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\centerline{\sc Problem Set Seven: Due Thursday, March 28}

See the course website for policy on collaboration. 
All references to the textbook refer to the February 21, 2024 edition of \href{http://math.stanford.edu/~vakil/216blog/FOAGfeb2124public.pdf}{\emph{Foundations of Algebraic Geometry}}, by Ravi Vakil.

\textbf{Textbook problems:}\\
\textbf{18.2.A} Check that the \v{C}ech complex is a complex.\\
\textbf{18.2.D} Check that sheaf cohomology commutes with localization. This is the key lemma which lets us define ``relative sheaf cohomology".\\

\begin{Problem}
This is a follow up to Problem 1 on the previous problem set. To summarize the results of that problem:
Let $X$ be a $\leq d$ dimensional subvariety of $\PP^{n-1}$. Let $\lambda_d$ be the rational map $\PP^{n-1} \to \PP^{d-1}$ given by $\lambda((x_1 : \cdots : x_n)) = (x_1 : \cdots : x_d)$, and let $Z_d = V_+(x_1, x_2, \ldots, x_d)$ be the locus where $\lambda_d$ is not defined. You showed that, if $X \cap Z_d = \emptyset$, then $\pi : X \to \PP^{d-1}$ is finite.
\begin{enumerate}
\item Let $k$ be an infinite field and let $X \subset \PP^n_k$ be a closed subscheme of dimension $\leq d-1$. Show that we can choose coordinates in $\PP_k^{n-1}$ such that $X \cap Z_d$ is empty.
\item Let $Y$ be a scheme and let $\fp$ be a point of $Y$ with infinite residue field $\kappa(\fp)$. Let $X$ be a closed subscheme of $\PP^{n-1}_Y$ such that the fiber $X \cap \PP^{n-1}_{\fp}$ has dimension at most $d-1$. Show that we can find an open neighborhood $U \ni \fp$ and choose coordinates in $\PP^{n-1}_U$ such that the map $\pi_d : X \cap \PP^{n-1}_U \to \PP^{d-1}_U$ is finite.
\item In particular, let $Y$ be a scheme such that $\kappa(\fp)$ is infinite for every $\fp \in Y$ and let $X$ be a closed subscheme of $\PP^{n-1}_Y$ such that $X \cap \PP^{n-1}_{\fp}$ is finite for every $\fp \in Y$. Show that the projection map $X \into \PP^{n-1}_Y \to Y$ is finite.
\item Let $k$ be an infinite field, let $X$ be a closed subscheme of $\PP^{n-1}_k$,  let $Y$ be a separated $k$-scheme, and let $\pi : X \to Y$ be a morphism of $k$-schemes such that every fiber of $\pi$ is finite. Show that $\pi$ is a finite map. (Thanks to the several students who pointed out the requirement that $Y$ be separated. Here is an example to demonstrate the issue. Let $X = \PP^1$ and let $Y$ be $\PP^1$ with a point doubled; let $\pi: X \to Y$ be the open inclusion whose image misses one of the two doubled points. In a neighborhood of the missed point, this is not a finite map.)
\end{enumerate}
%\textbf{Comment:} Ravi proves this as an application of sheaf cohomology in Theorem 18.1.6 and remarks that it is one of several results that ``can also be shown directly, with-
%out the use of cohomology, but with much more elbow grease". I saw this and felt that sheaf cohomology was dramatic overkill, and this result was far too useful to wait for it. Having written the problem now, I'm surprised by how much elbow grease it took \dots
\end{Problem}

\begin{Problem}
 Let $\cO^{\ast}$ be the sheaf of abelian groups where $\cO^{\ast}(U)$ is the unit group of the ring $\cO(U)$. In this problem, we will relate the \v{C}ech cohomology of $\cO^{\ast}$ to the Picard group. Note that $\cO^{\ast}$ is a sheaf of abelian groups but \textbf{not} a quasi-coherent sheaf (or a sheaf of $\cO_X$-modules at all!), so you'll have to work with definitions without much theory to aid you.
\begin{enumerate}
\item Let $U_i$ be an open cover of $X$, so we have a corresponding \v{C}ech complex 
\[ \prod_i \cO^{\ast}(U_i) \overset{d^0}{\longrightarrow} \prod_{i<j} \cO^{\ast}(U_i \cap U_j) \overset{d^1}{\longrightarrow} \prod_{i<j<k} \cO^{\ast}(U_i \cap U_j \cap U_k).  \]
Let $\phi_{ij} \in \Ker(d^1)$. Show that it makes sense to define a line bundle $\cL(\phi)$ by gluing the trivial bundle over $U_i$ to the trivial bundle over $U_j$ by the gluing map $\phi_{ij}$. Hint: Why does it matter that $\phi_{ij} \in \Ker(d^1)$?
\item Suppose that $\phi$ and $\psi$ are two classes in $\Ker(d^1)$. Show that the line bundles $\cL(\phi)$ and $\cL(\psi)$ are isomorphic if and only if $[\phi]$ and $[\psi]$ define the same class in the \v{C}ech cohomology group $H^1(X, \cO^{\ast}, U_i)$. Conclude that we get an injection  $H^1_{\cU}(X, \cO^{\ast}) \into \text{Pic}(X)$.
\item Show that $\Pic(X)$ is the union of the images of all the \v{C}ech groups  $H^1_{\cU}(X, \cO^{\ast})$, as $\cU$ ranges over all open covers.
\item In particular, suppose that $U_i$ is an open cover where $U_i = \text{Spec}(A_i)$ and each $A_i$ is a unique factorization domain. Show that $H^1_{\cU}(X, \cO^{\ast}) \cong \text{Pic}(X)$.
\end{enumerate}
\end{Problem}




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