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\begin{document}
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	\textbf{18.721 Introduction to Algebraic Geometry\hspace{228pt}Fall~2021}\\\vspace{4pt}
	\textbf{Problem Set \#8\hspace{220pt}Due: 5 Nov 2021}
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\subsection*{Description}

Solutions are to be prepared in typeset form (typically via latex) and submitted electronically as a pdf file via \href{https://www.gradescope.com/courses/281407}{Gradescope} by midnight on the due date.
Late problem sets will not be accepted -- your lowest score is dropped, so you can afford to skip one problem set without penalty.

Collaboration is permitted/encouraged, but you must write up your own solutions and explicitly identify your collaborators, as well as any resources you consulted that are not listed above. If there are none, please indicate this by writing \textbf{Sources consulted:\ none} at the top of your submission.

\medskip

\noindent
\begin{enumerate}

	\item Prove that, when a variety $X$ is covered by countably many constructible sets, a finite number of those sets will cover $X$.

	\item 
	\begin{enumerate}
		\item Let $X$ be a proper affine variety. Show that $X$ is a point.
		\item Let $f \colon X \to Y$ be surjective morphism of varieties, and suppose $X$ is a proper variety. Prove that $Y$ is proper. 
		\item Let $X$ and $Y$ be varieties. Show that $X \times Y$ is proper if and only if both $X$ and $Y$ are proper.
	\end{enumerate}

	\item Consider the space $Z$ defined in Example 5.5.7 of the 22 October version of the lecture notes.
	\begin{enumerate}
		\item Consider the subspace $Z_{\mathrm{irred}}$ of irreducible polynomials. Prove that $Z_{\mathrm{irred}}$ is open in $Z$.
		\item Consider the projection map $\pi \colon \Sigma \to Z$. Show that $\pi^{-1}(z)$ has a component of dimension greater than 0 if and only if the polynomial $f$ corresponding to $z$ is not square-free.
		\item Show how the semicontinuity of the fibre dimension implies that the subspace $Z_{\textrm{sq-free}}$ of squarefree polynomials is open in $Z$.
	\end{enumerate}

	\item Finish the proof of Theorem 5.6.2. {\em Hint: see also exercises 5.7.16 and 5.7.18 in the lecture notes (but you are not obliged to use these).}

	\item Let $u \colon Y \to X$ be a surjective morphism of varieties and define $\Delta \colon X \to \mathbb{Z}$ by $$\Delta(x) = \max_{y \in f^{-1}(x)}\delta(y).$$ Suppose $Y$ is proper, prove that $\Delta$ is upper semicontinuous.
	
	\item Let $P = (0 : 0 : 1) \in \mathbb{P}^2$. Determine $\mathcal{O}_{\mathbb{P}^2}(\mathbb{P}^2 \setminus \{P\})$.

\end{document}
