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\begin{document}
\pagestyle{plain}
\title{Intro to Algebraic Geometry, Problem Set 11}
\date{December 3, 1999.}
\maketitle

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

Due Thursday December 9 in class.  Hand in 6 of these problems.
Ask someone about the ones you skip.  Problem 8 has a tricky step,
but the punchlines are worth it, so I encourage you to try it.
(You might want to try it earlier than the rest, to give yourself
time to talk to me about it.)
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).

I'll use ``line bundles'' and ``invertible sheaves'' interchangeably.

\begin{enumerate}
\item[1.]  {\em Secretly trivial line bundles.}
Suppose $U_i$ is a finite open cover of $X$, and for each $i$, $h_i$
is an invertible function on $U_i$.  Prove that the data of the open
cover $U_i$ and the transition functions $f_{ij} = h_i/h_j$ defines a
line bundle, and prove that this is isomorphic to the trivial line
bundle.  In fancy language, this shows that a zero-element of $H^1(X,
\oh^*)$ indeed induces a trivial line bundle.  (One possibility: 
Use the proposition, proved in Class 23, that any invertible sheaf on
$X$ with a nowhere vanishing section is isomorphic to the trivial
sheaf $\oh_X$.)  Conversely, suppose that the line bundle given by
$U_i$ and $f_{ij}$ is trivial.  Prove that there are invertible
functions $h_i$ on the $U_i$ such that $f_{ij} = h_i/h_j$ on $U_i \cap
U_j$.

\item[2.] Give (with proof) the identification between the vector space 
of global sections of $\oh_{\proj^n}(m)$ (where $\proj^n$ has
projective coordinates $x_0$, \dots, $x_n$) and the vector space of
polynomials in $x_0$, \dots, $x_n$ of homogeneous degree $m$.

\item[3.] {\em Torsion in the Picard group.}
Let $X$ be the variety $\proj^2$ minus an irreducible conic.  Let
$\oh_X(m)$ be the restriction of $\oh_{\proj^2}(m)$ to $X$.  Show that
$\oh_X(2)$ is trivial, but that $\oh_X(1)$ isn't.  Hence $\oh_X(1)$ is
a 2-torsion element of $\Pic X$.  (Hint: Show that $\oh_X(2)$ has a
global section vanishing nowhere, and that every global section of
$\oh_X(1)$ vanishes somewhere.  Use the proposition, proved in Class
23,  that any invertible sheaf on $X$ with a nowhere vanishing section
is isomorphic to the trivial sheaf $\oh_X$.

\item[4.]   Prove that $\oh_{\proj^1}(p)$ is isomorphic to $\oh_{\proj^1}(1)$.
{\em A small addition, thanks to a good question in Thursday's class.}
If you choose coordinates so $p = [1;0]$, then the global sections of
$\oh_{\proj^1}(p)$ can be identified with linear combinations of the
rational functions $1$ and $x_0/x_1$.  You might think that the global
section corresponding to $1$ has no zeros or poles, but this isn't
true!  (For example, you know from Problem Set 10 that the sums of
orders of zeros and poles of $\oh_{\proj^1}(1)$ is 1.)  So: find the
zeros and poles corresponding to $1$.

\item[5.]  Consider the morphism $\pi:  \proj^1 \rightarrow \proj^3$ 
given by $[u; v] \mapsto [u^4; u^3 v; u v^3; v^4]$ which expresses $\proj^1$
as a closed subvariety $C$ of $\proj^3$.  (You don't need to verify that.)
By Problem 2, the space of global sections of $\oh_{\proj^3}(1)$ has dimension
4.  Show that this induces a 4-dimensional vector space of global sections
$\pi^* \oh_{\proj^3}(1)$.  (Hint: show that no nonzero section of $\oh_{\proj^3}(1)$ vanishes on all of $C$.  This may involve exhibiting 4 points of $C$ that
are not coplanar.)  Show that $\pi^* \oh_{\proj^3}(1) \cong \oh_{\proj^1}(4)$,
so (by Problem 2), $\pi^* \oh_{\proj^3}(1)$ has a 5-dimensional vector space of global sections.

Hence if $\cL$ is an invertible sheaf on a variety $Y$, and $X$ is a
subvariety of $Y$, then the restriction of $Y$ to $X$ may have more
sections than those sections restricted from $Y$.

\item[6.] {\em Pullbacks of divisors and pullbacks of line bundles.}
Suppose $\pi: X \rightarrow Y$ is a dominant morphism of nonsingular curves,
and $D$ is a divisor $\sum_{q \in Y } n_q q$ on $Y$.  Define the
divisor $\pi^* D$ on $X$ by $\sum_{p \in X} m_p n_{\pi(p)} p$
where $m_p$ is defined as follows.  Let $u$ be a uniformizer in the
stalk $\oh_{Y, \pi(p)}$.  Then $u$ pulls back to give an element
$\pi^* u$ of $\oh_{X,p}$.  Then $m_p = v_p(\pi^* u)$.  Prove that
$m_p$ is independent of the choice of $u$.  Then prove that $\pi^*
\oh_Y(D) \cong \oh_X(\pi^* D)$.

\item[7.]  Suppose $X$ is a nonsingular projective curve, and $\cL$ is an invertible
sheaf on $X$ of degree $d$.  Prove that the sections of $\cL$ form a
{\em finite-dimensional} $\kb$-vector space.  (Hint: Prove it if
$d<0$.  Show by induction that if $d \geq 0$, then $\cL$ has at most
$d+1$ linearly independent sections.)  This fact is true more
generally, whenever $X$ is a projective scheme.  It is not necessarily
true if $X$ isn't projective (consider $\A^1$ for example).

\item[8.] {\em Fun with elliptic curves.}
Let $C$ be the curve $y^2 z = x^3 - x z^2$ in $\proj^2$.  (Everything will
work for $y^2 z = x^3 - \al x z^2$ for any nonzero $\al$.)
Let $\pi:  C \rightarrow \proj^2$ be the inclusion.
Let $\cL = \pi^* \oh(1)$ (an invertible sheaf on $C$). 

\item[](a)  Show that $\deg \cL = 3$.

\item[](b) Show that $\cL$ has a 3-dimensional family of global sections,
and that they are all restrictions of global sections of
$\oh_{\proj^2}(1)$.  Hence any nonzero global section of $\cL$
vanishes at collinear points (where $C$ is considered as lying in the
plane).  This step might get tricky.  If you get stuck after setting
up what you want to prove, come by on Wednesday afternoon or earlier.

\item[](c)  Use (a), (b) and Problem 2 to show that $C$ is not isomorphic to $\proj^1$.

\item[](d) {\em A nontrivial degree 0 line bundle.}
Let $p$ and $q$ be two distinct points of $C$.  Let $\cM = \oh_C(p-q)$
(a degree 0 line bundle).  Show that $\cM$ is not isomorphic to the
trivial line bundle as follows.  Let $q$, $r$, $s$ be distinct
collinear points on $C$ so that $p \neq r$, $p \neq s$.  Show that
$\oh_C(q+r+s) \cong \cL$.  Hence $\cL \otimes \cM \cong \oh_C(p+r+s)$,
so there is a global section of $\cL \otimes \cM$ with divisor
$p+r+s$.  Show that $\cL \otimes \cM$ can't be isomorphic to $\cL$
using (b), and hence that $\cM$ isn't trivial.  (Once again, this
proves that $C$ is not isomorphic to $\proj^1$, as the only degree 0
invertible sheaves on $\proj^1$ is the trivial sheaf.)

\item[](e) {\em The group law on the elliptic curve.}
We can put a group structure on the curve as follows.  Call the point
$[0;1;0]$ $o$; this will be the zero-point.  Prove that $\oh(3o) \cong
\cL$.  Define a map $i:  C \rightarrow \Cl(C)$ by $p \mapsto p-o$.  Show that this
map is injective (as sets), using (d).  Given two points $p$ and $q$
on $C$, define a point $r$ as follows.  Let $L$ be the line joining
$p$ and $q$ in the plane; say it meets $C$ at $r'$ as well.  Let $M$
be the line joining $o$ and $r'$; let $r$ be the third point of
intersection.  (Questions to ask yourself: what if $L$ is tangent to
$C$ at $p$?  What if $p=q$?  Think about these and similar issues.  In
writing up the problem, feel free to avoid them, but say clearly
what cases you aren't dealing with, and what assumptions you make.)
Prove that $i(p)+i(q) = i(r)$.  Given a point $p \in C$, describe a point
$p'$ such that $i(p)+i(p') = i(o)$ (with proof).  Show that this
gives the points of $C$ the structure of a group.  (We will see,
on the last day of class or in PS12, that the image of $i$
are precisely the degree 0 line bundles.)
 
\end{enumerate}

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

