F} \rightarrow 0\). Then use
  (6.11A) and (Ex. 6.5) to show that it has a finite locally free
  resolution
\item
  For each \(\mathcal{F}\), choose a finite locally free resolution
  \(\mathcal{E}\). \(\rightarrow \mathcal{F} \rightarrow 0\), and let
  \(\delta(\mathcal{F})=\sum(-1)^i \gamma\left(\mathcal{E}_i\right)\) in
  \(K_1(X)\). Show that \(\delta(\mathcal{F})\) is independent of the
  resolution chosen, that it defines a homomorphism of \(K(X)\) to
  \(K_1(X)\), and finally, that it is an inverse to \(\varepsilon\).
\end{enumerate}

\hypertarget{iii.6.10.-duality-for-a-finite-flat-morphism.}{%
\subsubsection{III.6.10. Duality for a Finite Flat
Morphism.}\label{iii.6.10.-duality-for-a-finite-flat-morphism.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(f: X \rightarrow Y\) be a finite morphism of noetherian schemes.
  For any quasicoherent \(\mathcal{O}_Y\)-module
  \(\mathcal{G}, \mathop{\mathcal{H}\! \mathit{om}}_Y(f_* {\mathcal{O}}_X, {\mathcal{G}})\)
  is a quasi-coherent \(f_* \mathcal{O}_X\)-module, hence corresponds to
  a quasi-coherent \(\mathcal{O}_X\)-module, which we call
  \(f ! \mathcal{G}\) (II, Ex. 5.17e).
\item
  Show that for any coherent \(\mathcal{F}\) on \(X\) and any
  quasi-coherent \(\mathcal{G}\) on \(Y\), there is a natural
  isomorphism
\item
  For each \(i \geqslant 0\), there is a natural map\footnote{Hint:
    First construct a map Then compose with a suitable map from
    \(f_* f^{!} \mathcal{G}\) to \(\mathcal{G}\).}
\item
  Now assume that \(X\) and \(Y\) are separated, \({\mathsf{Coh}}(X)\)
  has enough locally frees, and assume that \(f_* \mathcal{O}_X\) is
  locally free on \(Y\) (this is equivalent to saying \(f\) flat-see
  §9). Show that \(\varphi_i\) is an isomorphism for all \(i\), all
  \(\mathcal{F}\) coherent on \(X\), and all \(\mathcal{G}\)
  quasi-coherent on \(Y\).\footnote{Hints: First do \(i=0\). Then do
    \(\mathcal{F}=\mathcal{O}_X\), using (Ex. 4.1). Then do
    \(\mathcal{F}\) locally free. Do the general case by induction on
    \(i\), writing \(\mathcal{F}\) as a quotient of a locally free
    sheaf.}
\end{enumerate}

\hypertarget{iii.7-serre-duality}{%
\subsection{III.7: Serre Duality}\label{iii.7-serre-duality}}

\hypertarget{iii.7.1.-special-case-of-kodaira-vanishing.}{%
\subsubsection{III.7.1. Special case of Kodaira
vanishing.}\label{iii.7.1.-special-case-of-kodaira-vanishing.}}

Let \(X\) be an integral projective scheme of dimension \(\geqslant 1\)
over a field \(k\), and let \(\mathcal{L}\) be an ample invertible sheaf
on \(X\). Then

\hypertarget{iii.7.2.}{%
\subsubsection{III.7.2.}\label{iii.7.2.}}

Let \(f: X \rightarrow Y\) be a finite morphism of projective schemes of
the same dimension over a field \(k\), and let \(\omega_Y^{\circ}\) be a
dualizing sheaf for \(Y\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(f^{\prime} \omega_Y^{\circ}\) is a dualizing sheaf for
  \(X\), where \(f^{\prime}\) is defined as in (Ex. 6.10).
\item
  If \(X\) and \(Y\) are both nonsingular, and \(k\) algebraically
  closed, conclude that there is a natural trace map
  \(t: f_* \omega_X \rightarrow \omega_Y\).
\end{enumerate}

\hypertarget{iii.7.3.}{%
\subsubsection{III.7.3.}\label{iii.7.3.}}

Let \(X=\mathbf{P}_k^n\). Show that \(H^q\left(X, \Omega_X^p\right)=0\)
for \(p \neq q\), \(k\) for \(p=q, 0 \leqslant p, q \leqslant n\).

\hypertarget{iii.7.4.-the-cohomology-class-of-a-subvariety.}{%
\subsubsection{III.7.4. * The Cohomology Class of a
Subvariety.}\label{iii.7.4.-the-cohomology-class-of-a-subvariety.}}

Let \(X\) be a nonsingular projective variety of dimension \(n\) over an
algebraically closed field \(k\). Let \(Y\) be a nonsingular subvariety
of codimension \(p\) (hence dimension \(n-p\) ). From the natural map
\(\Omega_X \otimes\) \(\mathcal{O}_Y \rightarrow \Omega_Y\) of
\((\mathrm{II}, 8.12)\) we deduce a map
\(\Omega_X^{n-p} \rightarrow \Omega_Y^{n-p}\). This induces a map on
cohomology Now \(\Omega_Y^{n-p}=\omega_Y\) is a dualizing sheaf for
\(Y\), so we have the trace map Composing, we obtain a linear map
\(H^{n-p}\left(X, \Omega_X^{n-p}\right) \rightarrow k\). By (7.13) this
corresponds to an element \(\eta(Y) \in\)
\(H^p\left(X, \Omega_X^p\right)\), which we call the \textbf{cohomology
class of \(Y\)}.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(P \in X\) is a closed point, show that \(t_X(\eta(P))=1\), where
  \(\eta(P) \in H^n\left(X, \Omega^n\right)\) and \(t_X\) is the trace
  map.
\item
  If \(X=\mathbf{P}^n\), identify \(H^p\left(X, \Omega^p\right)\) with
  \(k\) by (Ex. 7.3), and show that
  \(\eta(Y)=(\operatorname{deg} Y) \cdot 1\), where deg \(Y\) is its
  degree as a projective variety (I, \(\S\) 7).\footnote{Hint: Cut with
    a hyperplane \(H \subseteq X\), and use Bertini's theorem (II, 8.18)
    to reduce to the case \(Y\) is a finite set of points.}
\item
  For any scheme \(X\) of finite type over \(k\), we define a
  homomorphism of sheaves of abelian groups
  \(d \log : \mathcal{O}_X^* \rightarrow \Omega_X\) by
  \(d \log (f)=f^{-1} \, df\). Here \(\mathcal{O}^*\) is a group under
  multiplication, and \(\Omega_X\) is a group under addition. This
  induces a map on cohomology which we denote by \(c\). See (Ex. 4.5).
\item
  Returning to the hypotheses above, suppose \(p=1\). Show that
  \(\eta(Y)=c(\mathcal{L}(Y))\), where \(\mathcal{L}(Y)\) is the
  invertible sheaf corresponding to the divisor \(Y\).
\end{enumerate}

\hypertarget{iii.8-higher-direct-images-of-sheaves}{%
\subsection{III.8: Higher Direct Images of
Sheaves}\label{iii.8-higher-direct-images-of-sheaves}}

\hypertarget{iii.8.1.}{%
\subsubsection{III.8.1.}\label{iii.8.1.}}

Let \(f: X \rightarrow Y\) be a continuous map of topological spaces.
Let \(\mathcal{F}\) be a sheaf of abelian groups on \(X\), and assume
that \(R^i f_*(\mathcal{F})=0\) for all \(i>0\). Show that there are
natural isomorphisms, for each \(i \geqslant 0\),\footnote{This is a
  degenerate case of the Leray spectral sequence-see Godement
  \([1, II, 4.17.1]\).}

\hypertarget{iii.8.2.}{%
\subsubsection{III.8.2.}\label{iii.8.2.}}

Let \(f: X \rightarrow Y\) be an affine morphism of schemes (II, Ex.
5.17) with \(X\) noetherian, and let \(\mathcal{F}\) be a quasi-coherent
sheaf on \(X\). Show that the hypotheses of (Ex. 8.1) are satisfied, and
hence that
\(H^i(X, \mathcal{F}) \cong H^i\left(Y, f_* \mathcal{F}\right)\) for
each \(i \geqslant 0\).

\hypertarget{iii.8.3.-the-projection-formula.}{%
\subsubsection{III.8.3. The Projection
Formula.}\label{iii.8.3.-the-projection-formula.}}

Let \(f: X \rightarrow Y\) be a morphism of ringed spaces, let
\(\mathcal{F}\) be an \(\mathcal{O}_X\)-module, and let \(\mathcal{E}\)
be a locally free \(\mathcal{O}_Y\)-module of finite rank. Prove the
projection formula (cf.~(II, Ex. 5.1))

\hypertarget{iii.8.4.}{%
\subsubsection{III.8.4.}\label{iii.8.4.}}

Let \(Y\) be a noetherian scheme, and let \(\mathcal{E}\) be a locally
free \(\mathcal{O}_Y\)-module of rank \(n+1\), \(n \geqslant 1\). Let
\(X=\mathbf{P}(\mathcal{E})\) (II, \(\S\) ), with the invertible sheaf
\(\mathcal{O}_X(1)\) and the projection morphism
\(\pi: X \rightarrow Y\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Then

  \begin{itemize}
  \tightlist
  \item
    \(\pi_*(\mathcal{O}(l)) \cong S^l(\mathcal{E})\) for
    \(l \geqslant 0, \pi_*(\mathcal{O}(l))=0\) for \(l<0\) (II, 7.11);
  \item
    \(R^i \pi_*(\mathcal{O}(l))=0\) for \(0<i<n\) and all
    \(l \in \mathbf{Z}\); and
  \item
    \(R^n \pi_*(\mathcal{O}(l))=0\) for \(l>-n-1\).
  \end{itemize}
\item
  Show there is a natural exact sequence cf.~(II, 8.13), and conclude
  that the \textbf{relative canonical sheaf}
  \(\omega_{X / Y}=\wedge^n \Omega_{X / Y}\) is isomorphic to
  \(\left(\pi^* \wedge^{n+1} \mathcal{E}\right)(-n-1)\). Show
  furthermore that there is a natural isomorphism
  \(R^n \pi_*\left(\omega_{X / Y}\right) \cong \mathcal{O}_Y\)
  (cf.~(7.1.1)).
\item
  Now show, for any \(l \in \mathbf{Z}\), that
\item
  Show that \(p_a(X)=(-1)^n p_a(Y)\) (use (Ex. 8.1)) and \(p_g(X)=0\)
  (use (II, 8.11)).
\item
  In particular, if \(Y\) is a nonsingular projective curve of genus
  \(g\), and \(\mathcal{E}\) a locally free sheaf of rank 2 , then \(X\)
  is a projective surface with \(p_a=-g, p_g=0\), and irregularity \(g\)
  (7.12.3). This kind of surface is called a \textbf{geometrically ruled
  surface} (V, §2).
\end{enumerate}

\hypertarget{iii.9-flat-morphisms}{%
\subsection{III.9: Flat Morphisms}\label{iii.9-flat-morphisms}}

\hypertarget{iii.9.1.}{%
\subsubsection{III.9.1.}\label{iii.9.1.}}

A flat morphism \(f: X \rightarrow Y\) of finite type of noetherian
schemes is open, i.e, for every open subset \(U \subseteq X, f(U)\) is
open in Y.\footnote{Hint: Show that \(f(U)\) is constructible and stable
  under generization (II, Ex. 3.18) and (II, Ex. 3.19).}

\hypertarget{iii.9.2.}{%
\subsubsection{III.9.2.}\label{iii.9.2.}}

Do the calculation of (9.8.4) for the curve of (I, Ex. 3.14). Show that
you get an embedded point at the cusp of the plane cubic curve.

\hypertarget{iii.9.3.}{%
\subsubsection{III.9.3.}\label{iii.9.3.}}

Some examples of flatness and nonflatness.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(f: X \rightarrow Y\) is a finite surjective morphism of
  nonsingular varieties over an algebraically closed field \(k\), then
  \(f\) is flat.
\item
  Let \(X\) be a union of two planes meeting at a point, each of which
  maps isomorphically to a plane \(Y\). Show that \(f\) is not flat. For
  example, let \(Y=\) \(\operatorname{Spec} k[x, y]\) and
\item
  Again let \(Y=\operatorname{Spec} k[x, y]\), but take
  \(X=\operatorname{Spec} k[x, y, z, w] /\left(z^2, z w, w^2, x z-y w\right)\).
  Show that \(X_{\text {red }} \cong Y, X\) has no embedded points, but
  that \(f\) is not flat.
\end{enumerate}

\hypertarget{iii.9.4.-open-nature-of-flatness.}{%
\subsubsection{III.9.4. Open Nature of
Flatness.}\label{iii.9.4.-open-nature-of-flatness.}}

Let \(f: X \rightarrow Y\) be a morphism of finite type of noetherian
schemes. Then \(\{x \in X \mathrel{\Big|}f\) is flat at \(x\}\) is an
open subset of \(X\) (possibly empty).\footnote{See Grothendieck EGA
  \(IV_3,11.1.1\).}

\hypertarget{iii.9.5.-very-flat-families.}{%
\subsubsection{III.9.5. Very Flat
Families.}\label{iii.9.5.-very-flat-families.}}

For any closed subscheme \(X \subseteq \mathbf{P}^n\), we denote by
\(C(X) \subseteq \mathbf{P}^{n+1}\) the projective cone over \(X\) (I,
Ex. 2.10). If \(I \subseteq k\left[x_0, \ldots, x_n\right]\) is the
(largest) homogeneous ideal of \(X\), then \(C(X)\) is defined by the
ideal generated by \(I\) in \(k\left[x_0, \ldots, x_{n+1}\right]\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Give an example to show that if \(\left\{X_t\right\}\) is a flat
  family of closed subschemes of \(\mathbf{P}^n\), then
  \(\left\{C\left(X_t\right)\right\}\) need not be a flat family in
  \(\mathbf{P}^{n+1}\).
\item
  To remedy this situation, we make the following definition. Let
  \(X \subseteq \mathbf{P}_T^n\) be a closed subscheme, where \(T\) is a
  noetherian integral scheme. For each \(t \in T\), let
  \(I_t \subseteq S_t=k(t)\left[x_0, \ldots, x_n\right]\) be the
  homogeneous ideal of \(X_t\) in \(\mathbf{P}_{k(t)}^n\). We say that
  the family \(\left\{X_t\right\}\) is \textbf{very flat} if for all
  \(d \geqslant 0\), is independent of \(t\). Here \((\quad)_d\) means
  the homogeneous part of degree \(d\).
\item
  If \(\left\{X_t\right\}\) is a very flat family in \(\mathbf{P}^n\),
  show that it is flat. Show also that
  \(\left\{C\left(X_t\right)\right\}\) is a very flat family in
  \(\mathbf{P}^{n+1}\), and hence flat.
\item
  If \(\left\{X_{(t)}\right\}\) is an algebraic family of projectively
  normal varieties in \(\mathbf{P}_k^n\), parametrized by a nonsingular
  curve \(T\) over an algebraically closed field \(k\), then
  \(\left\{X_{(t)}\right\}\) is a very flat family of schemes.
\end{enumerate}

\hypertarget{iii.9.6.}{%
\subsubsection{III.9.6.}\label{iii.9.6.}}

Let \(Y \subseteq \mathbf{P}^n\) be a nonsingular variety of dimension
\(\geqslant 2\) over an algebraically closed field \(k\). Suppose
\(\mathbf{P}^{n-1}\) is a hyperplane in \(\mathbf{P}^n\) which does not
contain \(Y\), and such that the scheme
\(Y^{\prime}=Y \cap \mathbf{P}^{n-1}\) is also nonsingular. Prove that
\(Y\) is a complete intersection in \(\mathbf{P}^n\) if and only if
\(Y^{\prime}\) is a complete intersection in
\(\mathbf{P}^{n-1}\).\footnote{Hint: See (II, Ex. 8.4) and use (9.12)
  applied to the affine cones over \(Y\) and \(Y^{\prime}\).}

\hypertarget{iii.9.7.}{%
\subsubsection{III.9.7.}\label{iii.9.7.}}

Let \(Y \subseteq X\) be a closed subscheme, where \(X\) is a scheme of
finite type over a field \(k\). Let \(D=k[t] / t^2\) be the ring of dual
numbers, and define an infinitesimal deformation of \(Y\) as a closed
subscheme of \(X\), to be a closed subscheme
\(Y^{\prime} \subseteq X \underset{\scriptscriptstyle {k} }{\times} D\),
which is flat over \(D\), and whose closed fibre is \(Y\).

Show that these \(Y^{\prime}\) are classified by
\(H^0\left(Y, \mathcal{N}_{Y / X}\right)\), where

\hypertarget{iii.9.8.}{%
\subsubsection{III.9.8. *}\label{iii.9.8.}}

Let \(A\) be a finitely generated \(k\)-algebra. Write \(A\) as a
quotient of a polynomial ring \(P\) over \(k\), and let \(J\) be the
kernel: Consider the exact sequence of (II, 8.4A) Apply the functor
\(\operatorname{Hom}_A(\cdot, A)\), and let \(T^1(A)\) be the cokernel:
Now use the construction of (II, Ex. 8.6) to show that \(T^1(A)\)
classifies infinitesimal deformations of \(A\), i.e., algebras
\(A^{\prime}\) flat over \(D=k[t] / t^2\), with
\(A^{\prime} \otimes_D k \cong A\). It follows that \(T^1(A)\) is
independent of the given representation of \(A\) as a quotient of a
polynomial ring \(P\).

\hypertarget{iii.9.9.}{%
\subsubsection{III.9.9.}\label{iii.9.9.}}

A \(k\)-algebra \(A\) is said to be \textbf{rigid} if it has no
infinitesimal deformations, or equivalently, by (Ex. 9.8) if
\(T^1(A)=0\). Let \(A=k[x, y, z, w] /(x, y) \cap(z, w)\), and show that
\(A\) is rigid. This corresponds to two planes in \(\mathbf{A}^4\) which
meet at a point.

\hypertarget{iii.9.10.}{%
\subsubsection{III.9.10.}\label{iii.9.10.}}

A scheme \(X_0\) over a field \(k\) is rigid if it has no infinitesimal
deformations.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(\mathbf{P}_k^1\) is rigid, using (9.13.2).
\item
  One might think that if \(X_0\) is rigid over \(k\), then every global
  deformation of \(X_0\) is locally trivial. Show that this is not so,
  by constructing a proper, flat morphism
  \(f: X \rightarrow \mathbf{A}^2\) over \(k\) algebraically closed,
  such that \(X_0 \cong \mathbf{P}_k^1\), but there is no open
  neighborhood \(U\) of 0 in \(\mathbf{A}^2\) for which
  \(f^{-1}(U) \cong U \times \mathbf{P}^1\).
\item
  * Show, however, that one can trivialize a global deformation of
  \(\mathbf{P}^1\) after a flat base extension, in the following sense:
  let \(f: X \rightarrow T\) be a flat projective morphism, where \(T\)
  is a nonsingular curve over \(k\) algebraically closed. Assume there
  is a closed point \(t \in T\) such that \(X_t \cong \mathbf{P}_k^1\).
  Then there exists a nonsingular curve \(T^{\prime}\), and a flat
  morphism \(g: T^{\prime} \rightarrow T\), whose image contains \(t\),
  such that if \(X^{\prime}=X \times_T T^{\prime}\) is the base
  extension, then the new family
  \(f^{\prime}: X^{\prime} \rightarrow T^{\prime}\) is isomorphic to
  \(\mathbf{P}_{T^{\prime}}^1 \rightarrow T^{\prime}\).
\end{enumerate}

\hypertarget{iii.9.11.}{%
\subsubsection{III.9.11.}\label{iii.9.11.}}

Let \(Y\) be a nonsingular curve of degree \(d\) in \(\mathbf{P}_k^n\),
over an algebraically closed field \(k\). Show that\footnote{Hint:
  Compare \(Y\) to a suitable projection of \(Y\) into \(\mathbf{P}^2\),
  as in (9.8.3) and (9.8.4).}

\hypertarget{iii.10-smooth-morphisms}{%
\subsection{III.10: Smooth Morphisms}\label{iii.10-smooth-morphisms}}

\hypertarget{iii.10.1.-smooth-neq-regular.-to_work}{%
\subsubsection{\texorpdfstring{III.10.1. Smooth \(\neq\)
Regular.}{III.10.1. Smooth \textbackslash neq Regular.}}\label{iii.10.1.-smooth-neq-regular.-to_work}}

Over a nonperfect field, smooth and regular are not equivalent. For
example, let \(k_0\) be a field of characteristic \(p>0\), let
\(k=k_0(t)\), and let \(X \subseteq \mathbf{A}_k^2\) be the curve
defined by \(y^2=x^p-t\). Show that every local ring of \(X\) is a
regular local ring, but \(X\) is not smooth over \(k\).

\hypertarget{iii.10.2.}{%
\subsubsection{III.10.2.}\label{iii.10.2.}}

Let \(f: X \rightarrow Y\) be a proper, flat morphism of varieties over
\(k\). Suppose for some point \(y \in Y\) that the fibre \(X_y\) is
smooth over \(k(y)\). Then show that there is an open neighborhood \(U\)
of \(y\) in \(Y\) such that \(f: f^{-1}(U) \rightarrow U\) is smooth.

\hypertarget{iii.10.3.-tale-morphisms.}{%
\subsubsection{III.10.3. Tale
Morphisms.}\label{iii.10.3.-tale-morphisms.}}

A morphism \(f: X \rightarrow Y\) of schemes of finite type over \(k\)
is \textbf{étale}if it is smooth of relative dimension 0 . It is
\textbf{unramified} if for every \(x \in X\), letting \(y=f(x)\), we
have \(\mathrm{m}_y \cdot \mathcal{O}_x=\mathfrak{m}_x\), and \(k(x)\)
is a separable algebraic extension of \(k(y)\).

Show that the following conditions are equivalent:

\begin{enumerate}
\def\labelenumi{(\roman{enumi})}
\tightlist
\item
  \(f\) is étale;
\item
  \(f\) is flat, and \(\Omega_{X / Y}=0\);
\item
  \(f\) is flat and unramified.
\end{enumerate}

\hypertarget{iii.10.4.}{%
\subsubsection{III.10.4.}\label{iii.10.4.}}

Show that a morphism \(f: X \rightarrow Y\) of schemes of finite type
over \(k\) is étale if and only if the following condition is satisfied:
for each \(x \in X\), let \(y=f(x)\). Let \(\widehat{\mathcal{O}}_x\)
and \(\widehat{\mathcal{O}}_y\) be the completions of the local rings at
\(x\) and \(y\). Choose fields of representatives (II, 8.25A)
\(k(x) \subseteq \widehat{\mathcal{O}}_x\) and
\(k(y) \subseteq \widehat{\mathcal{O}}_y\) so that
\(k(y) \subseteq k(x)\) via the natural map
\(\widehat{\mathcal{O}}_y \rightarrow \widehat{\mathcal{O}}_x\).

Then our condition is that for every \(x \in X, k(x)\) is a separable
algebraic extension of \(k(y)\), and the natural map is an isomorphism.

\hypertarget{iii.10.5.-uxe9tale-neighborhoods.}{%
\subsubsection{III.10.5. Étale
Neighborhoods.}\label{iii.10.5.-uxe9tale-neighborhoods.}}

If \(x\) is a point of a scheme \(X\), we define an \textbf{étale
neighborhood} of \(x\) to be an étale morphism \(f: U \rightarrow X\),
together with a point \(x^{\prime} \in U\) such that
\(f\left(x^{\prime}\right)=x\).

As an example of the use of étale neighborhoods, prove the following: if
\(\mathcal{F}\) is a coherent sheaf on \(X\), and if every point of
\(X\) has an étale neighborhood \(f: U \rightarrow X\) for which
\(f^* \mathcal{F}\) is a free \(\mathcal{O}_U\)-module, then
\(\mathcal{F}\) is locally free on \(X\).

\hypertarget{iii.10.6.}{%
\subsubsection{III.10.6.}\label{iii.10.6.}}

Let \(Y\) be the plane nodal cubic curve \(y^2=x^2(x+1)\). Show that
\(Y\) has a finite étale covering \(X\) of degree 2, where \(X\) is a
union of two irreducible components, each one isomorphic to the
normalization of \(Y\) (Fig. 12).

\includegraphics{figures/2022-10-23_00-23-42.png}

\hypertarget{iii.10.7.-serre.-a-linear-system-with-moving-singularities.}{%
\subsubsection{III.10.7. (Serre). A linear system with moving
singularities.}\label{iii.10.7.-serre.-a-linear-system-with-moving-singularities.}}

Let \(k\) be an algebraically closed field of characteristic 2. Let
\(P_1, \ldots, P_7 \in \mathbf{P}_k^2\) be the seven points of the
projective plane over the prime field \(\mathbf{F}_2 \subseteq k\). Let
\(D\) be the linear system of all cubic curves in \(X\) passing through
\(P_1, \ldots, P_7\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(D\) is a linear system of dimension 2 with base points
  \(P_1, \ldots, P_7\), which determines an inseparable morphism of
  degree 2 from \(X-\left\{P_i\right\}\) to \(\mathbf{P}^2\).
\item
  Every curve \(C \in D\) is singular.

  More precisely, either \(C\) consists of 3 lines all passing through
  one of the \(P_i\), or \(C\) is an irreducible cuspidal cubic with
  cusp \(P \neq\) any \(P_i\).

  Furthermore, the correspondence \(C \mapsto\) the singular point of
  \(C\) is a \(1-1\) correspondence between \(D\) and \(\mathbf{P}^2\).
  Thus the singular points of elements of \(D\) move all over.
\end{enumerate}

\hypertarget{iii.10.8.-a-linear-system-with-moving-singularities-contained-in-the-base-locus-any-characteristic.}{%
\subsubsection{III.10.8. A linear system with moving singularities
contained in the base locus (any
characteristic).}\label{iii.10.8.-a-linear-system-with-moving-singularities-contained-in-the-base-locus-any-characteristic.}}

In affine 3 -space with coordinates \(x, y, z\), let \(C\) be the conic
\((x-1)^2+\) \(y^2=1\) in the \(x y\)-plane, and let \(P\) be the point
\((0,0, t)\) on the \(z\)-axis. Let \(Y_t\) be the closure in
\(\mathbf{P}^3\) of the cone over \(C\) with vertex \(P\).

Show that as \(t\) varies, the surfaces \(\left\{Y_t\right\}\) form a
linear system of dimension 1, with a moving singularity at \(P\). The
base locus of this linear system is the conic \(C\) plus the \(z\)-axis.

\hypertarget{iii.10.9.}{%
\subsubsection{III.10.9.}\label{iii.10.9.}}

Let \(f: X \rightarrow Y\) be a morphism of varieties over \(k\). Assume
that \(Y\) is regular, \(X\) is Cohen-Macaulay, and that every fibre of
\(f\) has dimension equal to
\(\operatorname{dim} X-\operatorname{dim} Y\). Then \(f\) is
flat.\footnote{Hint: Imitate the proof of (10.4), using (II, 8.21A).}

\hypertarget{iii.11-the-theorem-on-formal-functions}{%
\subsection{III.11: The Theorem on Formal
Functions}\label{iii.11-the-theorem-on-formal-functions}}

\hypertarget{iii.11.1.}{%
\subsubsection{III.11.1.}\label{iii.11.1.}}

Show that the result of \((11.2)\) is false without the projective
hypothesis. For example, let \(X=\mathbf{A}_k^n\), let
\(P=(0, \ldots, 0)\), let \(U=X-P\), and let \(f: U \rightarrow X\) be
the inclusion. Then the fibres of \(f\) all have dimension 0 , but
\(R^{n-1} f_* \mathcal{O}_U \neq 0\).

\hypertarget{iii.11.2.}{%
\subsubsection{III.11.2.}\label{iii.11.2.}}

Show that a projective morphism with finite fibres (= quasi-finite (II,
Ex. 3.5)) is a finite morphism.

\hypertarget{iii.11.3.-improved-bertinis-theorem.}{%
\subsubsection{III.11.3. Improved Bertini's
Theorem.}\label{iii.11.3.-improved-bertinis-theorem.}}

Let \(X\) be a normal, projective variety over an algebraically closed
field \(k\). Let \(D\) be a linear system (of effective Cartier
divisors) without base points, and assume that \(D\) is \textbf{not
composite with a pencil}, which means that if
\(f: X \rightarrow \mathbf{P}_k^n\) is the morphism determined by
\(\mathrm{D}\), then \(\operatorname{dim} f(X) \geqslant 2\).

Then show that every divisor in \(\mathrm{D}\) is connected.\footnote{See
  (10.9.1). Hints: Use (11.5), (Ex. 5.7) and (7.9).}

\hypertarget{iii.11.4.-principle-of-connectedness.}{%
\subsubsection{III.11.4. Principle of
Connectedness.}\label{iii.11.4.-principle-of-connectedness.}}

Let \(\left\{X_t\right\}\) be a flat family of closed subschemes of
\(\mathbf{P}_k^n\) parametrized by an irreducible curve \(T\) of finite
type over \(k\). Suppose there is a nonempty open set \(U \subseteq T\),
such that for all closed points \(t \in U, X_t\) is connected. Then
prove that \(X_t\) is connected for all \(t \in T\).

\hypertarget{iii.11.5.}{%
\subsubsection{III.11.5. *}\label{iii.11.5.}}

Let \(Y\) be a hypersurface in \(X=\mathbf{P}_k^N\) with
\(N \geqslant 4\). Let \(\widehat{X}\) be the formal completion of \(X\)
along \(Y\) (II, \(\S\) ). Prove that the natural map
\(\operatorname{Pic}\widehat{X} \rightarrow \operatorname{Pic}Y\) is an
isomorphism.\footnote{Hint: Use (II, Ex. 9.6), and then study the maps
  \(\operatorname{Pic}X_{n+1} \rightarrow \operatorname{Pic}X_n\) for
  each \(n\) using (Ex. 4.6) and (Ex. 5.5).}

\hypertarget{iii.11.6.}{%
\subsubsection{III.11.6.}\label{iii.11.6.}}

Again let \(Y\) be a hypersurface in \(X=\mathbf{P}_k^N\), this time
with \(N \geqslant 2\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\mathcal{F}\) is a locally free sheaf on \(X\), show that the
  natural map is an isomorphism.
\item
  Show that the following conditions are equivalent:

  \begin{enumerate}
  \def\labelenumii{(\roman{enumii})}
  \item
    For each locally free sheaf \(\mathcal{F}\) on \(\widehat{X}\),
    there exists a coherent sheaf \(\mathscr{F}\) on \(X\) such that
    \(\mathcal{F} \cong \widehat{\mathscr{F}}\) (i.e., \(\mathcal{F}\)
    is algebraizable);
  \item
    For each locally free sheaf \(\mathcal{F}\) on \(\widehat{X}\),
    there is an integer \(n_0\) such that \(\mathcal{F}(n)\) is
    generated by global sections for all \(n \geqslant n_0\).\footnote{Hint:
      For (ii) \(\Rightarrow\) (i), show that one can find sheaves
      \(\mathcal{E}_0, \mathcal{E}_1\) on \(X\), which are direct sums
      of sheaves of the form \(\mathcal{O}\left(-q_i\right)\), and an
      exact sequence
      \(\widehat{\mathcal{E}}_1 \rightarrow \widehat{\mathcal{E}}_0 \rightarrow \tilde{{\mathcal{F}}} \rightarrow 0\)
      on \(\widehat{X}\). Then apply (a) to the sheaf
      \(\mathop{\mathcal{H}\! \mathit{om}}\left(\mathcal{E}_1, \mathcal{E}_0\right)\).}
  \end{enumerate}
\item
  Show that the conditions (i) and (ii) of (b) imply that the natural
  map \(\operatorname{Pic}X \rightarrow \operatorname{Pic}\widehat{X}\)
  is an isomorphism.\footnote{Note. In fact, (i) and (ii) always hold if
    \(N \geqslant 3\). This fact, coupled with (Ex. 11.5) leads to
    Grothendieck's proof \([SGA \, 2]\) of the Lefschetz theorem which
    says that if \(Y\) is a hypersurface in \(\mathbf{P}_k^N\) with
    \(N \geqslant 4\), then \(\operatorname{Pic}Y \cong \mathbf{Z}\),
    and it. is generated by \(\mathcal{O}_Y(1)\). See Hartshorne
    \([ 5, Ch. IV]\) for more details.}
\end{enumerate}

\hypertarget{iii.11.7.}{%
\subsubsection{III.11.7.}\label{iii.11.7.}}

Now let \(Y\) be a curve in \(X=\mathbf{P}_k^2\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Use the method of (Ex. 11.5) to show that Pic
  \(\widehat{X} \rightarrow\) Pic \(Y\) is surjective, and its kernel is
  an infinite-dimensional vector space over \(k\).
\item
  Conclude that there is an invertible sheaf \({\mathcal{L}}\) on
  \(\widehat{X}\) which is not algebraizable.
\item
  Conclude also that there is a locally free sheaf \(\mathcal{F}\) on
  \(\widehat{X}\) so that no twist \(\mathcal{F}(n)\) is generated by
  global sections. Cf. (II, 9.9.1)
\end{enumerate}

\hypertarget{iii.11.8.}{%
\subsubsection{III.11.8.}\label{iii.11.8.}}

Let \(f: X \rightarrow Y\) be a projective morphism, let \(\mathcal{F}\)
be a coherent sheaf on \(X\) which is flat over \(Y\), and assume that
\(H^i\left(X_y, \mathcal{F}_y\right)=0\) for some \(i\) and some
\(y \in Y\). Then show that \(R^i f_*(\mathcal{F})\) is \(0\) in a
neighborhood of \(y\).

\hypertarget{iii.12-the-semicontinuity-theorem}{%
\subsection{III.12: The Semicontinuity
Theorem}\label{iii.12-the-semicontinuity-theorem}}

\hypertarget{iii.12.1.}{%
\subsubsection{III.12.1.}\label{iii.12.1.}}

Let \(Y\) be a scheme of finite type over an algebraically closed field
\(k\). Show that the function is upper semicontinuous of the set of
closed points \(Y\).

\hypertarget{iii.12.2.}{%
\subsubsection{III.12.2.}\label{iii.12.2.}}

Let \(\left\{X_t\right\}\) be a family of hypersurfaces of the same
degree in \(\mathbf{P}_k^n\). Show that for each \(i\), the function
\(h^i\left(X_t, \mathcal{O}_{X_t}\right)\) is a constant function of
\(t\).

\hypertarget{iii.12.3.}{%
\subsubsection{III.12.3.}\label{iii.12.3.}}

Let \(X_1 \subseteq \mathbf{P}_k^4\) be the \textbf{rational normal
quartic curve} (which is the 4-uple embedding of \(\mathbf{P}^1\) in
\(\mathbf{P}^4\) ). Let \(X_0 \subseteq \mathbf{P}_k^3\) be a
nonsingular rational quartic curve, such as the one in (I, Ex. 3.18b).

Use (9.8.3) to construct a flat family \(\left\{X_t\right\}\) of curves
in \(\mathbf{P}^4\), parametrized by \(T=\mathbf{A}^1\), with the given
fibres \(X_1\) and \(X_0\) for \(t=1\) and \(t=0\).

Let \(\mathcal{I} \subseteq \mathcal{O}_{\mathbf{P}^4 \times T}\) be the
ideal sheaf of the total family \(X \subseteq \mathbf{P}^4 \times T\).
Show that \(\mathcal{I}\) is flat over \(T\).

Then show that and also This gives another example of cohomology groups
jumping at a special point.

\hypertarget{iii.12.4.}{%
\subsubsection{III.12.4.}\label{iii.12.4.}}

Let \(Y\) be an integral scheme of finite type over an algebraically
closed field \(k\). Let \(f: X \rightarrow Y\) be a flat projective
morphism whose fibres are all integral schemes. Let
\(\mathcal{L}, \mathcal{M}\) be invertible sheaves on \(X\), and assume
for each \(y \in Y\) that \(\mathcal{L}_y \cong \mathcal{M}_y\) on the
fibre \(X_y\).

Then show that there is an invertible sheaf \(\mathcal{N}\) on \(Y\)
such that
\(\mathcal{L} \cong \mathcal{M} \otimes f^* \mathcal{N}\).\footnote{Hint:
  Use the results of this section to show that
  \(f_*\left(\mathcal{L} \otimes \mathcal{M}^{-1}\right)\) is locally
  free of rank 1 on \(Y\).}

\hypertarget{iii.12.5.}{%
\subsubsection{III.12.5.}\label{iii.12.5.}}

Let \(Y\) be an integral scheme of finite type over an algebraically
closed field \(k\). Let \(\mathcal{E}\) be a locally free sheaf on
\(Y\), and let \(X=\mathbf{P}(\mathcal{E})\) -- see \((II, \S 7)\).

Then show that
\(\operatorname{Pic}X \cong( \operatorname{Pic}Y) \times \mathbf{Z}\).
This strengthens (II, Ex. 7.9).

\hypertarget{iii.12.6.}{%
\subsubsection{III.12.6. *}\label{iii.12.6.}}

Let \(X\) be an integral projective scheme over an algebraically closed
field \(k\), and assume that \(H^1\left(X, \mathcal{O}_X\right)=0\). Let
\(T\) be a connected scheme of finite type over \(k\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\mathcal{L}\) is an invertible sheaf on \(X \times T\), show that
  the invertible sheaves \(\mathcal{L}_t\) on \(X=X \times\{t\}\) are
  isomorphic, for all closed points \(t \in T\).
\item
  Show that \(\operatorname{Pic}(X \times T)=\) Pic \(X \times\) Pic
  \(T\). (Do not assume that \(T\) is reduced!)\footnote{Hint: Apply
    (12.11) with \(i=0,1\) for suitable invertible sheaves on
    \(X \times T\).}

  Cf. (IV, Ex. 4.10) and (V, Ex. 1.6) for examples where
  \(\operatorname{Pic}(X \times T) \neq \operatorname{Pic} X \times\)
  Pic T.
\end{enumerate}

\hypertarget{iv-curves}{%
\section{IV: Curves}\label{iv-curves}}

\hypertarget{iv.1-riemann-roch}{%
\subsection{IV.1: Riemann-Roch}\label{iv.1-riemann-roch}}

\hypertarget{section-59}{%
\subsubsection{1.1.}\label{section-59}}

Let \(X\) be a curve, and let \(P \in X\) be a point. Then there exists
a nonconstant rational function \(f \in K(X)\), which is regular
everywhere except at \(P\).

\hypertarget{section-60}{%
\subsubsection{1.2.}\label{section-60}}

Again let \(X\) be a curve, and let \(P_1, \ldots P_r \in X\) be points.
Then there is a rational function \(f \in K(X)\) having poles (of some
order) at each of the \(P_1\), and regular elsewhere.

\hypertarget{section-61}{%
\subsubsection{1.3.}\label{section-61}}

Let \(X\) be an integral, separated, regular, one-dimensional scheme of
finite type over \(k\), which is \textbf{not} proper over \(k\). Then
\(X\) is affine.\footnote{Hint: Embed \(X\) in a (proper) curve
  \(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) over
  \(k\), and use (Ex. 1.2) to construct a morphism
  \(f: \mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu \rightarrow \mathbf{P}^1\)
  such that \(f^{-1}\left(\mathbf{A}^1\right)=X\)}

\hypertarget{section-62}{%
\subsubsection{1.4.}\label{section-62}}

Show that a separated, one-dimensional scheme of finite type over \(k\),
none of whose irreducible components is proper over \(k\), is
affine.\footnote{Hint: Combine (Ex. 1.3) with (III, Ex. 3.1, Ex. 3.2,
  Ex. 4.2).}

\hypertarget{section-63}{%
\subsubsection{1.5.}\label{section-63}}

For an effective divisor \(D\) on a curve \(X\) of genus \(g\), show
that \(\operatorname{dim}|D| \leqslant \operatorname{deg} D\).
Furthermore, equality holds if and only if \(D=0\) or \(g=0\).

\hypertarget{section-64}{%
\subsubsection{1.6.}\label{section-64}}

Let \(X\) be a curve of genus \(g\). Show that there is a finite
morphism \(f: X \rightarrow \mathbf{P}^1\) of degree\footnote{Recall
  that the degree of a finite morphism of curves \(f: X \rightarrow Y\)
  is defined as the degree of the field extension \([K(X): K(Y)]\)
  (II.6).} \(\leqslant g+1\).

\hypertarget{section-65}{%
\subsubsection{1.7.}\label{section-65}}

A curve \(X\) is called \textbf{hyperelliptic} if \(g \geqslant 2\) and
there exists a finite morphism \(f: X \rightarrow \mathbf{P}^1\) of
degree 2.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(X\) is a curve of genus \(g=2\), show that the canonical divisor
  defines a complete linear system \(|K|\) of degree 2 and dimension 1,
  without base points. Use (II, 7.8.1) to conclude that \(X\) is
  hyperelliptic.
\item
  Show that the curves constructed in (1.1.1) all admit a morphism of
  degree 2 to \(\mathbf{P}^1\). Thus there exist hyperelliptic curves of
  any genus \(g \geqslant 2\).\footnote{Note: we will see later (Ex.
    3.2) that there exist non-hyperelliptic curves. See also (V, Ex.
    2.10).}
\end{enumerate}

\hypertarget{p_a-of-a-singular-curve.}{%
\subsubsection{\texorpdfstring{1.8. \(p_a\) of a Singular
Curve.}{1.8. p\_a of a Singular Curve.}}\label{p_a-of-a-singular-curve.}}

Let \(X\) be an integral projective scheme of dimension 1 over \(k\),
and let \(\tilde{X}\) be its normalization (II, Ex. 3.8). Then there is
an exact sequence of sheaves on \(X\), where \(\tilde{\mathcal{O}}_P\)
is the integral closure of \({\mathcal{O}}_P\). For each \(P \in X\),
let
\(\delta_P=\operatorname{length}(\tilde{\mathcal{O}}_P/{\mathcal{O}}_P)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that
  \(p_a(X)=p_a(\tilde{X})+\sum_{p \in X} \delta_p\).\footnote{Hint: Use
    (III, Ex. 4.1) and (III, Ex. 5.3).}
\item
  If \(p_a(X)=0\), show that \(X\) is already nonsingular and in fact
  isomorphic to \(\mathbf{P}^1\).\footnote{This strengthens (1.3.5).}
\item
  * If \(P\) is a node or an ordinary cusp (I, Ex. 5.6, Ex. 5.14), show
  that \(\delta_P=1\).\footnote{Hint: Show first that \(\delta_P\)
    depends only on the analytic isomorphism class of the singularity at
    \(P\). Then compute \(\delta_P\) for the node and cusp of suitable
    plane cubic curves. See \((\mathrm{V}, 3.9 .3)\) for another method.}
\end{enumerate}

\hypertarget{riemann-roch-for-singular-curves.}{%
\subsubsection{1.9. * Riemann-Roch for Singular
Curves.}\label{riemann-roch-for-singular-curves.}}

Let \(X\) be an integral projective scheme of dimension 1 over \(k\).
Let \(X_{\mathrm{reg}}\) be the set of regular points of \(X\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(D=\sum n_i P_i\) be a divisor with support in
  \(X_{\mathrm{reg}}\), i.e., all \(P_i \in X_{\mathrm{reg}}\). Then
  define deg \(D=\sum n_i\). Let \(\mathscr{L}(D)\) be the associated
  invertible sheaf on \(X\), and show that
\item
  Show that any Cartier divisor on \(X\) is the difference of two very
  ample Cartier divisors. \footnote{Use (II, Ex. 7.5).}
\item
  Conclude that every invertible sheaf \(\mathscr{L}\) on \(X\) is
  isomorphic to \(\mathscr{L}(D)\) for some divisor \(D\) with support
  in \(X_{\mathrm{reg}}\).
\item
  Assume furthermore that \(X\) is a locally complete intersection in
  some projective space. Then by (III, 7.11) the dualizing sheaf
  \(\omega_X\) is an invertible sheaf on \(X\), so we can define the
  canonical divisor \(K\) to be a divisor with support in
  \(X_{\mathrm{reg}}\) corresponding to \(\omega_X\). Then the formula
  of a. becomes
\end{enumerate}

\hypertarget{section-66}{%
\subsubsection{1.10.}\label{section-66}}

Let \(X\) be an integral projective scheme of dimension 1 over \(k\),
which is locally complete intersection, and has \(p_a=1\). Fix a point
\(P_0 \in X_{\text {reg. }}\). Imitate (1.3.7) to show that the map
\(P \rightarrow \mathscr{L}\left(P-P_0\right)\) gives a one-to-one
correspondence between the points of \(X_{\mathrm{reg}}\) and the
elements of the group \(\operatorname{Pic}X\).\footnote{This generalizes
  (II, 6.11.4) and (II, Ex. 6.7).}

\hypertarget{iv.2-hurwitz}{%
\subsection{IV.2: Hurwitz}\label{iv.2-hurwitz}}

\hypertarget{section-67}{%
\subsubsection{2.1}\label{section-67}}

Use (2.5.3) to show that \(\mathbf{P}^n\) is simply connected.

\hypertarget{classification-of-curves-of-genus-2-.}{%
\subsubsection{2.2 Classification of Curves of Genus 2
.}\label{classification-of-curves-of-genus-2-.}}

Fix an algebraically closed field \(k\) of characteristic \(\neq 2\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(X\) is a curve of genus 2 over \(k\), the canonical linear system
  \(|K|\) determines a finite morphism \(f: X \rightarrow \mathbf{P}^1\)
  of degree 2 (Ex. 1.7). Show that it is ramified at exactly 6 points,
  with ramification index 2 at each one. Note that \(f\) is uniquely
  determined, up to an automorphism of \(\mathbf{P}^1\), so \(X\)
  determines an (unordered) set of 6 points of \(\mathbf{P}^1\), up to
  an automorphism of \(\mathbf{P}^1\).
\item
  Conversely, given six distinct elements
  \(\alpha_1, \ldots, \alpha_6 \in k\), let \(K\) be the extension of
  \(k(x)\) determined by the equation
  \(z^2=\left(x-\alpha_1\right) \cdots\left(x-\alpha_6\right)\). Let
  \(f: X \rightarrow \mathbf{P}^1\) be the corresponding morphism of
  curves. Show that \(g(X)=2\), the map \(f\) is the same as the one
  determined by the canonical linear system, and \(f\) is ramified over
  the six points \(x=\alpha_i\) of \(\mathbf{P}^1\), and nowhere else.
  (Cf. (II, Ex. 6.4).)
\item
  Using (I, Ex. 6.6), show that if \(P_1, P_2, P_3\) are three distinct
  points of \(\mathbf{P}^1\), then there exists a unique \(\varphi \in\)
  Aut \(\mathbf{P}^1\) such that
  \(\varphi\left(P_1\right)=0, \varphi\left(P_2\right)=1, \varphi\left(P_3\right)=\infty\).
  Thus in (a), if we order the six points of \(\mathbf{P}^1\), and then
  normalize by sending the first three to \(0,1, x\), respectively, we
  may assume that \(X\) is ramified over
  \(0,1, \infty, \beta_1, \beta_2, \beta_3\), where
  \(\beta_1, \beta_2, \beta_3\) are three distinct elements of
  \(k, \neq 0,1\).
\item
  Let \(\Sigma_6\) be the symmetric group on 6 letters. Define an action
  of \(\Sigma_6\) on sets of three distinct elements
  \(\beta_1, \beta_2, \beta_3\) of \(k, \neq 0,1\), as follows: reorder
  the set \(0,1, \infty, \beta_1, \beta_2, \beta_3\) according to a
  given element \(\sigma \in \Sigma_6\), then renormalise as in (c) so
  that the first three become \(0,1, \infty\) again. Then the last three
  are the new \(\beta_1^{\prime}, \beta_2^{\prime}, \beta_3^{\prime}\).
\item
  Summing up, conclude that there is a one-to-one correspondence between
  the set of isomorphism classes of curves of genus 2 over \(k\), and
  triples of distinct elements \(\beta_1, \beta_2, \beta_3\) of
  \(k, \neq 0,1\), modulo the action of \(\Sigma_6\) described in (d).
  In particular, there are many non-isomorphic curves of genus 2 . We
  say that curves of genus 2 depend on three parameters, since they
  correspond to the points of an open subset of \(\mathbf{A}_k^3\)
  modulo a finite group.
\end{enumerate}

\hypertarget{plane-curves.}{%
\subsubsection{2.3 Plane Curves.}\label{plane-curves.}}

Let \(X\) be a curve of degree \(d\) in \(\mathbf{P}^2\). For each point
\(P \in X\), let \(T_P(X)\) be the tangent line to \(X\) at \(P\) (I,
Ex. 7.3). Considering \(T_P(X)\) as a point of the dual projective plane
\(\left(\mathbf{P}^2\right)^*\), the map \(P \rightarrow T_P(X)\) gives
a morphism of \(X\) to its \textbf{dual curve} \(X^*\) in
\(\left(\mathbf{P}^2\right)^*\) (I, Ex. 7.3).

Note that even though \(X\) is nonsingular, \(X^*\) in general will have
singularities. We assume char \(k=0\) below.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Fix a line \(L \subseteq \mathbf{P}^2\) which is not tangent to \(X\).
  Define a morphism \(\varphi: X \rightarrow L\) by
  \(\varphi(P)=T_P(X) \cap L\), for each point \(P \in X\). Show that
  \(\varphi\) is ramified at \(P\) if and only if either

  \begin{enumerate}
  \def\labelenumii{(\arabic{enumii})}
  \tightlist
  \item
    \(P \in L\), or
  \item
    \(P\) is an inflection point of \(X\), which means that the
    intersection multiplicity (I, Ex. 5.4) of \(T_P(X)\) with \(X\) at
    \(P\) is \(\geqslant 3\). Conclude that \(X\) has only finitely many
    inflection points.
  \end{enumerate}
\item
  A line of \(\mathbf{P}^2\) is a \textbf{multiple tangent} of \(X\) if
  it is tangent to \(X\) at more than one point. It is a
  \textbf{bitangent} if it is tangent to \(X\) at exactly two points. If
  \(L\) is a multiple tangent of \(X\), tangent to \(X\) at the points
  \(P_1, \ldots, P_r\), and if none of the \(P_i\) is an inflection
  point, show that the corresponding point of the dual curve \(X^*\) is
  an ordinary \(r\)-fold point, which means a point of multiplicity
  \(r\) with distinct tangent directions (I, Ex. 5.3). Conclude that
  \(X\) has only finitely many multiple tangents.
\item
  Let \(O \in \mathbf{P}^2\) be a point which is not on \(X\), nor on
  any inflectional or multiple tangent of \(X\). Let \(L\) be a line not
  containing \(O\). Let \(\psi: X \rightarrow L\) be the morphism
  defined by projection from \(O\). Show that \(\psi\) is ramified at a
  point \(\mathrm{P} \in X\) if and only if the line \(O P\) is tangent
  to \(X\) at \(P\), and in that case the ramification index is 2. Use
  Hurwitz's theorem and (I, Ex. 7.2) to conclude that there are exactly
  \(d(d-1)\) tangents of \(X\) passing through \(O\). Hence the degree
  of the dual curve (sometimes called the \textbf{class} of \(X)\) is
  \(d(d-1)\).
\item
  Show that for all but a finite number of points of \(X\), a point
  \(O\) of \(X\) lies on exactly \((d+1)(d-2)\) tangents of \(X\), not
  counting the tangent at \(O\).
\item
  Show that the degree of the morphism \(\varphi\) of a. is \(d(d-1)\).
  Conclude that if \(d \geqslant 2\), then \(X\) has \(3 d(d-2)\)
  inflection points, properly counted. (If \(T_P(X)\) has intersection
  multiplicity \(r\) with \(X\) at \(P\), then \(P\) should be counted
  \(r-2\) times as an inflection point. If \(r=3\) we call it an
  ordinary inflection point.) Show that an ordinary inflection point of
  \(X\) corresponds to an ordinary cusp of the dual curve \(X^*\).
\item
  Now let \(X\) be a plane curve of degree \(d \geqslant 2\), and assume
  that the dual curve \(X^*\) has only nodes and ordinary cusps as
  singularities (which should be true for sufficiently general \(X)\).
  Then show that \(X\) has exactly \(\frac{1}{2} d(d-2)(d-3)(d+3)\)
  bitangents.\footnote{Hint: Show that \(X\) is the normalization of
    \(X^*\). Then calculate \(p_a\left(X^*\right)\) two ways: once as a
    plane curve of degree \(d(d-1)\), and once using (Ex. 1.8).}
\item
  For example, a plane cubic curve has exactly 9 inflection points, all
  ordinary. The line joining any two of them intersects the curve in a
  third one.
\item
  A plane quartic curve has exactly 28 bitangents. (This holds even if
  the curve has a tangent with four-fold contact, in which case the dual
  curve \(X^*\) has a tacnode.)
\end{enumerate}

\hypertarget{a-funny-curve-in-characteristic-p.}{%
\subsubsection{\texorpdfstring{2.4 A Funny Curve in Characteristic
\(p\).}{2.4 A Funny Curve in Characteristic p.}}\label{a-funny-curve-in-characteristic-p.}}

Let \(X\) be the plane quartic curve \(x^3 y+y^3 z+ z^3 x = 0\) over a
field of characteristic 3 . Show that \(X\) is nonsingular, every point
of \(X\) is an inflection point, the dual curve \(X^*\) is isomorphic to
\(X\), but the natural map \(X \rightarrow X^*\) is purely inseparable.

\hypertarget{automorphisms-of-a-curve-of-genus-geqslant-2.}{%
\subsubsection{\texorpdfstring{2.5 Automorphisms of a Curve of Genus
\(\geqslant 2\).}{2.5 Automorphisms of a Curve of Genus \textbackslash geqslant 2.}}\label{automorphisms-of-a-curve-of-genus-geqslant-2.}}

Prove the theorem of Hurwitz that a curve \(X\) of genus
\(g \geqslant 2\) over a field of characteristic 0 has at most
\(84(g-1)\) automorphisms.

We will see later (Ex. 5.2) or (V, Ex. 1.11) that the group \(G=\) Aut
\(X\) is finite. So let \(G\) have order \(n\). Then \(G\) acts on the
function field \(K(X)\). Let \(L\) be the fixed field. Then the field
extension \(L \subseteq K(X)\) corresponds to a finite morphism of
curves \(f: X \rightarrow Y\) of degree \(n\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(P \in X\) is a ramification point, and \(e_P=r\), show that
  \(f^{-1} f(P)\) consists of exactly \(n / r\) points, each having
  ramification index \(r\). Let \(P_1, \ldots, P_s\) be a maximal set of
  ramification points of \(X\) lying over distinct points of \(Y\), and
  let \(e_{P_1}=r_i\). Then show that Hurwitz's theorem implies that
\item
  Since \(g \geqslant 2\), the left hand side of the equation is \(>0\).
  Show that if \(g(Y) \geqslant 0\),
  \(s \geqslant 0, r_i \geqslant 2, i=1, \ldots, s\) are integers such
  that then the minimum value of this expression is \(1 / 42\). Conclude
  that \(n \leqslant 84(g-1)\).\footnote{See (Ex. 5.7) for an example
    where this maximum is achieved. Note: It is known that this maximum
    is achieved for infinitely many values of \(g\) (Macbeath \[1\]).
    Over a field of characteristic \(p>0\), the same bound holds,
    provided \(p>g+1\), with one exception, namely the hyperelliptic
    curve \(y^2=x^p-x\), which has \(p=2 g+1\) and
    \(2 p\left(p^2-1\right)\) automorphisms (Roquette). For other bounds
    on the order of the group of automorphisms in characteristic \(p\),
    see Singh and Stichtenoth.}
\end{enumerate}

\hypertarget{f_-for-divisors.}{%
\subsubsection{\texorpdfstring{2.6 \(f_*\) for
Divisors.}{2.6 f\_* for Divisors.}}\label{f_-for-divisors.}}

Let \(f: X \rightarrow Y\) be a finite morphism of curves of degree
\(n\). We define a homomorphism
\(f_*: \operatorname{Div} X \rightarrow\) Div \(Y\) by
\(f_*\left(\sum n_i P_i\right)=\sum n_i f\left(P_i\right)\) for any
divisor \(D=\sum n_i P_i\) on \(X\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  For any locally free sheaf \(\mathscr{E}\) on \(Y\), of rank \(r\), we
  define
  \(\operatorname{det}\mathscr{E}=\wedge^r \mathscr{E} \in \operatorname{Pic}Y\)
  (II, Ex. 6.11). In particular, for any invertible sheaf
  \(\mathscr{M}\) on \(X, f_* \mathscr{M}\) is locally free of rank
  \(n\) on \(Y\), so we can consider
  \(\operatorname{det}f_* \mathscr{M} \in \operatorname{Pic}Y\). Show
  that for any divisor \(D\) on \(X\),\footnote{Hint: First consider an
    effective divisor \(D\), apply \(f_*\) to the exact sequence
    \(0 \rightarrow \mathscr{L}(-D) \rightarrow\)
    \({\mathcal{O}}_X \rightarrow \mathcal{O}_D \rightarrow 0\), and use
    (II, Ex. 6.11).{]}} Note in particular that
  \(\operatorname{det}\left(f_* \mathscr{L}(D)\right) \neq \mathscr{L}\left(f_* D\right)\)
  in general!
\item
  Conclude that \(f_* D\) depends only on the linear equivalence class
  of \(D\), so there is an induced homomorphism
  \(f_*: \operatorname{Pic}X \rightarrow \operatorname{Pic}Y\). Show
  that \(f_* f^*: \operatorname{Pic}Y \rightarrow \operatorname{Pic}Y\)
  is just multiplication by \(n\).
\item
  Use duality for a finite flat morphism (III, Ex. 6.10) and (III, Ex.
  7.2) to show that
\item
  Now assume that \(f\) is separable, so we have the ramification
  divisor \(R\). We define the \textbf{branch divisor} \(B\) to be the
  divisor \(f_* R\) on \(Y\). Show that
\end{enumerate}

\hypertarget{etale-covers-of-degree-2.}{%
\subsubsection{2.7 Etale Covers of Degree
2.}\label{etale-covers-of-degree-2.}}

Let \(Y\) be a curve over a field \(k\) of characteristic \(\neq 2\). We
show there is a one-to-one correspondence between finite étale morphisms
\(f: X \rightarrow Y\) of degree 2, and 2-torsion elements of
\(\operatorname{Pic}Y\), i.e., invertible sheaves \(\mathscr{L}\) on
\(Y\) with \(\mathscr{L}^2 \cong \mathcal{O}_Y\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Given an étale morphism \(f: X \rightarrow Y\) of degree 2, there is a
  natural map \({\mathcal{O}}_Y \rightarrow\) \(f_* {\mathcal{O}}_X\).
  Let \(\mathscr{L}\) be the cokernel. Then \(\mathscr{L}\) is an
  invertible sheaf on
  \(Y, \mathscr{L} \cong \operatorname{det} f_* {\mathcal{O}}_X\), and
  so \(\mathscr{L}^2 \cong \mathcal{O}_Y\) by (Ex. 2.6). Thus an étale
  cover of degree 2 determines a 2-torsion element in
  \(\operatorname{Pic}Y\).
\item
  Conversely, given a 2-torsion element \(\mathscr{L}\) in
  \(\operatorname{Pic}Y\), define an
  \({\mathcal{O}}_Y{\hbox{-}}\)algebra structure on
  \(\mathcal{O}_Y \oplus \mathscr{L}\) by
  \(\langle a, b\rangle \cdot\left\langle a^{\prime}, b^{\prime}\right\rangle=\left\langle a a^{\prime}+\varphi\left(b \otimes b^{\prime}\right), a b^{\prime}+a^{\prime} b\right\rangle\),
  where \(\varphi\) is an isomorphism of
  \(\mathscr{L} \otimes \mathscr{L} \rightarrow \mathcal{O}_Y\). Then
  take
  \(X=\operatorname{Spec}\left({\mathcal{O}}_Y \oplus \mathscr{L}\right)\)
  (II, Ex. 5.17). Show that \(X\) is an étale cover of \(Y\).
\item
  Show that these two processes are inverse to each
  other.\footnote{Note. This is a special case of the more general fact
    that for \((n\), char \(k)=1\), the étale Galois covers of \(Y\)
    with group \(\mathbf{Z} / n \mathbf{Z}\) are classified by the étale
    cohomology group
    \(H_{\mathrm{et}}^1(Y, \mathbf{Z} / n \mathbf{Z})\), which is equal
    to the group of \(n\)-torsion points of \(\operatorname{Pic}Y\). See
    Serre \[6\].}\footnote{Hint: Let \(\tau: X \rightarrow X\) be the
    involution which interchanges the points of each fibre of \(f\). Use
    the trace map \(a \mapsto a+\tau(a)\) from
    \(f_* \mathcal{O}_X \rightarrow \mathcal{O}_Y\) to show that the
    sequence of \(\mathcal{O}_{Y^{-}}\) modules in a. is split exact.}
\end{enumerate}

\hypertarget{iv.3-embeddings-in-projective-space}{%
\subsection{IV.3: Embeddings in Projective
Space}\label{iv.3-embeddings-in-projective-space}}

\hypertarget{ii.3.1.}{%
\subsubsection{II.3.1.}\label{ii.3.1.}}

If \(X\) is a curve of genus 2 , show that a divisor \(D\) is very ample
\(\Leftrightarrow \operatorname{deg} D \geqslant 5\). This strengthens
(3.3.4).

\hypertarget{ii.3.2.}{%
\subsubsection{II.3.2.}\label{ii.3.2.}}

Let \(X\) be a plane curve of degree 4 .

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that the effective canonical divisors on \(X\) are exactly the
  divisors \(X.L\), where \(L\) is a line in \(\mathbf{P}^2\).
\item
  If \(D\) is any effective divisor of degree 2 on \(X\), show that
  \(\operatorname{dim}|D|=0\).
\item
  Conclude that \(X\) is not hyperelliptic (Ex. 1.7).
\end{enumerate}

\hypertarget{ii.3.3.}{%
\subsubsection{II.3.3.}\label{ii.3.3.}}

If \(X\) is a curve of genus \(\geqslant 2\) which is a complete
intersection (II, Ex. 8.4) in some \(\mathbf{P}^n\), show that the
canonical divisor \(K\) is very ample. Conclude that a curve of genus 2
can never be a complete intersection in any \(\mathbf{P}^n\). Cf. (Ex.
5.1).

\hypertarget{ii.3.4.-the-rational-normal-curve.}{%
\subsubsection{II.3.4. The Rational Normal
Curve.}\label{ii.3.4.-the-rational-normal-curve.}}

Let \(X\) be the \(d\)-uple embedding (I, Ex. 2.12) of \(\mathbf{P}^1\)
in \(\mathbf{P}^d\), for any \(d \geqslant 1\). We call \(X\) the
\textbf{rational normal curve} of degree \(d\) in \(\mathbf{P}^d\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(X\) is projectively normal, and that its homogeneous ideal
  can be generated by forms of degree 2 .
\item
  If \(X\) is any curve of degree \(d\) in \(\mathbf{P}^n\), with
  \(d \leqslant n\), which is not contained in any \(\mathbf{P}^{n-1}\),
  show that in fact \(d=n, g(X)=0\), and \(X\) differs from the rational
  normal curve of degree \(d\) only by an automorphism of
  \(\mathbf{P}^d\). Cf. (II. 7.8.5).
\item
  In particular, any curve of degree 2 in any \(\mathbf{P}^n\) is a
  conic in some \(\mathbf{P}^2\).
\item
  A curve of degree 3 in any \(\mathbf{P}^n\) must be either a plane
  cubic curve, or the twisted cubic curve in \(\mathbf{P}^3\).
\end{enumerate}

\hypertarget{ii.3.5.}{%
\subsubsection{II.3.5.}\label{ii.3.5.}}

Let \(X\) be a curve in \(\mathbf{P}^3\), which is not contained in any
plane.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(O \notin X\) is a point, such that the projection from \(O\)
  induces a birational morphism \(\varphi\) from \(X\) to its image in
  \(\mathbf{P}^2\), show that \(\varphi(X)\) must be
  singular.\footnote{Hint: Calculate
    \(\operatorname{dim} H^0\left(X, \mathcal{O}_X(1)\right)\) two ways.}
\item
  If \(X\) has degree \(d\) and genus \(g\), conclude that
  \(g<\frac{1}{2}(d-1)(d-2)\). (Use (Ex. 1.8).)
\item
  Now let \(\left\{X_t\right\}\) be the flat family of curves induced by
  the projection (III, 9.8.3) whose fibre over \(t=1\) is \(X\), and
  whose fibre \(X_0\) over \(t=0\) is a scheme with support
  \(\varphi(X)\). Show that \(X_0\) always has nilpotent elements. Thus
  the example (III, 9.8.4) is typical.
\end{enumerate}

\hypertarget{ii.3.6.}{%
\subsubsection{II.3.6.}\label{ii.3.6.}}

Curves of Degree 4.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  If \(X\) is a curve of degree 4 in some \(\mathbf{P}^n\), show that
  either

  \begin{enumerate}
  \def\labelenumii{(\arabic{enumii})}
  \tightlist
  \item
    \(g=0\), in which case \(X\) is either the rational normal quartic
    in \(\mathbf{P}^4\) (Ex. 3.4) or the rational quartic curve in
    \(\mathbf{P}^3\) (II, 7.8.6), or
  \item
    \(X \subseteq \mathbf{P}^2\), in which case \(g=3\), or
  \item
    \(X \subseteq \mathbf{P}^3\) and \(g=1\).
  \end{enumerate}
\item
  In the case \(g=1\), show that \(X\) is a complete intersection of two
  irreducible quadric surfaces in \(\mathbf{P}^3\) (I, Ex.
  5.11).\footnote{Hint: Use the exact sequence
    \(0 \rightarrow \mathcal{I}_X \rightarrow\)
    \(\mathcal{O}_{\mathbf{P}^3} \rightarrow \mathcal{O}_X \rightarrow 0\)
    to compute
    \(\operatorname{dim} H^0\left(\mathbf{P}^3, \mathcal{I}_X(2)\right)\),
    and thus conclude that \(X\) is contained in at least two
    irreducible quadric surfaces.}
\end{enumerate}

\hypertarget{ii.3.7.}{%
\subsubsection{II.3.7.}\label{ii.3.7.}}

In view of (3.10), one might ask conversely, is every plane curve with
nodes a projection of a nonsingular curve in \(\mathbf{P}^3\) ? Show
that the curve \(x y+x^4+y^4=0\) (assume char \(k \neq 2\) ) gives a
counterexample.

\hypertarget{ii.3.8.}{%
\subsubsection{II.3.8.}\label{ii.3.8.}}

We say a (singular) integral curve in \(\mathbf{P}^n\) is
\textbf{strange} if there is a point which lies on all the tangent lines
at nonsingular points of the curve.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  There are many singular strange curves, e.g., the curve given
  parametrically by \(x=t, y=t^p, z=t^{2 p}\) over a field of
  characteristic \(p>0\).
\item
  Show, however, that if char \(k=0\), there aren't even any singular
  strange curves besides \(\mathbf{P}^1\).
\end{enumerate}

\hypertarget{ii.3.9.-bertinis-lemma.}{%
\subsubsection{II.3.9. Bertini's Lemma.}\label{ii.3.9.-bertinis-lemma.}}

Prove the following lemma of Bertini: if \(X\) is a curve of degree
\(d\) in \(\mathbf{P}^3\), not contained in any plane, then for almost
all planes \(H \subseteq \mathbf{P}^3\) (meaning a Zariski open subset
of the dual projective space
\(\left.\left(\mathbf{P}^3\right)^*\right)\), the intersection
\(X \cap H\) consists of exactly \(d\) distinct points, no three of
which are collinear.

\hypertarget{ii.3.10.-not-every-secant-is-a-multisecant.}{%
\subsubsection{II.3.10. Not every secant is a
multisecant.}\label{ii.3.10.-not-every-secant-is-a-multisecant.}}

Generalize the statement that ``not every secant is a multisecant'' as
follows. If \(X\) is a curve in \(\mathbf{P}^n\), not contained in any
\(\mathbf{P}^{n-1}\), and if char \(k=0\), show that for almost all
choices of \(n-1\) points \(P_1, \ldots, P_{n-1}\) on \(X\), the linear
space \(L^{n-2}\) spanned by the \(P_i\) does not contain any further
points of \(X\).

\hypertarget{ii.3.11}{%
\subsubsection{II.3.11}\label{ii.3.11}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(X\) is a nonsingular variety of dimension \(r\) in
  \(\mathbf{P}^n\), and if \(n>2 r+1\), show that there is a point
  \(O \notin X\), such that the projection from \(O\) induces a closed
  immersion of \(X\) into \(\mathbf{P}^{n-1}\).
\item
  If \(X\) is the Veronese surface in \(\mathbf{P}^5\), which is the
  2-uple embedding of \(\mathbf{P}^2\) (I, Ex. 2.13), show that each
  point of every secant line of \(X\) lies on infinitely many secant
  lines. Therefore, the secant variety of \(X\) has dimension 4, and so
  in this case there is a projection which gives a closed immersion of
  \(X\) into \(\mathrm{P}^4\) (II, Ex. 7.7).\footnote{A theorem of
    Severi \([1]\) states that the Veronese surface is the only surface
    in \(\mathbf{P}^5\) for which there is a projection giving a closed
    immersion into \(\mathbf{P}^4\). Usually one obtains a finite number
    of double points with transversal tangent planes.}
\end{enumerate}

\hypertarget{ii.3.12.}{%
\subsubsection{II.3.12.}\label{ii.3.12.}}

For each value of \(d=2,3,4,5\) and \(r\) satisfying
\(0 \leqslant r \leqslant \frac{1}{2}(d-1)(d-2)\), show that there
exists an irreducible plane curve of degree \(d\) with \(r\) nodes and
no other singularities.

\hypertarget{iv.4-elliptic-curves}{%
\subsection{IV.4: Elliptic Curves}\label{iv.4-elliptic-curves}}

\hypertarget{ii.4.1.}{%
\subsubsection{II.4.1.}\label{ii.4.1.}}

Let \(X\) be an elliptic curve over \(k\), with char \(k \neq 2\), let
\(P \in X\) be a point, and let \(R\) be the graded ring
\(R=\bigoplus_{n \geqslant 0} H^0\left(X, \mathcal{O}_X(n P)\right)\).
Show that for suitable choice of \(t, x, y\) as a graded ring, where
\(k[t, x, y]\) is graded by setting
\(\operatorname{deg} t=1, \operatorname{deg} x=2\),
\(\operatorname{deg} y=3\).

\hypertarget{ii.4.2.}{%
\subsubsection{II.4.2.}\label{ii.4.2.}}

If \(D\) is any divisor of degree \(\geqslant 3\) on the elliptic curve
\(X\), and if we embed \(X\) in \(\mathbf{P}^n\) by the complete linear
system \(|D|\), show that the image of \(X\) in \(\mathbf{P}^n\) is
projectively normal.\footnote{Note. It is true more generally that if
  \(D\) is a divisor of degree \(\geqslant 2 g+1\) on a curve of genus
  \(g\), then the embedding of \(X\) by \(|D|\) is projectively normal
  (Mumford \([4\), p.~55\(])\)}

\hypertarget{ii.4.3.}{%
\subsubsection{II.4.3.}\label{ii.4.3.}}

Let the elliptic curve \(X\) be embedded in \(\mathbf{P}^2\) so as to
have the equation \(y^2=\) \(x(x-1)(x-\lambda)\). Show that any
automorphism of \(X\) leaving \(P_0=(0,1,0)\) fixed is induced by an
automorphism of \(\mathbf{P}^2\) coming from the automorphism of the
affine \((x, y)\)-plane given by In each of the four cases of (4.7),
describe these automorphisms of \(\mathbf{P}^2\) explicitly, and hence
determine the structure of the group
\(G=\operatorname{Aut}\left(X, P_0\right)\).

\hypertarget{ii.4.4.}{%
\subsubsection{II.4.4.}\label{ii.4.4.}}

Let \(X\) be an elliptic curve in \(\mathbf{P}^2\) given by an equation
of the form Show that the \(j\)-invariant is a rational function of the
\(a_i\), with coefficients in \(\mathbf{Q}\). In particular, if the
\(a_i\) are all in some field \(k_0 \subseteq k\), then \(j \in k_0\)
also. Furthermore, for every \(\alpha \in k_0\), there exists an
elliptic curve defined over \(k_0\) with \(j\)-invariant equal to
\(\alpha\).

\hypertarget{ii.4.5.}{%
\subsubsection{II.4.5.}\label{ii.4.5.}}

Let \(X, P_0\) be an elliptic curve having an endomorphism
\(f: X \rightarrow X\) of degree 2 .

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If we represent \(X\) as a 2-1 covering of \(\mathbf{P}^1\) by a
  morphism \(\pi: X \rightarrow \mathbf{P}^1\) ramified at \(P_0\), then
  as in (4.4), show that there is another morphism
  \(\pi^{\prime}: X \rightarrow \mathbf{P}^1\) and a morphism
  \(g: \mathbf{P}^1 \rightarrow \mathbf{P}^1\), also of degree 2 , such
  that \(\pi \circ f=g \circ \pi^{\prime}\).
\item
  For suitable choices of coordinates in the two copies of
  \(\mathbf{P}^1\), show that \(g\) can be taken to be the morphism
  \(x \rightarrow x^2\).
\item
  Now show that \(g\) is branched over two of the branch points of
  \(\pi\), and that \(g^{-1}\) of the other two branch points of \(\pi\)
  consists of the four branch points of \(\pi^{\prime}\). Deduce a
  relation involving the invariant \(\lambda\) of \(X\).
\item
  Solving the above, show that there are just three values of \(j\)
  corresponding to elliptic curves with an endomorphism of degree 2 ,
  and find the corresponding values of \(\lambda\) and \(j\).\footnote{Answers:
    \(j=2^6 \cdot 3^3 ; j=2^6 \cdot 5^3 ; j=-3^3 \cdot 5^3\).}
\end{enumerate}

\hypertarget{ii.4.6.}{%
\subsubsection{II.4.6.}\label{ii.4.6.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(X\) be a curve of genus \(g\) embedded birationally in
  \(\mathbf{P}^2\) as a curve of degree \(d\) with \(r\) nodes.
  Generalize the method of (Ex. 2.3) to show that \(X\) has inflection
  points. A node does not count as an inflection point. Assume char
  \(k=0\).
\item
  Now let \(X\) be a curve of genus \(g\) embedded as a curve of degree
  \(d\) in \(\mathbf{P}^n, n \geqslant 3\), not contained in any
  \(\mathbf{P}^{n-1}\). For each point \(P \in X\), there is a
  hyperplane \(H\) containing \(P\), such that \(P\) counts at least
  \(n\) times in the intersection \(H \cap X\). This is called an
  \textbf{osculating} hyperplane at \(P\). It generalizes the notion of
  tangent line for curves in \(\mathbf{P}^2\).

  If \(P\) counts at least \(n+1\) times in \(H \cap X\), we say \(H\)
  is a \textbf{hyperosculating hyperplane}, and that \(P\) is a
  \textbf{hyperosculation point}. Use Hurwitz's theorem as above, and
  induction on \(n\), to show that \(X\) has hyperosculation points.
\item
  If \(X\) is an elliptic curve, for any \(d \geqslant 3\), embed \(X\)
  as a curve of degree \(d\) in \(\mathbf{P}^{d-1}\), and conclude that
  \(X\) has exactly \(d^2\) points of order \(d\) in its group law.
\end{enumerate}

\hypertarget{ii.4.7.-the-dual-of-a-morphism.}{%
\subsubsection{II.4.7. The Dual of a
Morphism.}\label{ii.4.7.-the-dual-of-a-morphism.}}

Let \(X\) and \(X^{\prime}\) be elliptic curves over \(k\), with base
points \(P_0, P_0^{\prime}\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(f: X \rightarrow X^{\prime}\) is any morphism, use (4.11) to show
  that \(f^*:\) Pic \(X^{\prime} \rightarrow\) Pic \(X\) induces a
  homomorphism
  \(\widehat{f}:\left(X^{\prime}, P_0^{\prime}\right) \rightarrow\left(X, P_0\right)\).
  We call this the \textbf{dual} of \(f\).
\item
  If \(f: X \rightarrow X^{\prime}\) and
  \(g: X^{\prime} \rightarrow X^{\prime \prime}\) are two morphisms,
  then \((g \circ f)\widehat{} = \widehat{f} \circ \widehat{g}\).
\item
  Assume \(f\left(P_0\right)=P_0^{\prime}\), and let
  \(n=\operatorname{deg} f\). Show that if \(Q \in X\) is any point, and
  \(f(Q)=Q^{\prime}\), then
  \(\widehat{f}\left(Q^{\prime}\right)=n_X(Q)\). (Do the separable and
  purely inseparable cases separately, then combine.) Conclude that
  \(f \circ \widehat{f}=n_{X^{\prime}}\) and
  \(\widehat{f} \circ f=n_X\).
\item
  * If \(f, g: X \rightarrow X^{\prime}\) are two morphisms preserving
  the base points \(P_0, P_0^{\prime}\), then
  \((f+g) \widehat{} = \widehat{f}+\widehat{g}\).\footnote{Hints: It is
    enough to show for any \(\mathcal{L} \in\) Pic \(X^{\prime}\), that
    \((f+g)^* \mathcal{L} \cong f^* \mathcal{L} \otimes g^* \mathcal{L}\).
    For any \(f\), let \(\Gamma_f: X \rightarrow X \times X^{\prime}\)
    be the graph morphism. Then it is enough to show (for
    \(\mathcal{L}^{\prime}=p_2^* \mathcal{L}\) ) that Let
    \(\sigma: X \rightarrow X \times X^{\prime}\) be the section
    \(x \rightarrow\left(x, P_0^{\prime}\right)\). Define a subgroup of
    \(\operatorname{Pic}\left(X \times X^{\prime}\right)\) as follows:
    Note that this subgroup is isomorphic to the group
    \(\operatorname{Pic}^{\circ}\left(X^{\prime} / X\right)\) used in
    the definition of the Jacobian variety. Hence there is a 1-1
    correspondence between morphisms \(f: X \rightarrow X^{\prime}\) and
    elements \(\mathcal{L}_f \in\) Pic \(_\sigma\) (this defines
    \(\mathcal{L}_f\) ). Now compute explicitly to show that
    \(\Gamma_g^*\left(\mathcal{L}_f\right)=\Gamma_f^*\left(\mathcal{L}_g\right)\)
    for any \(f, g\).\\
    ~\\
    Use the fact that
    \(\mathcal{L}_{f+g}=\mathcal{L}_f \otimes \mathcal{L}_g\), and the
    fact that for any \(\mathcal{L}\) on \(X^{\prime}\),
    \(p_2^* \mathcal{L} \in \mathrm{Pic}_\sigma^{\circ}\) to prove the
    result.}
\end{enumerate}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{4}
\item
  Using (d), show that for any \(n \in \mathbf{Z}, \widehat{n}_X=n_X\).
  Conclude that deg \(n_X=n^2\).
\item
  Show for any \(f\) that
  \(\operatorname{deg} \widehat{f}=\operatorname{deg} f\).
\end{enumerate}

\hypertarget{ii.4.8.-the-algebraic-fundamental-group.}{%
\subsubsection{II.4.8. The Algebraic Fundamental
Group.}\label{ii.4.8.-the-algebraic-fundamental-group.}}

For any curve \(X\), the \textbf{algebraic fundamental group}
\(\pi_1(X)\) is defined as
\(\cocolim \operatorname{Gal}\left(K^{\prime} / K\right)\), where \(K\)
is the function field of \(X\), and \(K^{\prime}\) runs over all Galois
extensions of \(K\) such that the corresponding curve \(X^{\prime}\) is
étale over \(X\) (III, Ex. 10.3).

Thus, for example, \(\pi_1\left(\mathbf{P}^1\right)=1\). (See 2.5.3)

Show that for an elliptic curve \(X\), where
\(\mathbf{Z}_l=\lim \mathbf{Z} / l^n\) is the \(l\)-adic
integers.\footnote{Hints: Any Galois étale cover \(X^{\prime}\) of an
  elliptic curve is again an elliptic curve. If the degree of
  \(X^{\prime}\) over \(X\) is relatively prime to \(p\), then
  \(X^{\prime}\) can be dominated by the cover \(n_X: X \rightarrow X\)
  for some integer \(n\) with \((n, p)=1\). The Galois group of the
  covering \(n_X\) is \(\mathbf{Z} / n \times \mathbf{Z} / n\). Étale
  covers of degree divisible by \(p\) can occur only if the Hasse
  invariant of \(X\) is not zero.}\footnote{Note: More generally,
  Grothendieck has shown \([\mathrm{SGA} 1, X, 2.6, p. 272]\) that the
  algebraic fundamental group of any curve of genus \(g\) is isomorphic
  to a quotient of the completion, with respect to subgroups of finite
  index, of the ordinary topological fundamental group of a compact
  Riemann surface of genus \(g\), i.e., a group with \(2 g\) generators
  \(a_1, \ldots, a_g, b_1, \ldots, b_g\) and the relation
  \(\left(a_1 b_1 a_1^{-1} b_1^{-1}\right) \cdots\)
  \(\left(a_g b_g a_g^{-1} b_g^{-1}\right)=1\).}

\hypertarget{ii.4.9.-isogenies.}{%
\subsubsection{II.4.9. Isogenies.}\label{ii.4.9.-isogenies.}}

We say two elliptic curves \(X, X^{\prime}\) are \textbf{isogenous} if
there is a finite morphism \(f: X \rightarrow X^{\prime}\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that isogeny is an equivalence relation.
\item
  For any elliptic curve \(X\), show that the set of elliptic curves
  \(X^{\prime}\) isogenous to \(X\), up to isomorphism, is
  countable.\footnote{Hint: \(X^{\prime}\) is uniquely determined by
    \(X\) and \(\operatorname{ker} f\).}
\end{enumerate}

\hypertarget{ii.4.10.}{%
\subsubsection{II.4.10.}\label{ii.4.10.}}

If \(X\) is an elliptic curve, show that there is an exact sequence
where \(R=\operatorname{End}\left(X, P_0\right)\). In particular, we see
that \(\operatorname{Pic}(X \times X)\) is bigger than the sum of the
Picard groups of the factors.\footnote{Cf. (III, Ex. 12.6), (V, Ex.
  1.6).}

\hypertarget{ii.4.11.}{%
\subsubsection{II.4.11.}\label{ii.4.11.}}

Let \(X\) be an elliptic curve over \(\mathbf{C}\), defined by the
elliptic functions with periods \(1, \tau\). Let \(R\) be the ring of
endomorphisms of \(X\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(f \in R\) is a nonzero endomorphism corresponding to complex
  multiplication by \(\alpha\), as in (4.18), show that
  \(\operatorname{deg} f=|\alpha|^2\).
\item
  If \(f \in R\) corresponds to \(\alpha \in \mathbf{C}\) again, show
  that the dual \(\widehat{f}\) of (Ex. 4.7) corresponds to the complex
  conjugate
  \(\mkern 1.5mu\overline{\mkern-1.5mu\alpha\mkern-1.5mu}\mkern 1.5mu\)
  of \(\alpha\).
\item
  If \(\tau \in \mathbf{Q}(\sqrt{-d})\) happens to be integral over
  \(\mathbf{Z}\), show that \(R=\mathbf{Z}[\tau]\).
\end{enumerate}

\hypertarget{ii.4.12.}{%
\subsubsection{II.4.12.}\label{ii.4.12.}}

Again let \(X\) be an elliptic curve over \(\mathbf{C}\) determined by
the elliptic functions with periods \(1, \tau\), and assume that
\(\tau\) lies in the region \(G\) of (4.15B).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(X\) has any automorphisms leaving \(P_0\) fixed other than
  \(\pm 1\), show that either \(\tau=i\) or \(\tau=\omega\), as in
  (4.20.1) and (4.20.2). This gives another proof of the fact (4.7) that
  there are only two curves, up to isomorphism, having automorphisms
  other than \(\pm 1\).
\item
  Now show that there are exactly three values of \(\tau\) for which
  \(X\) admits an endomorphism of degree 2. Can you match these with the
  three values of \(j\) determined in (Ex. 4.5)?\footnote{Answers:
    \(\tau=i ; \tau=\sqrt{-2} ; \tau=\frac{1}{2}(-1+\sqrt{-7})\).}
\end{enumerate}

\hypertarget{ii.4.13.}{%
\subsubsection{II.4.13.}\label{ii.4.13.}}

If \(p=13\), there is just one value of \(j\) for which the Hasse
invariant of the corresponding curve is 0 . Find it.\footnote{Answer:
  \(j=5(\bmod 13)\).}

\hypertarget{ii.4.14.}{%
\subsubsection{II.4.14.}\label{ii.4.14.}}

The Fermat curve \(X: x^3+y^3=z^3\) gives a nonsingular curve in
characteristic \(p\) for every \(p \neq 3\). Determine the set
\(\mathfrak{P}=\left\{p \neq 3 \mathrel{\Big|}X_{(p)}\right.\) has Hasse
invariant 0\(\}\), and observe (modulo Dirichlet's theorem) that it is a
set of primes of density \(\frac{1}{2}\).

\hypertarget{ii.4.15.}{%
\subsubsection{II.4.15.}\label{ii.4.15.}}

Let \(X\) be an elliptic curve over a field \(k\) of characteristic
\(p\). Let \(F^{\prime}: X_p \rightarrow X\) be the \(k\)-linear
Frobenius morphism (2.4.1). Use (4.10.7) to show that the dual morphism
\(\widehat{F}^{\prime}: X \rightarrow X_p\) is separable if and only if
the Hasse invariant of \(X\) is 1 .

Now use (Ex. 4.7) to show that if the Hasse invariant is 1, then the
subgroup of points of order \(p\) on \(X\) is isomorphic to
\(\mathbf{Z} / p\); if the Hasse invariant is 0 , it is 0 .

\hypertarget{ii.4.16.}{%
\subsubsection{II.4.16.}\label{ii.4.16.}}

Again let \(X\) be an elliptic curve over \(k\) of characteristic \(p\),
and suppose \(X\) is defined over the field \(\mathbf{F}_q\) of
\(q=p^r\) elements, i.e., \(X \subseteq \mathbf{P}^2\) can be defined by
an equation with coefficients in \(\mathbf{F}_q\). Assume also that
\(X\) has a rational point over \(\mathbf{F}_q\). Let
\(F^{\prime}: X_q \rightarrow X\) be the \(k\)-linear Frobenius with
respect to \(q\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(X_q \cong X\) as schemes over \(k\), and that under this
  identification, \(F^{\prime}: X \rightarrow X\) is the map obtained by
  the \(q\) th-power map on the coordinates of points of \(X\), embedded
  in \(\mathbf{P}^2\).
\item
  Show that \(1_X-F^{\prime}\) is a separable morphism and its kernel is
  just the set \(X\left(\mathbf{F}_q\right)\) of points of \(X\) with
  coordinates in \(\mathbf{F}_q\).
\item
  Using (Ex. 4.7), show that \(F^{\prime}+\widehat{F}^{\prime}=a_X\) for
  some integer \(a\), and that \(N=\) \(q-a+1\), where
  \(N={\sharp}X\left(\mathbf{F}_q\right)\).
\item
  Use the fact that \(\operatorname{deg}\left(m+n F^{\prime}\right)>0\)
  for all \(m, n \in \mathbf{Z}\) to show that
  \(|a| \leqslant 2 \sqrt{q}\). This is Hasse's proof of the analogue of
  the Riemann hypothesis for elliptic curves (App. C, Ex. 5.6).
\item
  Now assume \(q=p\), and show that the Hasse invariant of \(X\) is 0 if
  and only if \(a \equiv 0(\bmod p)\). Conclude for \(p \geqslant 5\)
  that \(X\) has Hasse invariant 0 if and only if \(N=p+1\).
\end{enumerate}

\hypertarget{ii.4.17.}{%
\subsubsection{II.4.17.}\label{ii.4.17.}}

Let \(X\) be the curve \(y^2+y=x^3-x\) of \((4.23 .8)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(Q=(a, b)\) is a point on the curve, compute the coordinates of
  the point \(P+Q\), where \(P=(0,0)\), as a function of \(a, b\). Use
  this formula to find the coordinates of
  \(n P, n=1,2, \ldots, 10\).\footnote{Check: \(6P = (6,14)\)}
\item
  This equation defines a nonsingular curve over \(\mathbf{F}_p\) for
  all \(p \neq 37\).
\end{enumerate}

\hypertarget{ii.4.18.}{%
\subsubsection{II.4.18.}\label{ii.4.18.}}

Let \(X\) be the curve \(y^2=x^3-7 x+10\). This curve has at least 26
points with integer coordinates. Find them (use a calculator), and
verify that they are all contained in the subgroup (maybe equal to all
of \(X(\mathbf{Q})\)?) generated by \(P=(1,2)\) and \(Q=(2,2)\).

\hypertarget{ii.4.19.}{%
\subsubsection{II.4.19.}\label{ii.4.19.}}

Let \(X, P_0\) be an elliptic curve defined over \(\mathbf{Q}\),
represented as a curve in \(\mathbf{P}^2\) defined by an equation with
integer coefficients. Then \(X\) can be considered as the fibre over the
generic point of a scheme
\(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) over
Spec \(\mathbf{Z}\). Let \(T \subseteq \operatorname{Spec} \mathbf{Z}\)
be the open subset consisting of all primes \(p \neq 2\) such that the
fibre \(X_{(p)}\) of
\(\mkern 1.5mu\overline{\mkern-1.5muX\mkern-1.5mu}\mkern 1.5mu\) over
\(p\) is nonsingular.

\begin{itemize}
\tightlist
\item
  For any \(n\), show that \(n_X: X \rightarrow X\) is defined over
  \(T\), and is a flat morphism.
\item
  Show that the kernel of \(n_X\) is also flat over \(T\).
\item
  Conclude that for any \(p \in T\), the natural map
  \(X(\mathbf{Q}) \rightarrow X_{(p)}\left(\mathbf{F}_p\right)\) induced
  on the groups of rational points, maps the \(n\)-torsion points of
  \(X(\mathbf{Q})\) injectively into the torsion subgroup of
  \(X_{(p)}\left(\mathbf{F}_p\right)\), for any \((n, p)=1\).
\end{itemize}

By this method one can show easily that the groups \(X(\mathbf{Q})\) in
(Ex. 4.17) and (Ex. 4.18) are torsion-free.

\hypertarget{ii.4.20.}{%
\subsubsection{II.4.20.}\label{ii.4.20.}}

Let \(X\) be an elliptic curve over a field \(k\) of characteristic
\(p>0\), and let \(R=\) \(\operatorname{End}\left(X, P_0\right)\) be its
ring of endomorphisms.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(X_p\) be the curve over \(k\) defined by changing the
  \(k\)-structure of \(X\) (2.4.1). Show that
  \(j\left(X_p\right)=j(X)^{1 / p}\). Thus \(X \cong X_p\) over \(k\) if
  and only if \(j \in \mathbf{F}_p\).
\item
  Show that \(p_X\) in \(R\) factors into a product
  \(\pi \widehat{\pi}\) of two elements of degree \(p\) if and only if
  \(X \cong X_p\). In this case, the Hasse invariant of \(X\) is 0 if
  and only if \(\pi\) and \(\widehat{\pi}\) are associates in \(R\)
  (i.e., differ by a unit). (Use (2.5).)
\item
  If \(\operatorname{Hasse}(X)=0\) show in any case
  \(j \in \mathbf{F}_{p^2}\).
\item
  For any \(f \in R\), there is an induced map
  \(f^*: H^1\left(\mathcal{O}_X\right) \rightarrow H^1\left(\mathcal{O}_X\right)\).
  This must be multiplication by an element \(\lambda_f \in k\). So we
  obtain a ring homomorphism \(\varphi: R \rightarrow k\) by sending
  \(f\) to \(\lambda_f\). Show that any \(f \in R\) commutes with the
  (nonlinear) Frobenius morphism \(F: X \rightarrow X\), and conclude
  that if Hasse \((X) \neq 0\), then the image of \(\varphi\) is
  \(\mathbf{F}_p\). Therefore, \(R\) contains a prime ideal
  \(\mathfrak{p}\) with \(R / p \cong { \mathbf{F} }_p\).
\end{enumerate}

\hypertarget{ii.4.21.}{%
\subsubsection{II.4.21.}\label{ii.4.21.}}

Let \(O\) be the ring of integers in a quadratic number field
\(\mathbf{Q}(\sqrt{-d})\). Show that any subring
\(R \subseteq O, R \neq \mathbf{Z}\), is of the form
\(R=\mathbf{Z}+f \cdot O\), for a uniquely determined integer
\(f \geqslant 1\). This integer \(f\) is called the \textbf{conductor}
of the ring \(R\).

\hypertarget{ii.4.22-.}{%
\subsubsection{II.4.22 *.}\label{ii.4.22-.}}

If \(X \rightarrow \mathbf{A}_{\mathbf{C}}^1\) is a family of elliptic
curves having a section, show that the family is trivial.\footnote{Hints:
  Use the section to fix the group structure on the fibres. Show that
  the points of order 2 on the fibres form an étale cover of
  \(\mathbf{A}_{\mathbf{C}}^1\), which must be trivial, since
  \(\mathbf{A}_{\mathbf{C}}^1\) is simply connected. This implies that
  \(\lambda\) can be defined on the family, so it gives a map
  \(\mathbf{A}_{\mathbf{C}}^1 \rightarrow \mathbf{A}_{\mathbf{C}}^1-\{0,1\}\).
  Any such map is constant, so \(\lambda\) is constant, so the family is
  trivial.}

\hypertarget{iv.5-the-canonical-embedding}{%
\subsection{IV.5: The Canonical
Embedding}\label{iv.5-the-canonical-embedding}}

\hypertarget{iv.5.1.}{%
\subsubsection{IV.5.1.}\label{iv.5.1.}}

Show that a hyperelliptic curve can never be a complete intersection in
any projective space. Cf. (Ex. 3.3).

\hypertarget{iv.5.2.}{%
\subsubsection{IV.5.2.}\label{iv.5.2.}}

If \(X\) is a curve of genus \(\geqslant 2\) over a field of
characteristic 0 , show that the group \(\mathop{\mathrm{Aut}}X\) of
automorphisms of \(X\) is finite.\footnote{Hint: If \(X\) is
  hyperelliptic, use the unique \(g_2^1\) and show that
  \(\mathop{\mathrm{Aut}}X\) permutes the ramification points of the 2
  -fold covering \(X \rightarrow \mathbf{P}^1\). If \(X\) is not
  hyperelliptic, show that \(\mathop{\mathrm{Aut}}X\) permutes the
  hyperosculation points (Ex. 4.6) of the canonical embedding. Cf. (Ex.
  2.5).}

\hypertarget{iv.5.3.-moduli-of-curves-of-genus-4.}{%
\subsubsection{IV.5.3. Moduli of Curves of Genus
4.}\label{iv.5.3.-moduli-of-curves-of-genus-4.}}

The hyperelliptic curves of genus 4 form an irreducible family of
dimension 7 . The nonhyperelliptic ones form an irreducible family of
dimension 9. The subset of those having only one \(g_3^1\) is an
irreducible family of dimension 8.\footnote{Hint: Use (5.2.2) to count
  how many complete intersections \(Q \cap F_3\) there are.}

\hypertarget{iv.5.4.}{%
\subsubsection{IV.5.4.}\label{iv.5.4.}}

Another way of distinguishing curves of genus \(g\) is to ask, what is
the least degree of a birational plane model with only nodes as
singularities (3.11)? Let \(X\) be nonhyperelliptic of genus 4 . Then:

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  if \(X\) has two \(g_3^1\),s, it can be represented as a plane quintic
  with two nodes, and conversely;
\item
  if \(X\) has one \(g_3^1\), then it can be represented as a plane
  quintic with a tacnode (I, Ex. 5.14d), but the least degree of a plane
  representation with only nodes is 6 .
\end{enumerate}

\hypertarget{iv.5.5.-curves-of-genus-5.}{%
\subsubsection{IV.5.5. Curves of Genus
5.}\label{iv.5.5.-curves-of-genus-5.}}

Assume \(X\) is not hyperelliptic.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  The curves of genus 5 whose canonical model in \(\mathbf{P}^4\) is a
  complete intersection \(F_2 . F_2 . F_2\) form a family of dimension
  12 .
\item
  \(X\) has a \(g_3^1\) if and only if it can be represented as a plane
  quintic with one node. These form an irreducible family of dimension
  11.\footnote{Hint: If \(D \in g_3^1\), use \(K-D\) to
    \(\operatorname{map} X \rightarrow \mathbf{P}^2\).}
\item
  * In that case, the conics through the node cut out the canonical
  system (not counting the fixed points at the node). Mapping
  \(\mathbf{P}^2 \rightarrow \mathbf{P}^4\) by this linear system of
  conics, show that the canonical curve \(X\) is contained in a cubic
  surface \(V \subseteq \mathbf{P}^4\), with \(V\) isomorphic to
  \(\mathbf{P}^2\) with one point blown up (II, Ex. 7.7).

  Furthermore, \(V\) is the union of all the trisecants of \(X\)
  corresponding to the \(g_3^1(5.5 .3)\), so \(V\) is contained in the
  intersection of all the quadric hypersurfaces containing \(X\). Thus
  \(V\) and the \(g_3^1\) are unique.\footnote{Note. Conversely, if
    \(X\) does not have a \(g_3^1\), then its canonical embedding is a
    complete intersection, as in (a). More generally, a classical
    theorem of Enriques and Petri shows that for any nonhyperelliptic
    curve of genus \(g \geqslant 3\), the canonical model is
    projectively normal, and it is an intersection of quadric
    hypersurfaces unless \(X\) has a \(g_3^1\) or \(g=6\) and \(X\) has
    a \(g_5^2\). See Saint-Donat \([1]\).}
\end{enumerate}

\hypertarget{iv.5.6.}{%
\subsubsection{IV.5.6.}\label{iv.5.6.}}

Show that a nonsingular plane curve of degree 5 has no \(g_3^1\). Show
that there are nonhyperelliptic curves of genus 6 which cannot be
represented as a nonsingular plane quintic curve.

\hypertarget{iv.5.7.}{%
\subsubsection{IV.5.7.}\label{iv.5.7.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Any automorphism of a curve of genus 3 is induced by an automorphism
  of \(\mathbf{P}^2\) via the canonical embedding.
\item
  * Assume char \(k \neq 3\). If \(X\) is the curve given by the group
  \(\mathop{\mathrm{Aut}}X\) is the simple group of order 168 , whose
  order is the maximum \(84(g-1)\) allowed by (Ex. 2.5).\footnote{See
    Burnside \([1, \S 232]\) or Klein \([1]\).}
\item
  * Most curves of genus 3 have no automorphisms except the
  identity.\footnote{Hint: For each \(n\), count the dimension of the
    family of curves with an automorphism \(T\) of order \(n\). For
    example, if \(n=2\), then for suitable choice of coordinates, \(T\)
    can be written as
    \(x \rightarrow-x, y \rightarrow y, z \rightarrow z\). Then there is
    an 8-dimensional family of curves fixed by \(T\); changing
    coordinates there is a 4-dimensional family of such \(T\), so the
    curves having an automorphism of degree 2 form a family of
    dimensional 12 inside the 14-dimensional family of all plane curves
    of degree 4.}\footnote{More generally it is true (at least over
    \(\mathbf{C}\) ) that for any \(g \geqslant 3\), a ``sufficiently
    general'' curve of genus \(g\) has no automorphisms except the
    identity-see Baily \[1\].}
\end{enumerate}

\hypertarget{iv.6-classification-of-curves-in-mathbfp3}{%
\subsection{\texorpdfstring{IV.6: Classification of Curves in
\({\mathbf{P}}^3\)}{IV.6: Classification of Curves in \{\textbackslash mathbf\{P\}\}\^{}3}}\label{iv.6-classification-of-curves-in-mathbfp3}}

\hypertarget{iv.6.1.}{%
\subsubsection{IV.6.1.}\label{iv.6.1.}}

A rational curve of degree 4 in \(\mathbf{P}^3\) is contained in a
unique quadric surface \(Q\), and \(Q\) is necessarily nonsingular.

\hypertarget{iv.6.2.}{%
\subsubsection{IV.6.2.}\label{iv.6.2.}}

A rational curve of degree 5 in \(\mathbf{P}^3\) is always contained in
a cubic surface, but there are such curves which are not contained in
any quadric surface.

\hypertarget{iv.6.3.}{%
\subsubsection{IV.6.3.}\label{iv.6.3.}}

A curve of degree 5 and genus 2 in \(\mathbf{P}^3\) is contained in a
unique quadric surface \(Q\). Show that for any abstract curve \(X\) of
genus 2 , there exist embeddings of degree 5 in \(\mathbf{P}^3\) for
which \(Q\) is nonsingular, and there exist other embeddings of degree 5
for which \(Q\) is singular.

\hypertarget{iv.6.4.}{%
\subsubsection{IV.6.4.}\label{iv.6.4.}}

There is no curve of degree 9 and genus 11 in
\(\mathbf{P}^3\).\footnote{Hint: Show that it would have to lie on a
  quadric surface, then use (6.4.1).}

\hypertarget{iv.6.5.}{%
\subsubsection{IV.6.5.}\label{iv.6.5.}}

If \(X\) is a complete intersection of surfaces of degrees \(a, b\) in
\(\mathbf{P}^3\), then \(X\) does not lie on any surface of degree
\(<\min (a, b)\).

\hypertarget{iv.6.6.}{%
\subsubsection{IV.6.6.}\label{iv.6.6.}}

Let \(X\) be a projectively normal curve in \(\mathbf{P}^3\), not
contained in any plane. If \(d=6\), then \(g=3\) or 4 . If \(d=7\), then
\(g=5\) or 6 . Cf. (II, Ex. 8.4) and (III, Ex. 5.6).

\hypertarget{iv.6.7.}{%
\subsubsection{IV.6.7.}\label{iv.6.7.}}

The line, the conic, the twisted cubic curve and the elliptic quartic
curve in \(\mathbf{P}^3\) have no multisecants. Every other curve in
\(\mathbf{P}^3\) has infinitely many multisecants.\footnote{Hint:
  Consider a projection from a point of the curve to \(\mathbf{P}^2\).}

\hypertarget{iv.6.8.}{%
\subsubsection{IV.6.8.}\label{iv.6.8.}}

A curve \(X\) of genus \(g\) has a nonspecial divisor \(D\) of degree
\(d\) such that \(|D|\) has no base points if and only if
\(d \geqslant g+1\).

\hypertarget{iv.6.9.}{%
\subsubsection{IV.6.9.}\label{iv.6.9.}}

* Let \(X\) be an irreducible nonsingular curve in \(\mathbf{P}^3\).
Then for each \(m \gg>0\), there is a nonsingular surface \(F\) of
degree \(m\) containing \(X\).\footnote{Hint: Let
  \(\pi: \tilde{\mathbf{P}} \rightarrow \mathbf{P}^3\) be the blowing-up
  of \(X\) and let \(Y=\pi^{-1}(X)\). Apply Bertini's theorem to the
  projective embedding of \(\tilde{\mathbf{P}}\) corresponding to
  \(\mathcal{I}_Y \otimes \pi^* \mathcal{O}_{\mathbf{P}^3}(m)\).}

\hypertarget{v-surfaces}{%
\section{V: Surfaces}\label{v-surfaces}}

\hypertarget{v.1-geometry-on-a-surface}{%
\subsection{V.1: Geometry on a
Surface}\label{v.1-geometry-on-a-surface}}

\hypertarget{v.1.1.}{%
\subsubsection{V.1.1.}\label{v.1.1.}}

Let \(C, D\) be any two divisors on a surface \(X\), and let the
corresponding invertible sheaves be \(\mathcal{L}, \mathcal{M}\). Show
that

\hypertarget{v.1.2.}{%
\subsubsection{V.1.2.}\label{v.1.2.}}

Let \(H\) be a very ample divisor on the surface \(X\), corresponding to
a projective embedding \(X \subseteq \mathbf{P}^N\). If we write the
Hilbert polynomial of \(X\) (III, Ex. 5.2) as show that
\(a=H^2, b=\frac{1}{2} H^2+1-\pi\), where \(\pi\) is the genus of a
nonsingular curve representing \(H\), and \(c=1+p_a\). Thus the degree
of \(X\) in \(\mathbf{P}^N\), as defined in (I, §7), is just \(H^2\).
Show also that if \(C\) is any curve in \(X\), then the degree of \(C\)
in \(\mathbf{P}^N\) is just \(C . H\)

\hypertarget{v.1.3.}{%
\subsubsection{V.1.3.}\label{v.1.3.}}

Recall that the arithmetic genus of a projective scheme \(D\) of
dimension 1 is defined as See III, Ex. 5.3.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(D\) is an effective divisor on the surface \(X\), use (1.6) to
  show that
\item
  \(p_a(D)\) depends only on the linear equivalence class of \(D\) on
  \(X\).
\item
  More generally, for any divisor \(D\) on \(X\), we define the virtual
  arithmetic genus (which is equal to the ordinary arithmetic genus if
  \(D\) is effective) by the same formula: \(2 p_a-2=D .(D+K)\). Show
  that for any two divisors \(C, D\) we have and
\end{enumerate}

\hypertarget{v.1.4.}{%
\subsubsection{V.1.4.}\label{v.1.4.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If a surface \(X\) of degree \(d\) in \(\mathbf{P}^3\) contains a
  straight line \(C=\mathbf{P}^1\), show that \(C^2=2-d\)
\item
  Assume char \(k=0\), and show for every \(d \geqslant 1\), there
  exists a nonsingular surface \(X\) of degree \(d\) in \(\mathbf{P}^3\)
  containing the line \(x=y=0\).
\end{enumerate}

\hypertarget{v.1.5.}{%
\subsubsection{V.1.5.}\label{v.1.5.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(X\) is a surface of degree \(d\) in \(\mathbf{P}^3\), then
\item
  If \(X\) is a product of two nonsingular curves \(C, C^{\prime}\), of
  genus \(g, g^{\prime}\) respectively, then Cf. (II, Ex. 8.3).
\end{enumerate}

\hypertarget{v.1.6.}{%
\subsubsection{V.1.6.}\label{v.1.6.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(C\) is a curve of genus \(g\), show that the diagonal
  \(\Delta \subseteq C \times C\) has self-intersection
  \(\Delta^2=2-2 g\). (Use the definition of \(\Omega_{C / k}\) in (II,
  §8).)
\item
  Let \(l=C \times \mathrm{pt}\) and \(m=\mathrm{pt} \times C\). If
  \(g \geqslant 1\), show that \(l, m\), and \(\Delta\) are linearly
  independent in \(\operatorname{Num}(C \times C)\). Thus
  \(\operatorname{Num}(C \times C)\) has rank \(\geqslant 3\), and in
  particular, Cf. (III, Ex. 12.6), (V, Ex. 4.10).
\end{enumerate}

\hypertarget{v.1.7.-algebraic-equivalence-of-divisors.}{%
\subsubsection{V.1.7. Algebraic Equivalence of
Divisors.}\label{v.1.7.-algebraic-equivalence-of-divisors.}}

Let \(X\) be a surface. Recall that we have defined an algebraic family
of effective divisors on \(X\), parametrized by a nonsingular curve
\(T\), to be an effective Cartier divisor \(D\) on \(X \times T\), flat
over \(T\) (III, 9.8.5). In this case, for any two closed points
\(0,1 \in T\), we say the corresponding divisors \(D_0, D_1\) on \(X\)
are prealgebraically equivalent.

Two arbitrary divisors are prealgebraically equivalent if they are
differences of prealgebraically equivalent effective divisors. Two
divisors \(D, D^{\prime}\) are algebraically equivalent if there is a
finite sequence \(D=D_0, D_1, \ldots, D_n=D^{\prime}\) with \(D_i\) and
\(D_{i+1}\) prealgebraically equivalent for each \(i\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that the divisors algebraically equivalent to 0 form a subgroup
  of Div \(X\).
\item
  Show that linearly equivalent divisors are algebraically
  equivalent.\footnote{Hint: If \((f)\) is a principal divisor on \(X\),
    consider the principal divisor \((t f-u)\) on
    \(X \times \mathbf{P}^1\), where \(t, u\) are the homogeneous
    coordinates on \(\mathbf{P}^1\).}
\item
  Show that algebraically equivalent divisors are numerically
  equivalent.\footnote{Hint: Use (III, 9.9) to show that for any very
    ample \(H\), if \(D\) and \(D^{\prime}\) are algebraically
    equivalent, then \(D . H=D^{\prime} . H\).}\footnote{Note. The
    theorem of Néron and Severi states that the group of divisors modulo
    algebraic equivalence, called the Néron-Severi group, is a finitely
    generated abelian group. Over \(\mathbf{C}\) this can be proved
    easily by transcendental methods (App. B, \(\S 5\) ) or as in (Ex.
    1.8) below. Over a field of arbitrary characteristic, see Lang and
    Néron \[1\] for a proof, and Hartshorne \[6\] for further
    discussion. Since \(\operatorname{Num}X\) is a quotient of the
    Néron-Severi group, it is also finitely generated, and hence free,
    since it is torsion-free by construction.}
\end{enumerate}

\hypertarget{v.1.8.-cohomology-class-of-a-divisor.}{%
\subsubsection{V.1.8. Cohomology Class of a
Divisor.}\label{v.1.8.-cohomology-class-of-a-divisor.}}

For any divisor \(D\) on the surface \(X\), we define its cohomology
class \(c(D) \in H^1\left(X, \Omega_X\right)\) by using the isomorphism
Pic \(X \cong\) \(H^1\left(X, \mathcal{O}_X^*\right.\) ) of (III, Ex.
4.5) and the sheaf homomorphism
\(d \log : \mathcal{O}^* \rightarrow \Omega_X\) (III, Ex. 7.4c). Thus we
obtain a group homomorphism
\(c: \operatorname{Pic} X \rightarrow H^1\left(X, \Omega_X\right)\). On
the other hand, \(H^1(X, \Omega)\) is dual to itself by Serre duality
(III, 7.13), so we have a nondegenerate bilinear map

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  Prove that this is compatible with the intersection pairing, in the
  following sense: for any two divisors \(D, E\) on \(X\), we have in
  \(k\).\footnote{Hint: Reduce to the case where \(D\) and \(E\) are
    nonsingular curves meeting transversally. Then consider the
    analogous map
    \(c: \operatorname{Pic} D \rightarrow H^1\left(D, \Omega_D\right)\),
    and the fact (III, Ex. 7.4) that \(c\) (point) goes to 1 under the
    natural isomorphism of \(H^1\left(D, \Omega_D\right)\) with \(k\).}
\end{enumerate}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item
  If char \(k=0\), use the fact that \(H^1\left(X, \Omega_X\right)\) is
  a finite-dimensional vector space to show that \(\operatorname{Num}X\)
  is a finitely generated free abelian group.
\end{enumerate}

\hypertarget{v.1.9.}{%
\subsubsection{V.1.9.}\label{v.1.9.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(H\) is an ample divisor on the surface \(X\), and if \(D\) is any
  divisor, show that
\item
  Now let \(X\) be a product of two curves \(X=C \times C^{\prime}\).
  Let \(l=C \times \mathrm{pt}\), and \(m=\) pt \(\times C^{\prime}\).
  For any divisor \(D\) on \(X\), let \(a=D . l, b=D . m\). Then we say
  \(D\) has type \((a, b)\). If \(D\) has type \((a, b)\), with
  \(a, b \in \mathbf{Z}\), show that and equality holds if and only if
  \(D \equiv b l+a m\).\footnote{Hint: Show that \(H=l+m\) is ample, let
    \(E=l-m\), let
    \(D^{\prime}=\left(H^2\right)\left(E^2\right) D-\left(E^2\right)(D \cdot H) H-\left(H^2\right)(D \cdot E) E\),
    and apply (1.9). This inequality is due to Castelnuovo and Severi.
    See Grothendieck \([2]\).}
\end{enumerate}

\hypertarget{v.1.10.-weils-proof-of-the-analogue-of-the-riemann-hypothesis-for-curves.}{%
\subsubsection{V.1.10. Weil's Proof of the Analogue of the Riemann
Hypothesis for
Curves.}\label{v.1.10.-weils-proof-of-the-analogue-of-the-riemann-hypothesis-for-curves.}}

Let \(C\) be a curve of genus \(g\) defined over the finite field
\(\mathbf{F}_q\), and let \(N\) be the number of points of \(C\)
rational over \(\mathbf{F}_q\). Then \(N=1-a+q\), with
\(|a| \leqslant 2 g \sqrt{q}\).

To prove this, we consider \(C\) as a curve over the algebraic closure
\(k\) of \(\mathbf{F}_q\). Let \(f: C \rightarrow C\) be the
\(k\)-linear Frobenius morphism obtained by taking \(q\) th powers,
which makes sense since \(C\) is defined over \(\mathbf{F}_q\), so
\(X_q \cong X\) (See \(V, 2.4 .1)\).

Let \(\Gamma \subseteq C \times C\) be the graph of \(f\), and let
\(\Delta \subseteq C \times C\) be the diagonal.

Show that \(\Gamma^2=q(2-2 g)\), and \(\Gamma . \Delta=N\). Then apply
(Ex. 1.9) to \(D=r \Gamma+s \Delta\) for all \(r\) and \(s\) to obtain
the result.\footnote{See (App. C, Ex. 5.7) for another interpretation of
  this result.}

\hypertarget{v.1.11.}{%
\subsubsection{V.1.11.}\label{v.1.11.}}

In this problem, we assume that \(X\) is a surface for which
\(\operatorname{Num}X\) is finitely generated (i.e., any surface, if you
accept the Néron-Severi theorem (Ex. 1.7)).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  If \(H\) is an ample divisor on \(X\), and \(d \in \mathbf{Z}\), show
  that the set of effective divisors \(D\) with \(D . H=d\), modulo
  numerical equivalence, is a finite set.\footnote{Hint: Use the
    adjunction formula, the fact that \(p_a\) of an irreducible curve is
    \(\geqslant 0\), and the fact that the intersection pairing is
    negative definite on \(H^{\perp}\) in \(\operatorname{Num} X\).}
\end{enumerate}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\setcounter{enumi}{1}
\tightlist
\item
  Now let \(C\) be a curve of genus \(g \geqslant 2\), and use (a) to
  show that the group of automorphisms of \(C\) is finite, as follows.
  Given an automorphism \(\sigma\) of \(C\), let
  \(\Gamma \subseteq X=C \times C\) be its graph. First show that if
  \(\Gamma \equiv \Delta\), then \(\Gamma=\Delta\), using the fact that
  \(\Delta^2<0\), since \(g \geqslant 2\) (Ex. 1.6). Then use (a). Cf.
  (V, Ex. 2.5).
\end{enumerate}

\hypertarget{v.1.12.}{%
\subsubsection{V.1.12.}\label{v.1.12.}}

If \(D\) is an ample divisor on the surface \(X\), and
\(D^{\prime} \equiv D\), then \(D^{\prime}\) is also ample. Give an
example to show, however, that if \(D\) is very ample, \(D^{\prime}\)
need not be very ample.

\hypertarget{v.2-ruled-surfaces}{%
\subsection{V.2: Ruled Surfaces}\label{v.2-ruled-surfaces}}

\hypertarget{v.2.1.-1}{%
\subsubsection{V.2.1.}\label{v.2.1.-1}}

If \(X\) is a birationally ruled surface, show that the curve \(C\),
such that \(X\) is birationally equivalent to \(C \times \mathbf{P}^1\),
is unique (up to isomorphism).

\hypertarget{v.2.2.-1}{%
\subsubsection{V.2.2.}\label{v.2.2.-1}}

Let \(X\) be the ruled surface \(\mathbf{P}(\mathcal{E})\) over a curve
\(C\). Show that \(\mathcal{E}\) is decomposable if and only if there
exist two sections \(C^{\prime}, C^{\prime \prime}\) of \(X\) such that
\(C^{\prime} \cap C^{\prime \prime}=\varnothing\).

\hypertarget{v.2.3.}{%
\subsubsection{V.2.3.}\label{v.2.3.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\mathcal{E}\) is a locally free sheaf of rank \(r\) on a
  (nonsingular) curve \(C\), then there is a sequence of subsheaves such
  that \(\mathcal{E}_i / \mathcal{E}_{i-1}\) is an invertible sheaf for
  each \(i=1, \ldots, r\). We say that \(\mathcal{E}\) is a successive
  extension of invertible sheaves.\footnote{Hint: Use (II, Ex. 8.2).}
\item
  Show that this is false for varieties of dimension \(\geqslant 2\). In
  particular, the sheaf of differentials \(\Omega\) on \(\mathbf{P}^2\)
  is not an extension of invertible sheaves.
\end{enumerate}

\hypertarget{v.2.4.}{%
\subsubsection{V.2.4.}\label{v.2.4.}}

Let \(C\) be a curve of genus \(g\), and let \(X\) be the ruled surface
\(C \times \mathbf{P}^1\). We consider the question, for what integers
\(s \in \mathbf{Z}\) does there exist a section \(D\) of \(X\) with
\(D^2=s\) ? First show that \(s\) is always an even integer, say
\(s=2 r\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(r=0\) and any \(r \geqslant g+1\) are always possible. Cf.
  (V, Ex. 6.8).
\item
  If \(g=3\), show that \(r=1\) is not possible, and just one of the two
  values \(r=2,3\) is possible, depending on whether \(C\) is
  hyperelliptic or not.
\end{enumerate}

\hypertarget{v.2.5.-values-of-e.-to_work}{%
\subsubsection{\texorpdfstring{V.2.5. Values of
\(e\).}{V.2.5. Values of e.}}\label{v.2.5.-values-of-e.-to_work}}

Let \(C\) be a curve of genus \(g \geqslant 1\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that for each \(0 \leqslant e \leqslant 2 g-2\) there is a ruled
  surface \(X\) over \(C\) with invariant \(e\), corresponding to an
  indecomposable \(\mathcal{E}\). Cf. (2.12).
\item
  Let \(e<0\), let \(D\) be any divisor of degree \(d=-e\), and let
  \(\xi \in H^1(\mathcal{L}(-D))\) be a nonzero element defining an
  extension Let \(H \subseteq|D+K|\) be the sublinear system of
  codimension 1 defined by ker \(\xi\), where \(\xi\) is considered as a
  linear functional on \(H^0(\mathcal{L}(D+K))\). For any effective
  divisor \(E\) of degree \(d-1\), let \(L_E \subseteq|D+K|\) be the
  sublinear system \(|D+K-E|+E\). Show that \(\mathcal{E}\) is
  normalized if and only if for each \(E\) as above,
  \(L_E \nsubseteq H\). Cf. proof of \((2.15)\).
\item
  Now show that if \(-g \leqslant e<0\), there exists a ruled surface
  \(X\) over \(C\) with invariant \(e\).\footnote{Hint: For any given
    \(D\) in (b), show that a suitable \(\xi\) exists, using an argument
    similar to the proof of (II, 8.18).}
\item
  For \(g=2\), show that \(e \geqslant-2\) is also necessary for the
  existence of \(X\).\footnote{Note. It has been shown that
    \(e \geqslant-g\) for any ruled surface (Nagata \([8]\)).}
\end{enumerate}

\hypertarget{v.2.6.-1}{%
\subsubsection{V.2.6.}\label{v.2.6.-1}}

Show that every locally free sheaf of finite rank on \(\mathbf{P}^1\) is
isomorphic to a direct sum of invertible sheaves.\footnote{Hint: Choose
  a subinvertible sheaf of maximal degree, and use induction on the
  rank.}

\hypertarget{v.2.7.-1}{%
\subsubsection{V.2.7.}\label{v.2.7.-1}}

On the elliptic ruled surface \(X\) of (2.11.6), show that the sections
\(C_0\) with \(C_0^2=1\) form a one-dimensional algebraic family,
parametrized by the points of the base curve \(C\), and that no two are
linearly equivalent.

\hypertarget{v.2.8.}{%
\subsubsection{V.2.8.}\label{v.2.8.}}

A locally free sheaf \(\mathcal{E}\) on a curve \(C\) is said to be
\textbf{stable} if for every quotient locally free sheaf we have
Replacing \(>\) by \(\geqslant\) defines \textbf{semistable}.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  A decomposable \(\mathcal{E}\) is never stable.
\item
  If \(\mathcal{E}\) has rank 2 and is normalized, then \(\mathcal{E}\)
  is stable (respectively, semistable) if and only if
  \(\deg \mathcal{E}>0\) (respectively, \(\geqslant 0\) ).
\item
  Show that the indecomposable locally free sheaves \(\mathcal{E}\) of
  rank 2 that are not semistable are classified, up to isomorphism, by
  giving

  \begin{enumerate}
  \def\labelenumii{(\arabic{enumii})}
  \tightlist
  \item
    an integer \(0<e \leqslant\) \(2 g-2\),
  \item
    an element \(\mathcal{L} \in \operatorname{Pic}C\) of degree \(-e\),
    and
  \item
    a nonzero \(\xi \in H^1\left(\mathcal{L} {}^{ \vee }\right)\),
    determined up to a nonzero scalar multiple.
  \end{enumerate}
\end{enumerate}

\hypertarget{v.2.9.}{%
\subsubsection{V.2.9.}\label{v.2.9.}}

Let \(Y\) be a nonsingular curve on a quadric cone \(X_0\) in
\(\mathbf{P}^3\). Show that either

\begin{itemize}
\tightlist
\item
  \(Y\) is a complete intersection of \(X_0\) with a surface of degree
  \(a \geqslant 1\), in which case \(\deg Y=\) \(2 a, g(Y)=(a-1)^2\),
  or,
\item
  \(\deg Y\) is odd, say \(2 a+1\), and \(g(Y)=a^2-a\).\footnote{Cf. (V,
    6.4.1). Hint: Use (2.11.4).}
\end{itemize}

\hypertarget{v.2.10.}{%
\subsubsection{V.2.10.}\label{v.2.10.}}

For any \(n>e \geqslant 0\), let \(X\) be the rational scroll of degree
\(d=2 n-e\) in \(\mathbf{P}^{d+1}\) given by (2.19). If
\(n \geqslant 2 e-2\), show that \(X\) contains a nonsingular curve
\(Y\) of genus \(g=d+2\) which is a canonical curve in this embedding.

Conclude that for every \(g \geqslant 4\), there exists a
nonhyperelliptic curve of genus \(g\) which has a \(g_3^1\). Cf. (V,
\(\S 5\)).

\hypertarget{v.2.11.}{%
\subsubsection{V.2.11.}\label{v.2.11.}}

Let \(X\) be a ruled surface over the curve \(C\), defined by a
normalized bundle \(\mathcal{E}\), and let \(\mathfrak e\) be the
divisor on \(C\) for which
\(\mathcal{L}(\mathfrak{e}) \cong \bigwedge^2 \mathcal{E}\) (See 2.8
.1). Let \(\mathfrak{b}\) be any divisor on \(C\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(|{\mathfrak{b}}|\) and \(|{\mathfrak{b}}+ {\mathfrak{e}}|\) have
  no base points, and if \({\mathfrak{b}}\) is nonspecial, then there is
  a section \(D \sim C_0+{\mathfrak{b}}f\), and \(|D|\) has no base
  points.
\item
  If \({\mathfrak{b}}\) and \({\mathfrak{b}}+ {\mathfrak{e}}\) are very
  ample on \(C\), and for every point \(P \in C\), we have
  \(\mathfrak{b}-P\) and \(\mathfrak{b}+\mathfrak{e}-P\) nonspecial,
  then \(C_0+\mathfrak{b} f\) is very ample.
\end{enumerate}

\hypertarget{v.2.12.}{%
\subsubsection{V.2.12.}\label{v.2.12.}}

Let \(X\) be a ruled surface with invariant \(e\) over an elliptic curve
\(C\), and let \(\mathfrak{b}\) be a divisor on \(C\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(\deg \mathfrak{b} \geqslant e+2\), then there is a section
  \(D \sim C_0+\mathfrak{b} f\) such that \(|D|\) has no base points.
\item
  The linear system \(\left|C_0+\mathfrak{b} f\right|\) is very ample if
  and only if \(\deg \mathfrak{b} \geqslant e+3\). Note. The case
  \(e=-1\) will require special attention.
\end{enumerate}

\hypertarget{v.2.13.}{%
\subsubsection{V.2.13.}\label{v.2.13.}}

For every \(e \geqslant-1\) and \(n \geqslant e+3\), there is an
elliptic scroll of degree \(d=2 n-e\) in \(\mathbf{P}^{d-1}\). In
particular, there is an elliptic scroll of degree 5 in \(\mathbf{P}^4\).

\hypertarget{v.2.14.}{%
\subsubsection{V.2.14.}\label{v.2.14.}}

Let \(X\) be a ruled surface over a curve \(C\) of genus \(g\), with
invariant \(e<0\), and assume that char \(k=p>0\) and \(g \geqslant 2\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(Y \equiv a C_0+b f\) is an irreducible curve \(\neq C_0, f\),
  then either

  \begin{itemize}
  \tightlist
  \item
    \(a=1, b \geqslant 0\), or
  \item
    \(2 \leqslant a \leqslant p-1, b \geqslant \frac{1}{2} a e\), or
  \item
    \(a \geqslant p, b \geqslant \frac{1}{2} a e+1-g\).
  \end{itemize}
\item
  If \(a>0\) and \(b>a\left(\frac{1}{2} e+(1 / p)(g-1)\right)\), then
  any divisor \(D \equiv a C_0+b f\) is ample. On the other hand, if
  \(D\) is ample, then \(a>0\) and \(b>\frac{1}{2} a e\).
\end{enumerate}

\hypertarget{v.2.15.-funny-behavior-in-characteristic-p.-to_work}{%
\subsubsection{\texorpdfstring{V.2.15. Funny behavior in characteristic
\(p\).}{V.2.15. Funny behavior in characteristic p.}}\label{v.2.15.-funny-behavior-in-characteristic-p.-to_work}}

Let \(C\) be the plane curve \(x^3 y+y^3 z+z^3 x=0\) over a field \(k\)
of characteristic 3 (V, Ex. 2.4).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that the action of the \(k\)-linear Frobenius morphism \(f\) on
  \(H^1\left(C, \mathcal{O}_C\right)\) is identically 0 (Cf. (V, 4.21)).
\item
  Fix a point \(P \in C\), and show that there is a nonzero
  \(\xi \in H^1(\mathcal{L}(-P))\) such that \(f^* \xi=0\) in
  \(H^1(\mathcal{L}(-3 P))\).
\item
  Now let \(\mathcal{E}\) be defined by \(\xi\) as an extension and let
  \(X\) be the corresponding ruled surface over \(C\). Show that \(X\)
  contains a nonsingular curve \(Y \equiv 3 C_0-3 f\), such that
  \(\pi: Y \rightarrow C\) is purely inseparable.

  Show that the divisor \(D=2 C_0\) satisfies the hypotheses of
  (2.21.b), but is not ample.
\end{enumerate}

\hypertarget{v.2.16.}{%
\subsubsection{V.2.16.}\label{v.2.16.}}

Let \(C\) be a nonsingular affine curve. Show that two locally free
sheaves \(\mathcal{E}, \mathcal{E}^{\prime}\) of the same rank are
isomorphic if and only if their classes in the Grothendieck group
\(K(X)\) (II, Ex. 6.10) and (II, Ex. 6.11) are the same. This is false
for a projective curve.

\hypertarget{v.2.17-.}{%
\subsubsection{V.2.17 *.}\label{v.2.17-.}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Let \(\varphi: \mathbf{P}_k^1 \rightarrow \mathbf{P}_k^3\) be the
  3-uple embedding (I, Ex. 2.12). Let \(\mathcal{I}\) be the sheaf of
  ideals of the twisted cubic curve \(C\) which is the image of
  \(\varphi\). Then \(\mathcal{I} / \mathcal{I}^2\) is a locally free
  sheaf of rank 2 on \(C\), so
  \(\varphi^*\left(\mathcal{I} / \mathcal{I}^2\right)\) is a locally
  free sheaf of rank 2 on \(\mathbf{P}^1\). By (2.14), therefore, for
  some \(l, m \in \mathbf{Z}\). Determine \(l\) and \(m\).
\item
  Repeat part (a) for the embedding
  \(\varphi: \mathbf{P}^1 \rightarrow \mathbf{P}^3\) given by
  \(x_0=t^4, x_1=t^3 u\), \(x_2=t u^3, x_3=u^4\), whose image is a
  nonsingular rational quartic curve.\footnote{Answer: If char
    \(k \neq 2\), then \(l=m=-7\); if char \(k=2\), then \(l, m=-6,-8\).}
\end{enumerate}

\hypertarget{v.3-monoidal-transformations}{%
\subsection{V.3: Monoidal
Transformations}\label{v.3-monoidal-transformations}}

\hypertarget{v.3.1.-1}{%
\subsubsection{V.3.1.}\label{v.3.1.-1}}

Let \(X\) be a nonsingular projective variety of any dimension, let
\(Y\) be a nonsingular subvariety, and let
\(\pi: \tilde{X} \rightarrow X\) be obtained by blowing up \(Y\). Show
that \(p_a(\tilde{X})=\) \(p_a(X)\)

\hypertarget{v.3.2.-1}{%
\subsubsection{V.3.2.}\label{v.3.2.-1}}

Let \(C\) and \(D\) be curves on a surface \(X\), meeting at a point
\(P\). Let \(\pi: \tilde{X} \rightarrow X\) be the monoidal
transformation with center \(P\).

Show that Conclude that \(C . D=\sum \mu_P(C) \cdot \mu_P(D)\), where
the sum is taken over all intersection points of \(C\) and \(D\),
including infinitely near intersection points.

\hypertarget{v.3.3.-1}{%
\subsubsection{V.3.3.}\label{v.3.3.-1}}

Let \(\pi: \tilde{X} \rightarrow X\) be a monoidal transformation, and
let \(D\) be a very ample divisor on \(X\). Show that \(2 \pi^* D-E\) is
ample on \(\tilde{X}\).\footnote{Hint: Use a suitable generalization of
  (I, Ex. 7.5) to curves in \(\mathbf{P}^n\).}

\hypertarget{v.3.4.-multiplicity-of-a-local-ring.}{%
\subsubsection{V.3.4. Multiplicity of a Local
Ring.}\label{v.3.4.-multiplicity-of-a-local-ring.}}

Let \(A\) be a noetherian local ring with maximal ideal
\({\mathfrak{m}}\). For any \(l>0\), let \(\psi(l)=\) length
\(\left(A / {\mathfrak{m}}^l\right)\). We call \(\psi\) the
\textbf{Hilbert-Samuel function of \(A\)}.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that there is a polynomial \(P_A(z) \in \mathbf{Q}[z]\) such that
  \(P_A(l)=\psi(l)\) for all \(l \gg 0\). This is the Hilbert-Samuel
  polynomial of \(A\).\footnote{Hint: Consider the graded ring
    \(\mathrm{gr}_{{\mathfrak{m}}} A=\bigoplus_{d \geqslant 0} {\mathfrak{m}}^d / {\mathfrak{m}}^{d+1}\),
    and apply \((\mathrm{I}, 7.5)\)}\footnote{See Nagata
    \([7, \text{Ch} III, \S 23]\) or Zariski-Samuel
    \([1, \text{vol} 2 , \text{Ch} VIII, \S 10]\).}
\item
  Show that \(\operatorname{deg} P_A=\operatorname{dim} A\).
\item
  Let \(n=\operatorname{dim} A\). Then we define the multiplicity of
  \(A\), denoted \(\mu(A)\), to be \((n !)\). (leading coefficient of
  \(P_A\) ). If \(P\) is a point on a noetherian scheme \(X\), we define
  the multiplicity of \(P\) on \(X, \mu_P(X)\), to be
  \(\mu\left(\mathcal{O}_{P, X}\right)\).
\item
  Show that for a point \(P\) on a curve \(C\) on a surface \(X\), this
  definition of \(\mu_P(C)\) coincides with the one in the text just
  before (3.5.2).
\item
  If \(Y\) is a variety of degree \(d\) in \(\mathbf{P}^n\), show that
  the vertex of the cone over \(Y\) is a point of multiplicity \(d\).
\end{enumerate}

\hypertarget{v.3.5.-1}{%
\subsubsection{V.3.5.}\label{v.3.5.-1}}

Let \(a_1, \ldots, a_r, r \geqslant 5\), be distinct elements of \(k\),
and let \(C\) be the curve in \(\mathbf{P}^2\) given by the (affine)
equation \(y^2=\prod_{i=1}^r\left(x-a_i\right)\). Show that the point
\(P\) at infinity on the \(y\)-axis is a singular point. Compute
\(\delta_P\) and \(g(\tilde{Y})\), where \(\tilde{Y}\) is the
normalization of \(Y\). Show in this way that one obtains hyperelliptic
curves of every genus \(g \geqslant 2\).

\hypertarget{v.3.6.-1}{%
\subsubsection{V.3.6.}\label{v.3.6.-1}}

Show that analytically isomorphic curve singularities (I, 5.6.1) are
equivalent in the sense of (3.9.4), but not conversely.

\hypertarget{v.3.7.-1}{%
\subsubsection{V.3.7.}\label{v.3.7.-1}}

For each of the following singularities at \((0,0)\) in the plane, give
an embedded resolution, compute \(\delta_P\), and decide which ones are
equivalent.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(x^3+y^5=0\).
\item
  \(x^3+x^4+y^5=0\).
\item
  \(x^3+y^4+y^5=0\).
\item
  \(x^3+y^5+y^6=0\).
\item
  \(x^3+x y^3+y^5=0\).
\end{enumerate}

\hypertarget{v.3.8.-1}{%
\subsubsection{V.3.8.}\label{v.3.8.-1}}

Show that the following two singularities have the same multiplicity,
and the same configuration of infinitely near singular points with the
same multiplicities, hence the same \(\delta_P\), but are not
equivalent.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  \(x^4-x y^4=0\).
\item
  \(x^4-x^2 y^3-x^2 y^5+y^8=0\).
\end{enumerate}

\hypertarget{v.4-the-cubic-surface-in-mathbfp3}{%
\subsection{\texorpdfstring{V.4: The Cubic Surface in
\({\mathbf{P}}^3\)}{V.4: The Cubic Surface in \{\textbackslash mathbf\{P\}\}\^{}3}}\label{v.4-the-cubic-surface-in-mathbfp3}}

\hypertarget{v.4.1.-1}{%
\subsubsection{V.4.1.}\label{v.4.1.-1}}

The linear system of conics in \(\mathbf{P}^2\) with two assigned base
points \(P_1\) and \(P_2\) (4.1) determines a morphism \(\psi\) of
\(X^{\prime}\) (which is \(\mathbf{P}^2\) with \(P_1\) and \(P_2\) blown
up) to a nonsingular quadric surface \(Y\) in \(\mathbf{P}^3\), and
furthermore \(X^{\prime}\) via \(\psi\) is isomorphic to \(Y\) with one
point blown up.

\hypertarget{v.4.2.-1}{%
\subsubsection{V.4.2.}\label{v.4.2.-1}}

Let \(\varphi\) be the quadratic transformation of \((4.2 .3)\),
centered at \(P_1, P_2, P_3\). If \(C\) is an irreducible curve of
degree \(d\) in \(\mathbf{P}^2\), with points of multiplicity
\(r_1, r_2, r_3\) at \(P_1, P_2, P_3\), then the strict transform
\(C^{\prime}\) of \(C\) by \(\varphi\) has degree and has points of
multiplicity

\begin{itemize}
\tightlist
\item
  \(d-r_2-r_3\) at \(Q_1\),
\item
  \(d-r_1-r_3\) at \(Q_2\) and
\item
  \(d-\) \(r_1-r_2\) at \(Q_3\).
\end{itemize}

The curve \(C\) may have arbitrary singularities.\footnote{Hint: Use
  (Ex. 3.2).}

\hypertarget{v.4.3.-1}{%
\subsubsection{V.4.3.}\label{v.4.3.-1}}

Let \(C\) be an irreducible curve in \(\mathbf{P}^2\). Then there exists
a finite sequence of quadratic transformations, centered at suitable
triples of points, so that the strict transform \(C^{\prime}\) of \(C\)
has only ordinary singularities, i.e., multiple points with all distinct
tangent directions (I, Ex. 5.14). Use (3.8).

\hypertarget{v.4.4.-1}{%
\subsubsection{V.4.4.}\label{v.4.4.-1}}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Use (4.5) to prove the following lemma on cubics: If \(C\) is an
  irreducible plane cubic curve, if \(L\) is a line meeting \(C\) in
  points \(P, Q, R\), and \(L^{\prime}\) is a line meeting \(C\) in
  points \(P^{\prime}, Q^{\prime}, R^{\prime}\), let
  \(P^{\prime \prime}\) be the third intersection of the line
  \(P P^{\prime}\) with \(C\), and define
  \(Q^{\prime \prime}, R^{\prime \prime}\) similarly. Then
  \(P^{\prime \prime}, Q^{\prime \prime}, R^{\prime \prime}\) are
  collinear.
\item
  Let \(P_0\) be an inflection point of \(C\), and define the group
  operation on the set of regular points of \(C\) by the geometric
  recipe "let the line \(P Q\) meet \(C\) at \(R\), and let \(P_0 R\)
  meet \(C\) at \(T\), then \(P+Q=T^{\prime \prime}\) as in (II, 6.10.2)
  and (II, 6.11.4). Use (a) to show that this operation is associative.
\end{enumerate}

\hypertarget{v.4.5.-1}{%
\subsubsection{V.4.5.}\label{v.4.5.-1}}

Prove Pascal's theorem: if
\(A, B, C, A^{\prime}, B^{\prime}, C^{\prime}\) are any six points on a
conic, then the points
\(P=A B^{\prime} \cdot A^{\prime} B, Q=A C^{\prime} \cdot A^{\prime} C\),
and \(R=B C^{\prime} \cdot B^{\prime} C\) are collinear (Fig. 22).

\includegraphics{figures/2022-10-22_21-13-10.png}

\hypertarget{v.4.6.-1}{%
\subsubsection{V.4.6.}\label{v.4.6.-1}}

Generalize (4.5) as follows: given 13 points \(P_1, \ldots, P_{13}\) in
the plane, there are three additional determined points
\(P_{14}, P_{15}, P_{16}\), such that all quartic curves through
\(P_1, \ldots, P_{13}\) also pass through \(P_{14}, P_{15}, P_{16}\).
What hypotheses are necessary on \(P_1, \ldots, P_{13}\) for this to be
true?

\hypertarget{v.4.7.-1}{%
\subsubsection{V.4.7.}\label{v.4.7.-1}}

If \(D\) is any divisor of degree \(d\) on the cubic surface (4.7.3),
show that Show furthermore that for every \(d>0\), this maximum is
achieved by some irreducible nonsingular curve.

\hypertarget{v.4.8.}{%
\subsubsection{V.4.8. *}\label{v.4.8.}}

Show that a divisor class \(D\) on the cubic surface contains an
irreducible curve \(\iff\) if it contains an irreducible nonsingular
curve \(\iff\) it is either

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  one of the 27 lines, or
\item
  a conic (meaning a curve of degree 2) with \(D^2=0\), or
\item
  \(D . L \geqslant 0\) for every line \(L\), and \(D^2>0\).\footnote{Hint:
    Generalize (4.11) to the surfaces obtained by blowing up \(2,3,4\),
    or 5 points of \(\mathbf{P}^2\), and combine with our earlier
    results about curves on \(\mathbf{P}^1 \times \mathbf{P}^1\) and the
    rational ruled surface \(X_1,(2.18)\).}
\end{enumerate}

\hypertarget{v.4.9.-1}{%
\subsubsection{V.4.9.}\label{v.4.9.-1}}

If \(C\) is an irreducible non-singular curve of degree \(d\) on the
cubic surface, and if the genus \(g>0\), then and this minimum value of
\(g>0\) is achieved for each \(d\) in the range given.

\hypertarget{v.4.10.-1}{%
\subsubsection{V.4.10.}\label{v.4.10.-1}}

A curious consequence of the implication (iv) \(\Rightarrow\) (iii) of
(4.11) is the following numerical fact: Given integers
\(a, b_1, \ldots, b_6\) such that \(b_i>0\) for each \(i, a-b_i-\)
\(b_j>0\) for each \(i, j\) and \(2 a-\sum_{i \neq j} b_i>0\) for each
\(j\), we must necessarily have \(a^2-\sum b_i^2>0\). Prove this
directly (for \(a, b_1, \ldots, b_6 \in \mathbf{R}\) ) using methods of
freshman calculus.

\hypertarget{v.4.11.-the-weyl-groups.}{%
\subsubsection{V.4.11. The Weyl
Groups.}\label{v.4.11.-the-weyl-groups.}}

Given any diagram consisting of points and line segments joining some of
them, we define an abstract group, given by generators and relations, as
follows:

\begin{itemize}
\tightlist
\item
  Each point represents a generator \(x_i\). The relations are
\item
  \(x_i^2=1\) for each \(i\);
\item
  \(\left(x_i x_j\right)^2=1\) if \(i\) and \(j\) are not joined by a
  line segment, and
\item
  \(\left(x_i x_j\right)^3=1\) if \(i\) and \(j\) are joined by a line
  segment.
\end{itemize}

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  The Weyl group \(\mathbf{A}_n\) is defined using the following diagram
  of \(n-1\) points, each joined to the next:
  \includegraphics{figures/2022-10-22_21-16-28.png} Show that it is
  isomorphic to the symmetric group \(\Sigma_n\) as follows:

  \begin{itemize}
  \tightlist
  \item
    Map the generators of \(\mathbf{A}_n\) to the elements
    \((12),(23), .., (n-1,n)\) of \(\Sigma_n\), to get a surjective
    homomorphism \(\mathbf{A}_n \rightarrow \Sigma_n\).
  \item
    Then estimate the number of elements of \(\mathbf{A}_n\) to show in
    fact it is an isomorphism.
  \end{itemize}
\item
  The Weyl group \(\mathbf{E}_6\) is defined using the diagram
  \includegraphics{figures/2022-10-22_21-17-05.png} Call the generators
  \(x_1, \ldots, x_5\) and \(y\). Show that one obtains a surjective
  homomorphism \(\mathbf{E}_6 \rightarrow G\), the group of
  automorphisms of the configuration of 27 lines \((4.10 .1)\), by
  sending \(x_1, \ldots, x_5\) to the permutations
  \((12),(23), \ldots,(56)\) of the \(E_i\), respectively, and \(y\) to
  the element associated with the quadratic transformation based at
  \(P_1, P_2, P_3\).
\item
  * Estimate the number of elements in \(\mathbf{E}_6\), and thus
  conclude that \(\mathbf{E}_6 \cong G\).\footnote{Note: See Manin
    \([3, \S 25,26]\) for more about Weyl groups, root systems, and
    exceptional curves.}
\end{enumerate}

\hypertarget{v.4.12.}{%
\subsubsection{V.4.12.}\label{v.4.12.}}

Use (4.11) to show that if \(D\) is any ample divisor on the cubic
surface \(X\), then \(H^1\left(X, \mathcal{O}_X(-D)\right)=0\). This is
Kodaira's vanishing theorem for the cubic surface (III, 7.15).

\hypertarget{v.4.13.}{%
\subsubsection{V.4.13.}\label{v.4.13.}}

Let \(X\) be the Del Pezzo surface of degree 4 in \(\mathbf{P}^4\)
obtained by blowing up 5.points of \(\mathbf{P}^2(4.7)\)

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(X\) contains 16 lines.
\item
  Show that \(X\) is a complete intersection of two quadric
  hypersurfaces in \(\mathbf{P}^4\) (the converse follows from (4.7.1)).
\end{enumerate}

\hypertarget{v.4.14.}{%
\subsubsection{V.4.14.}\label{v.4.14.}}

Using the method of (4.13.1), verify that there are nonsingular curves
in \(\mathbf{P}^3\) with
\(d=8, g=6,7 ; d=9, g=7,8,9 ; d=10, g=8,9,10,11\). Combining with (IV,
§6), this completes the determination of all posible \(g\) for curves of
degree \(d \leqslant 10\) in \(\mathbf{P}^3\).

\hypertarget{v.4.15.}{%
\subsubsection{V.4.15.}\label{v.4.15.}}

Let \(P_1, \ldots, P_r\) be a finite set of (ordinary) points of
\(\mathbf{P}^2\), no 3 collinear. We define an \textbf{admissible
transformation} to be a quadratic transformation (4.2.3) centered at
some three of the \(P_i\) (call them \(P_1, P_2, P_3\) ).

This gives a new \(\mathbf{P}^2\), and a new set of \(r\) points, namely
\(Q_1, Q_2, Q_3\), and the images of \(P_4, \ldots, P_r\).

We say that \(P_1, \ldots, P_r\) are \textbf{in general position} if no
three are collinear, and furthermore after any finite sequence of
admissible transformations, the new set of \(r\) points also has no
three collinear.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  A set of 6 points is in general position if and only if no three are
  collinear and not all six lie on a conic.
\item
  If \(P_1, \ldots, P_r\) are in general position, then the \(r\) points
  obtained by any finite sequence of admissible transformations are also
  in general position.
\item
  Assume the ground field \(k\) is uncountable. Then given
  \(P_1, \ldots, P_r\) in general position, there is a dense subset
  \(V \subseteq \mathbf{P}^2\) such that for any
  \(P_{r+1} \in V, P_1, \ldots, P_{r+1}\) will be in general
  position.\footnote{Hint: Prove a lemma that when \(k\) is uncountable,
    a variety cannot be equal to the union of a countable family of
    proper closed subsets.}
\item
  Now take \(P_1, \ldots, P_r \in \mathbf{P}^2\) in general position,
  and let \(X\) be the surface obtained by blowing up
  \(P_1, \ldots, P_r\). If \(r=7\), show that \(X\) has exactly 56
  irreducible nonsingular curves \(C\) with \(g=0, C^2=-1\), and that
  these are the only irreducible curves with negative self-intersection.
  Ditto for \(r=8\), the number being 240 .
\item
  * For \(r=9\), show that the surface \(X\) defined in (d) has
  infinitely many irreducible nonsingular curves \(C\) with \(g=0\) and
  \(C^2=-1\).\footnote{Hint: Let \(L\) be the line joining \(P_1\) and
    \(P_2\). Show that there exist finite sequences of admissible
    transformations such that the strict transform of \(L\) becomes a
    plane curve of arbitrarily high degree. This example is apparently
    due to Kodaira-see Nagata \([5, II, p. 283]\).}
\end{enumerate}

\hypertarget{v.4.16.}{%
\subsubsection{V.4.16.}\label{v.4.16.}}

For the Fermat cubic surface \(x_0^3+x_1^3+x_2^3+x_3^3=0\), find the
equations of the 27 lines explicitly, and verify their incidence
relations. What is the group of automorphisms of this surface?

\hypertarget{v.5-birational-transformations}{%
\subsection{V.5: Birational
Transformations}\label{v.5-birational-transformations}}

\hypertarget{v.5.1.}{%
\subsubsection{V.5.1.}\label{v.5.1.}}

Let \(f\) be a rational function on the surface \(X\). Show that it is
possible to "resolve the singularities of \(f^{\prime \prime}\) in the
following sense: there is a birational morphism \(g\) :
\(X^{\prime} \rightarrow X\) so that \(f\) induces a morphism of
\(X^{\prime}\) to \(\mathbf{P}^1\).\footnote{Hints: Write the divisor of
  \(f\) as \((f)=\sum n_i C_i\). Then apply embedded resolution (3.9) to
  the curve \(Y=\bigcup C_i\). Then blow up further as necessary
  whenever a curve of zeros meets a curve of poles until the zeros and
  poles of \(f\) are disjoint.}

\hypertarget{v.5.2.}{%
\subsubsection{V.5.2.}\label{v.5.2.}}

Let \(Y \cong \mathbf{P}^1\) be a curve in a surface \(X\), with
\(Y^2<0\). Show that \(Y\) is contractible (5.7.2) to a point on a
projective variety \(X_0\) (in general singular).

\hypertarget{v.5.3.}{%
\subsubsection{V.5.3.}\label{v.5.3.}}

If \(\pi: \tilde{X} \rightarrow X\) is a monoidal transformation with
center \(P\), show that
\(H^1\left(\tilde{X}, \Omega_{\tilde{X}}\right) \cong\)
\(H^1\left(X, \Omega_X\right) \oplus k\). This gives another proof of
(5.8).\footnote{Hints: Use the projection formula (III, Ex. 8.3) and
  (III, Ex. 8.1) to show that
  \(H^i\left(X, \Omega_X\right) \cong H^i\left(\tilde{X}, \pi^* \Omega_X\right)\)
  for each i. Next use the exact sequence and a local calculation with
  coordinates to show that there is a natural isomorphism
  \(\Omega_{\tilde{X} / X} \cong \Omega_E\), where \(E\) is the
  exceptional curve. Now use the cohomology sequence of the above
  sequence (you will need every term) and Serre duality to get the
  result.}

\hypertarget{v.5.4.}{%
\subsubsection{V.5.4.}\label{v.5.4.}}

Let \(f: X \rightarrow X^{\prime}\) be a birational morphism of
nonsingular surfaces.

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  If \(Y \subseteq X\) is an irreducible curve such that \(f(Y)\) is a
  point, then \(Y \cong \mathbf{P}^1\) and \(Y^2<0\)
\item
  Let \(P^{\prime} \in X^{\prime}\) be a fundamental point of
  \(f^{-1}\), and let \(Y_1, \ldots, Y_r\) be the irreducible components
  of \(f^{-1}\left(P^{\prime}\right)\). Show that the matrix
  \(\left|Y_i . Y_j\right|\) is negative definite.
\end{enumerate}

\hypertarget{v.5.5.}{%
\subsubsection{V.5.5.}\label{v.5.5.}}

Let \(C\) be a curve, and let \(\pi: X \rightarrow C\) and
\(\pi^{\prime}: X^{\prime} \rightarrow C\) be two geometrically ruled
surfaces over \(C\). Show that there is a finite sequence of elementary
transformations (5.7.1) which transform \(X\) into
\(X^{\prime}\).\footnote{Hints: First show if \(D \subseteq X\) is a
  section of \(\pi\) containing a point \(P\), and if \(\tilde{D}\) is
  the strict transform of \(D\) by \(\mathrm{elm}_P\), then
  \(\widetilde{D}^2=D^2-1\) (Fig. 23).\\
  Next show that \(X\) can be transformed into a geometrically ruled
  surface \(X^{\prime \prime}\) with invariant \(e \gg 0\). Then use
  (2.12), and study how the ruled surface \(\mathbf{P}(\mathcal{E})\)
  with \(\mathcal{E}\) decomposable behaves under
  \(\operatorname{elm}_P\).}

\hypertarget{v.5.6.}{%
\subsubsection{V.5.6.}\label{v.5.6.}}

Let \(X\) be a surface with function field \(K\). Show that every
valuation \(\operatorname{ring} R\) of \(K / k\) is one of the three
kinds described in (II, Ex. 4.12).\footnote{Hint: In case (3), let
  \(f \in R\). Use (Ex. 5.1) to show that for all
  \(i \gg 0, f \in \mathcal{O}_{X_i}\), so in fact \(f \in R_0\).}

\hypertarget{v.5.7.}{%
\subsubsection{V.5.7.}\label{v.5.7.}}

Let \(Y\) be an irreducible curve on a surface \(X\), and suppose there
is a morphism \(f: X \rightarrow X_0\) to a projective variety \(X_0\)
of dimension 2 , such that \(f(Y)\) is a point \(P\) and
\(f^{-1}(P)=Y\). Then show that \(Y^2<0\).\footnote{Hint: Let \(|H|\) be
  a very ample (Cartier) divisor class on \(X_0\), let \(H_0 \in|H|\) be
  a divisor containing \(P\), and let \(H_1 \in|H|\) be a divisor not
  containing \(P\). Then consider \(f^* H_0, f^* H_1\) and
  \(\tilde{H}_0=f^*\left(H_0-P\right)^{-}\).}

\hypertarget{v.5.8.-a-surface-singularity.}{%
\subsubsection{V.5.8. A surface
singularity.}\label{v.5.8.-a-surface-singularity.}}

Let \(k\) be an algebraically closed field, and let \(X\) be the surface
in \(\mathbf{A}_k^3\) defined by the equation \(x^2+y^3+z^5=0\). It has
an isolated singularity at the origin \(P=(0,0,0)\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that the affine ring \(A=k[x, y, z] /\left(x^2+y^3+z^5\right)\)
  of \(X\) is a unique factorization domain, as follows. Let
  \(t=z^{-1} ; u=t^3 x\), and \(v=t^2 y\). Show that \(z\) is
  irreducible in \(A ; t \in k[u, v]\), and
  \(A\left[z^{-1}\right]=k\left[u, v, t^{-1}\right]\). Conclude that
  \(A\) is a UFD.
\item
  Show that the singularity at \(P\) can be resolved by eight successive
  blowings-up. If \(\tilde{X}\) is the resulting nonsingular surface,
  then the inverse image of \(P\) is a union of eight projective lines,
  which intersect each other according to the Dynkin
  \(\operatorname{diagram} \mathbf{E}_8\) :
  \includegraphics{figures/2022-10-22_21-57-34.png}
\end{enumerate}

\hypertarget{v.6-classification-of-surfaces}{%
\subsection{V.6: Classification of
Surfaces}\label{v.6-classification-of-surfaces}}

\hypertarget{v.6.1.}{%
\subsubsection{V.6.1.}\label{v.6.1.}}

Let \(X\) be a surface in \(\mathbf{P}^n, n \geqslant 3\), defined as
the complete intersection of hypersurfaces of degrees
\(d_1, \ldots, d_{n-2}\), with each \(d_i \geqslant 2\). Show that for
all but finitely many choices of
\(\left(n, d_1, \ldots, d_{n-2}\right)\), the surface \(X\) is of
general type. List the exceptional cases, and where they fit into the
classification picture.

\hypertarget{v.6.2.}{%
\subsubsection{V.6.2.}\label{v.6.2.}}

Prove the following theorem of Chern and Griffiths. Let \(X\) be a
nonsingular surface of degree \(d\) in
\(\mathbf{P}_{\mathbf{C}}^{n+1}\), which is not contained in any
hyperplane. If \(d<2 n\), then \(p_g(X)=0\). If \(d=2 n\), then either
\(p_g(X)=0\), or \(p_g(X)=1\) and \(X\) is a K3 surface.\footnote{Hint:
  Cut \(X\) with a hyperplane and use Clifford's theorem (IV, 5.4). For
  the last statement, use the Riemann-Roch theorem on \(X\) and the
  Kodaira vanishing theorem (III, 7.15).}

\newpage
\printbibliography[title=Bibliography]

\end{document}
