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\begin{document}

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\centerline{\sc Problem Set Nine: Due Thursday, April 11}

See the course website for policy on collaboration. 

\textbf{Textbook problems:}\\
\textbf{18.5.B} A nice concrete application of Serre duality. You'll want to prove Problem~1 on this worksheet first.\\
\textbf{18.6.G, 18.6.H, 18.6.J} These are all excellent practical problems about degree. I think they should all be short.

\begin{Problem}
An important consequence of Serre duality. Let $k$ be a field and let $X \subset \PP^r_k$ be a smooth, closed, geometrically irreducible subvariety of dimension $d$. Let $\cE$ be a locally free sheaf on $X$. Show that, for $0 \leq q \leq d-1$ and $N$ sufficiently large, we have $H^q(X, \cE \otimes \cO(-N)) = 0$. (Remark: We showed this without Serre duality in the case $X = \PP^1$, do you remember how?)
\end{Problem}

\begin{Problem}
Let $k$ be a field and let $X$ be an integral (Definition 5.2.4) closed subscheme of $\PP^n_k$.
Show that $H^0(X, \cO_X)$ is a field and $[H^0(X, \cO_X):k]<\infty$. Hint: This comes down to an algebraic fact, which you should prove: If a finite dimensional $k$-algebra is an integral domain, it is a field.
\end{Problem}


\begin{Problem}
Let $f$ and $g$ be relatively prime homogeneous polynomials in  $k[x_0, x_1, x_2, x_3]$ of degrees $a$ and $b$. Let $\Gamma$ be the closed subscheme of $\PP^3_k$ given by $f=g=0$. If $\Gamma$ is regular and irreducible, then what is its genus?
\end{Problem}
\end{document}


