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\begin{document}

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\centerline{\sc Problem Set Six: Due Thursday, March 14}

See the course website for policy on collaboration. 
All references to FOAG 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{Problems from the textbook:} \\
\textbf{16.2.C, 16.2.E and 16.2.F} We didn't get around to mentioning ``very ample" in class; ``very ample" means ``pulled back from a projective embedding". But I think there are enough tools in the text for you to put it together.

\begin{Problem}
This problem is rewritten because the old version was confusing to many. It is meant to be logically equivalent to the old one.
Let $X$ be a scheme and let $D$ be an effective Cartier divisor, meaning a closed subscheme which is locally principal. Let $\cO_D$ be the structure sheaf of $D$, considered as a sheaf on $X$.  Show that there is a short exact sequence of sheaves
\[ 0 \to \cO_X(-D) \to \cO_X \to \cO_D \to 0 \]
on $X$.
\end{Problem}

\begin{Problem}
Here are some useful examples to illustrate subtleties of finiteness and degree:
\begin{enumerate}
\item Let $\alpha$ be the obvious inclusion $\Spec k[x, x^{-1}] \to \Spec k[x]$. Show that all the fibers of $\alpha$ are finite, but that $k[x, x^{-1}]$ is not a finitely generated $k[x]$-module.
\item Let $\beta$ be the obvious morphism $\Spec k[t] \to \Spec k[t^2, t^3]$. Show that $k[t]$ is a finitely generated $k[t^2, t^3]$-module (so $\beta$ is a finite map) but $k[t]$ is not a free $k[t^2, t^3]$-module. Hint: Compute the length of various fibers of $\beta$.
\end{enumerate}
\end{Problem}


\begin{Problem}
Here is an example of a curve which we will use over and over again -- the hyperelliptic curve of genus $g$. Let $k$ be a field, not of characteristic $2$. Let $f(x) = f_{2g+1} x^{2g+1} + f_{2g} x^{2g} +  \cdots + f_2 x^2 + f_1 x$ be a polynomial of degree $2g+1$ with $\text{GCD}(f(x), f'(x)) = 1$. 

We set
\[ \begin{array}{rcl}
A_1 &=& k[x_1,y_1]/\langle y_1^2 - ( f_{2g+1} x_1^{2g+1} + f_{2g} x_1^{2g} +  \cdots + f_2 x_1^2 + f_1 x_1) \\
 A_2 &=&  k[x_2,y_2]/\langle y_2^2 - ( f_{1} x_2^{2g+1} + f_{2} x_1^2 +  \cdots + f_{2g} x_2^2 + f_{2g+1} x_2) . \\
 \end{array}\]

\begin{enumerate}
\item Check that $\Spec A_1$ and $\Spec A_2$ are smooth over $k$.
\item Give an isomorphism  $(A_1)_{x_1} \cong (A_2)_{x_2}$.
\end{enumerate}
We define $X$ to be the gluing of $\Spec A_1$ and $\Spec A_2$ along the isomorphism in $(2)$.
\begin{enumerate}[resume]
\item Check that the obvious maps $\Spec(A_1) \to \Spec k[x_1]$ and $\Spec(A_2) \to \Spec k[x_2]$ glue to give a finite map $\pi : X \to \PP_k^1$.
\end{enumerate}
We define $p_i$ to be the point $\langle x_i, y_i \rangle$ in $\Spec A_i$.
\begin{enumerate}[resume]
\item Check that $\pi^{\ast}( \cO(1)) \cong \cO(2 p_1) \cong \cO(2 p_2)$.
\item Compute the valuations $v_{p_2}(x_1)$, $v_{p_2}(y_1)$, $v_{p_2}(x_2)$, $v_{p_2}(y_2)$.
\item For $d \leq 2g$, show that $\{ 1, x_1, x_1^2, \ldots, x_1^{\lfloor d/2 \rfloor} \}$ is a basis of $\Gamma(X, \cO(d p_2))$. For $d \geq 2g+1$, show that $\{ 1, x_1, x_1^2, \ldots, x_1^{\lfloor d/2 \rfloor}, y_1, x_1 y_1, \ldots, x_1^{ \lfloor (d-2g-1)/2 \rfloor } \}$ is a basis for $\Gamma(X, \cO(d p_2))$. Here $\lfloor z \rfloor$ means $z$ rounded down to the nearest integer.
\end{enumerate}

\end{Problem}




\end{document}


