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	\large
	\textbf{18.721 Introduction to Algebraic Geometry\hspace{228pt}Fall~2021}\\\vspace{4pt}
	\textbf{Problem Set \#4\hspace{220pt}Due: 8 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 $X = \Specmax(A)$ be an affine algebraic variety, and let $f, g$ be elements of its function field $K = \mathrm{Frac}(A)$.
		\begin{enumerate}
			\item Prove that there is a Zariski open subset $U \subset X$ such that both $f$ and $g$ are regular on $U$.
			\item Suppose that for every $p \in U$, we have $f(p) = g(p)$. Prove that $f = g$.
	\end{enumerate}

\item
Let $X = \Specmax(A)$ be an affine variety. Show that homomorphisms $X \to \mathbb{A}^1$ correspond exactly to elements of $A$.

\item
Let $n$ be an integer. Describe an affine variety $X$ such that for any affine variety $Y = \Specmax{B}$, the set of homomorphisms $Y \to X$ can be naturally identified with the set of invertible $n \times n$ matrices with coefficients in $B$ and having trace 0.

\item
Consider the action of the Klein four group $V_4$ on $A = \C[x,y]$, where the first generator acts by swapping $x$ and $y$, and the second generator sends $x$ to $-x$ and $y$ to $-y$. Determine the ring of invariants $A^{V_4}$.

\item
Consider the action of the symmetric group $S_3$ on $A = \C[x,y,z]$. You may assume that $A^{S_3} = \C[s_1, s_2, s_3]$ with $s_1 = x+y+z$, $s_2 = xy+yz+zx$, and $s_3 = xyz$.
\begin{enumerate}
	\item Prove that $\Frac(A^{S_3}) = \C(s_1, s_2, s_3)$ has transcendence degree 3 over $\C$.
	\item Find a polynomial $f \in A^{S_3}[t]$ of degree 3 such that $B \colonequals A^{S_3}[x] \subset A$ is isomorphic to $A^{S_3}[t] / (f)$.
	\item Find a polynomial $g \in B[t]$ of degree 2 such that $C \colonequals B[y] \subset A$ is isomorphic to $B[t] / (g)$.
	\item Prove that $C = A$, and deduce that $A$ is a free $A^{S_3}$-module of rank 6 with basis $1,x,x²,y,xy,x²y$.
	\item Let $\sigma \in S_3$ be the cycle $(123)$. Explain how you can reduce the task of finding $A^{\langle \sigma \rangle}$ to a linear algebra problem over $A^{S_3}$. {\em You are not asked to actually solve this linear algebra problem. If you like to do that, we recommend using a computer algebra package.}
\end{enumerate}
	\item Show that the diagonal in $\P^n \times \P^n$ is not closed in the product topology.
\end{enumerate}


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