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\begin{document}
\begin{abstract}
This is the second problem set of the course MAT-248B Spring 2026. It is due Friday May 8 at 3:00pm submitted via Gradescope.
\end{abstract}

\title{MAT 248B: Problem Set 2}
\author{Due to Monday May 8th}

\maketitle
\vspace{-0.7cm}

{\bf Task}: Solve one of the problems below and submit it through Gradescope by Friday May 8th at 3pm. Be rigorous and precise in writing your solutions.\\

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{\bf Problem 1}. Consider the affine subvariety $Y=\{x^5=y^3\}\sse\C^2$ and its coordinate ring
$$\SO_Y(Y)\cong \C[x,y]/(x^5-y^3).$$

\begin{enumerate}[$(a)$]
    \item Describe the cotangent sheaf $\Omega_Y$ as a sheaf $\SO_Y$-module.\\
    
    \item At each point of $Y$, compute the dimension of the stalk $(\Omega_{Y})_p$ and find a basis of $(\Omega_{Y})_p$ as an $\SO_{Y,p}$-module.\\

    \item Show that the conormal sheaf of $Y$ in $\C^2$ is locally free.\\
\end{enumerate}

{\bf Problem 2}. Consider the projective curve $\{x^3-yz^2=0\}\sse\P^2_k$. Find the cohomologies of its tangent sheaf and its normal sheaf.\\

{\bf Problem 3}. Give an instance of an affine subvariety $Y\sse\C^n$ such that the conormal sheaf is {\it not} locally free.\\

{\bf Problem 4}. Show that the sheaf $\SO_{\P^1}(n)$ on $\P^1$ is coherent for any $n\in\Z$.\\

{\bf Problem 5}. Let $C$ be a projective curve and $f:C\lr\P^1$ a given finite morphism of degree $d$ (e.g.~a degree $d$ simple branched cover). Compute the cohomology of $\SO_C$ using $f$.\\

{\bf Problem 6}. Consider the projective surface $X=\P^1_k\times\P^1_k$, compute $H^*(X,\SO_X)$ using Cech cohomology.\\

{\bf Problem 7}. Compute the cohomology of the structure sheaf $\SO_X$ of the algebraic variety $X=\mathbb{A}^n_k\setminus\{(0,\ldots,0)\}$, i.e.~$X$ is affine $n$-space with the origin removed, $n\geq 2$. Conclude that $X$ is not affine.\\

{\bf Problem 8}. Exercise 4.9 from Hartshorne's Chapter III.\\

{\bf Problem 9}. Give an instance of a (necessarily singular) projective variety that does {\it not} admit a dualizing sheaf.


\end{document}