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\begin{document}
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	\large
	\textbf{18.721 Introduction to Algebraic Geometry\hspace{228pt}Fall~2021}\\\vspace{4pt}
	\textbf{Problem Set \#7\hspace{220pt}Due: 29 Oct 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 Let $X$ be a variety and let $C_0 \supsetneq C_1 \supsetneq \ldots \supsetneq C_k$ be some chain of irreducible closed subvarieties of $X$. Prove that this chain can be extended to a chain of length $\mathrm{dim}(X)$, by putting new irreducible closed subvarieties between, before, or after the existing ones.

	\item Let $f \colon X \to Y$ be a finite morphism. Prove that for every $y \in Y$ the fibre $f^{-1}(y)$ is finite.

	\item For each of the following statement determine whether the statement is true. Give a proof or disproof.
	\begin{enumerate}
		\item Chevalley's finiteness theorem (corollary 4.6.8 in the 15 Oct version of the notes) also holds when $Y$ is affine.
		\item If $f \colon X \to Y$ is a surjective morphism of varieties and $\mathrm{dim}(X) = \mathrm{dim}(Y)$, then $f$ is finite.
		\item Let $f \colon X \to Y$ be a surjective morphism of curves. Suppose that $X$ is smooth. Then $Y$ is smooth.
		\item Let $f \colon X \to Y$ be a surjective morphism of curves. Suppose that $Y$ is smooth. Then $X$ is smooth.
	\end{enumerate}

	\item Prove that, if $v$ is a (discrete) valuation on a field $K$ that contains the complex numbers, every nonzero complex number $c$ has value zero.

	\item Prove that every nonconstant morphism $\mathbb{P}^2 \to \mathbb{P}^2$ is finite. Do this by showing the fibres cannot have positive dimension.

\end{document}
