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\begin{document}
\pagestyle{plain}
\title{Intro to Algebraic Geometry, Problem Set 12}
\date{December 2, 1999.}
\maketitle

{\parskip=12pt % closing bracket is just before the bibliography 

Due Monday December 13 by noon at my office.  Read all of the problem
set, and hand in six of them, including one from each section.  You're
strongly encouraged to collaborate (although write up solutions
separately), and you're also strongly encouraged to ask me questions
(if you're stuck, or if the question is vaguely worded, or if you want
to try out an argument).

{\bf Line bundles.}
\begin{enumerate}
\item Suppose $C$ is a nonsingular curve, $p$ is a point of $C$, and $\cL$ is 
an invertible sheaf on $C$.  Describe a natural identification of
global sections of $\cL \otimes \oh_C(-p)$ (denoted $\cL(-p)$ for
short) with the global sections of $\cL$ vanishing at $p$.  If $C$ is
a projective nonsingular curve, prove that $h^0(C, \cL) - h^0(C,
\cL(-p))$ is 0 or 1.
\end{enumerate}

{\bf Differentials.}
\begin{enumerate}

\item[2.]
  Suppose $U$ is an affine variety, and $r \in A(U)$ is a regular
function.  Show $d (r^n ) = n r^{n-1} dr$ if $n$ is a non-negative
integer.  If $r$ is an invertible regular function (i.e. $r \in
A(U)^*$) show that $d r^n = n r^{n-1} dr$ for any integer $n$.

\item[3.] {\em The canonical sheaf of projective space.} 
For some $m$, $\wedge^n \Omega^1_{\proj^n} \cong \oh_{\proj^n}(m)$.  Find 
$m$ as a function of $n$.  (We haven't actually defined the canonical
sheaf of a higher-dimensional nonsingular variety, but think of
sections on each affine open with as being of the form $f d x_1 \wedge
\dots \wedge d x_n$, where $f$ and $x_i$ are regular functions on the open. 
Make whatever assumptions you need.)


\item[4.]  Suppose $X$ is a nonsingular curve, and $p$ a point on $X$.
Let $U$ be an affine neighborhood of $p$, and let $\fm$ be the maximal
ideal of $A(U)$ corresponding to $p$.  Then we earlier identified the
cotangent space to $X$ at $p$ with $\fm / \fm^2$.  As $\Omega^1$ is
the ``cotangent sheaf'', there should be a natural surjective map (of
$A(U)$-modules) from $\Omega^1(U) \rightarrow \fm / \fm^2$.
Describe this map explicitly.  (Hence there is a natural map from the
stalk $\Omega^1_p$ to $\fm/
\fm^2$.)

\item[5.] {\em Pullback of differentials.}  Suppose $\pi:  X \rightarrow Y$ is a morphism of nonsingular curves.
Show that there is a natural map from global differentials on $Y$
(i.e. global sections of $\Omega^1_Y$) to global differentials on $X$.
The naturality of your construction will show that there is a natural
morphism of invertible sheaves on $X$, $\pi^{-1} \Omega^1_Y \rightarrow \Omega^1_X$.
(Equivalently, there is a natural morphism of sheaves on $Y$, $\Omega^1_Y \rightarrow 
\pi_* \Omega^1_X$; but $\pi_* \Omega^1_X$ isn't an invertible sheaf, so we don't yet
have the language to effectively discuss it.)


\item[6.]  {\em Residues.}  In complex analysis, a differential on a one-dimensional
complex manifold with a simple pole at a point $p$ has a
naturally-defined residue at that point: in any analytic coordinate
$z$ at $p$ (i.e. $p$ corresponds to $z=0$), if the differential is of
the form $(a_{-1}/z + f(z)) dz$ where $f(z)$ is analytic, then the
residue is $a_{-1}$, and this is {\em independent of the choice of
analytic coordinate}.  Show that the same is true in algebraic
geometry: suppose $X$ is a nonsingular curve, $p$ a point of $X$, and
$s$ a differential on $X$ with simple pole at $p$; define the residue
of $s$ at $p$ (and show that it is well-defined, i.e. that it doesn't
depend on any choice).
\end{enumerate}

{\bf Genus.}
\begin{enumerate}
\item[7.] Suppose $C$ is a nonsingular projective curve, and $\cL$ an invertible
sheaf on $C$ with degree 1, and $h^0(C,\cL)=2$.  Show that $C$ has
genus 0.

\item[8.] \label{hyp1}
{\em Hyperelliptic curves.} Assume $\ch \kb \neq 2$. Suppose
$f(x_0,x_1)$ is a polynomial of homogeneous degree $n$ with 
no double zeros, where $n$ is
even.  Let $C_0$ be the affine plane curve given by $y^2 = f(1,x_1)$,
with the (generically 2-to-1) morphism $C_0 \rightarrow U_0$.  Let $C_1$ be
the affine plane curve given by $z^2 = f(x_0,1)$, with the morphism
$C_1
\rightarrow U_1$.  Check that $C_0$ and $C_1$ are nonsingular.  Show
that you can glue together $C_0$ and $C_1$ (and the double covers) so
as to give a double cover $C \rightarrow \proj^1$.  (For computational
convenience, you may assume that neither $[0;1]$ nor $[1;0]$ are zeros
of $f$.)  What goes wrong if $n$ is odd? Show that the genus
of $C$ is $n/2-1$.  (This is a special case of the {\em
Riemann-Hurwitz formula}.)  This provides examples of curves of any
genus.  (Not for credit:  What goes wrong if $\ch \kb  = 2$?)

\item[9.] 
\label{hyp2} {\em Hyperelliptic curves take 2.}  Again, assume $\ch \kb \neq 2$.  
Suppose $C$ is a
projective nonsingular curve of positive genus, and $\cL$ is a line
bundle of degree 2.  Prove that $h^0(C,\cL) \leq 2$.  If equality
holds, prove that $|\cL|$ gives a morphism $C \rightarrow \proj^1$.
(It is a fact that all such morphisms are of the form described in the
previous question, and vice versa.)
 
\item[10.] {\em Nonsingular plane curves.}  Suppose $C$ is a nonsingular degree $d$ curve in $\proj^2$.  
Then $\Omega^1_C \cong \oh_C(d-3)$.  Hence prove that the genus of $C$
is $\binom {d-1} 2$.
\end{enumerate}

{\bf The Riemann-Roch formula.}
\begin{enumerate}
\item[11.] {\em Genus 1 and 2 curves.}  
Prove that all projective nonsingular genus 1 curves are hyperelliptic
in the sense of Problem 9.  Prove that all projective
nonsingular genus 2 curves are hyperelliptic (use $\cL=\Omega^1$).
Hence (modulo the parenthetical fact in Problem 9 you have
now seen explicit descriptions of all genus 2 curves (see Problem
8).

\item[12.] {\em Genus 3 curves.}   Suppose $C$ is a genus 3 projective
nonsingular curve.  Suppose furthermore that $C$ is {\em not}
hyperelliptic (see above), i.e. that for each degree 2 invertible
sheaf $\cL$, $h^0(C,\cL) <2$.  Prove that $C$ can be expressed
as a nonsingular plane quartic.  (Hint:  Consider $|\Omega^1_C|$.)
(A mild addition to your argument will show that $C$ can be
expressed as a nonsingular plane quartic in essentially only
one way.)  Hence you can describe what all genus 3 curves look like.

\item[13.] {\em Dimension of the Picard variety.}  The set of degree $d$
line bundles on an nonsingular projective algebraic curve $C$ of genus
$g$ can also be given the structure of an algebraic variety $X$.
Here's a calculation that will convince you that the dimension of $X$
should be $g$.  Suppose $d>2g-2$.  Calculate the number of sections
any degree $d$ line bundle must have (or, more precisely, the
dimension of the vector space of sections).  Each section (up to
multiple by non-zero constant) gives a degree $d$ divisor on $C$, 
and distinct degree $d$ divisors on $C$ give rise to distinct
sections of degree $d$ line bundles (up to non-zero constant).
Explain why the degree $d$ divisors on $C$ should be parametrized
by a dimension $d$ variety.    You now know the dimension of
the ``space of degree $d$ divisors'', and you know the dimension of
the ``space of degree $d$ divisors corresponding to a fixed degree $d$
line bundle''; use this to calculate the dimension of the ``space of
degree $d$ line bundles''.

Now if $d'$ is any integer, show that the space of degree $d'$ line
bundles is isomorphic to the space of degree $d$ line bundles.  (Hint:
let $\cL$ be any line bundle of degree $d-d'$; tensoring by $\cL$ will
give the isomorphism.)
\end{enumerate}

} % end of parskip; it started just before the introduction
\end{document}

