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\begin{document}
\begin{abstract}
This problem set corresponds to the first week of the course MAT-248B Spring 2026. It is due Monday Apr 13 at 3:00pm submitted via Gradescope.
\end{abstract}

\title{MAT 248B: Problem Set 1}
\author{Due to Monday Apr 13}

\maketitle
\vspace{-0.7cm}

{\bf Task}: Solve one of the problems below and submit it through Gradescope by Monday Apr 13 at 3pm. Be rigorous and precise in writing your solutions.\\

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{\bf Problem 1}. Find a basis of the $k$-vector space of sections for the line bundles $T\P_k^1$ and $\Omega\P_k^1$, i.e.~the tangent and cotangent bundles of $\P_k^1$.\\

{\bf Problem 2}. Show that there is no surjective map $\pi:\Sigma_g\lr\Sigma_{g'}$ between two projective curves $\Sigma_g,\Sigma_{g'}$ of genus $g,g'$, respectively, if $g'>g$.\\

{\bf Problem 3}. Consider an algebraic curve $C_d\sse\P_k^2$ of degree $d$, e.g.
$$C_d=\{x^d+y^d+z^d=0\}.$$
Compute the normal bundle of $C_d$ inside $\P_k^2$.\\

{\bf Problem 4}. Let $\pi:\mbox{Bl}_0(\A_k^2)\to\A_k^2$ be the blow-up of the affine plane $\A_k^2$ along the origin $0\in \A_k^2$. Find the normal bundle of the exceptional divisor $E\cong\P^1_k\sse \mbox{Bl}_0(\A_k^2)$.\\

{\bf Problem 5}. Give an instance of an algebraic curve $C$ defined over $k$ and points $p_1,p_2,p_3,q_1,q_2,q_3\in C$ such that
$$\mbox{dim}_k\glsec(C,\SO_{C}(p_1+p_2+p_3))\neq \mbox{dim}_k\glsec(C,\SO_{C}(q_1+q_2+q_3)).$$

{\bf Problem 6}. Find an algebraic variety $X$ and two vector bundles $\mathcal{E}_1\to X$ and $\mathcal{E}_2\to X$ such that $\mathcal{E}_1\cong\mathcal{E}_2$ as smooth vector bundles, but $\mathcal{E}_1\not\cong\mathcal{E}_2$ as algebraic vector bundles.\\

{\bf Problem 7}. Consider an algebraic variety $X$ and the diagonal embedding $\Delta:X\to X\times X$, $\Delta(p)=(p,p)$. Show that the normal bundle of $\Delta$ is algebraically isomorphic to the tangent bundle $TX$.\\

{\bf Problem 8}. Consider the embedding $\iota_3:\P_k^1\lr\P^3_k$ given by the global sections of $\SO_{\P_k^1}(3)$, i.e. $\iota_3([s:t])=[s^3:s^2t:st^2:t^3]$. Show that the normal bundle of $\iota_3(\P_k^1)$ is isomorphic to $\SO_{\P_k^1}(5)\oplus \SO_{\P_k^1}(5)$.\\

{\bf Problem 9}. Let $\mathcal{E}\to\P_k^1$ be a vector bundle. Show that there exist $a_1,\ldots,a_r\in\Z$ such that
$$\mathcal{E}\cong\SO_{\P_k^1}(a_1)\oplus\SO_{\P_k^1}(a_2)\oplus\cdots\oplus\SO_{\P_k^1}(a_r).$$
That is, any vector bundle over $\P_k^1$ splits as a sum of line bundles.
\end{document}