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\begin{document}
\pagestyle{plain}
\title{Intro to Algebraic Geometry, Problem Set 3}
\date{September 28, 1999.}
\maketitle

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

Due Thursday October 7 in class.  No lates will be accepted, so
Ana-Maria will be able to return it in time for the next Tuesday's class.  Do
any 7 of the following 9 problems, including Problem 9 (which is, I
think, the hardest).  You're strongly encouraged to collaborate
(although obviously 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).
Ask someone about the solutions to the two problems you don't hand in too.


{\bf More straightforward problems.} 
\begin{enumerate}

\item  Prove that every prevariety $X$ is an irreducible
topological space.  (In particular, by a result on problem set 2,
every open set is dense.)  Prove that every prevariety is a Noetherian
space.

\item Let $X$ be the prevariety that is the line with the ``double
origin''.  Find the global sections of $\oh_X$.

\item Find the global sections of $\oh_{\proj^n}$.  

\item Prove that $\A^2 \setminus \{ (0,0) \}$ is not an affine variety.

\end{enumerate}

{\bf Sheaf problems.}
\begin{enumerate}


\item[5.] Suppose $f:  X \rightarrow Y$
is a (continuous) morphism of topological spaces, and $F$ is a sheaf
on $X$.  To each open set $U$ on $Y$, define $G(U) = F(f^{-1} (U))$.
Show that $G$ is a sheaf on $Y$.  (The sheaf $G$ is denoted $f_* F$,
and is called the {\em pushforward} of $F$.  There is also a {\em
pullback}, but it is more complicated to define.)

\item[6.] In class, I mentioned that 
you can recover a sheaf from knowing its sections over the elements of
a base, and the restriction maps between them.  This exercise makes
this precise.

\item[](a)  Suppose $F$ is a sheaf on a topological space $X$, with open subset $U$
that is the union of open sets  $U_i$ (where $i$ varies over some set $A$).  Show that 
$$F(U) \cong \{ (f_i \in F(U_i))_{i \in A} | \res_{U_i, U_i \cap U_j} f_i  =
\res_{U_j, U_i \cap U_j} f_j \text { for all $i$, $j \in A$ } \}.$$

\item[](b)  Suppose furthermore that $U_i \cap U_j$ is the union of open sets $U_{ijk}$ where
$k$ varies over some set $A_{ij}$.
Show that 
$$F(U) \cong \{ (f_i \in F(U_i))_{i \in A} | \res_{U_i, U_{ijk}} f_i  =
\res_{U_j, U_{ijk}} f_j \text { for all $i$, $j \in A$, $k \in A_{ij}$ } \}.$$

\item[](c)  Explain how you can recover a sheaf from knowing its sections over the elements of a base of the topology, and 
the restriction maps between them.  (Translation:  convince Ana-Maria you understand how
(b) above gives this fact.)  

\end{enumerate}

{\bf Theory.}

\begin{enumerate}

\item[7.]   In the course of proving that if $X$ is an affine variety and $f$ is a regular
function, then $D(f)$ is an affine variety, we used a fact we didn't prove.  We'll correct
this oversight now.  Suppose $X$ lies in $\A^n$ with co-ordinates $x_1$, \dots, $x_n$,
and the corresponding (radical) ideal is $I=I(X)$.  As $f \in \R/I$, let $f_1$ be an
element of $\R$ restricting to $f$ (i.e. choose a polynomial ``representing'' $f$).
Let $J$ be the ideal in $\kb[x_1, \dots, x_{n+1}]$ generated by $I$ and $1-f_1 x_{n+1}$.
(We showed that $J$ is prime.)  Define a morphism (of affine algebraic sets)
from $V(J)$ to $X$ by
$$
\pi:  (x_1, \dots, x_{n+1}) \mapsto (x_1, \dots, x_n).$$
Show that $\pi$ gives a homeomorphism from $V(J)$ to $D(f)$.

\item[8.]  {\em Projection from a point.}  Explain how the map
$\proj^n \setminus \{ (1;0; \dots; 0) \} \rightarrow \proj^{n-1}$ given
by $(x_0; x_1; \dots; x_n) \mapsto (x_1; \dots; x_n)$  corresponds
to a morphism of prevarieties.

\item[9.] Let $f(x_0, x_1, \dots, x_n)$ be a polynomial of {\em homogeneous degree $d$} (i.e.
$x_i$ has degree 1, and each monomial appearing in $f$ has total degree $d$).
\item[](a) Note that the set $S = \{ f(x_0, \dots, x_n) =0 : (x_0 ; \dots; x_n) \in \proj^n \}$ is well-defined:
if $\lambda \in \kb^*$, $f(x_0, \dots, x_n) = 0$ if and only if $f(\lambda x_0, \dots, \lambda x_n) = 0$.
Show that $S$ is a closed
subset of $\proj^n$.  
\item[](b)  Let $U$ be its complement; calculate
$\oh_{\proj^n}(U)$.  (Hint:  calculate the sections over the $n+1$ standard affine open sets, and 
try to patch them together.)
\item[](c)  (Not for credit; tricky but doable given what you know.) Think about how you would prove that 
$(U,\oh_{\proj^n}|_U)$ is an affine variety if the degree of $f$ is
positive.  (We'll do this in class; if you understand this, you'll have a good
handle on projective varieties.)

\end{enumerate}


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

