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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 \#5\hspace{220pt}Due: 15 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 $X = \Spec A$ be an affine variety. The following conditions defined close subsets $Y$ of $X \times \P^n$. Prove that they are equivalent.
		\begin{enumerate}
			\item Y is the locus of zeros of a finite number of homogeneous polynomials in $A[x_0, \ldots, x_n]$.
			\item For the standard affine opens $U_i \subset \P^n$, the intersection $Y \cap (X \times U_i)$ is a closed subset of the affine variety $X \times U_i$.
		\end{enumerate}
	\item Determine all the morphisms $\P^2 \to \P^1$.

\item
	Consider the Veronese embedding $\P^2_{x,y,z} \to \P^5_{u}$ by monomials of degree 2 defined by 
	$(u_0, u_1, u_2, u_3, u_4, u_5) = (z^2, y^2, x^2, yz, xz, xy)$.
	If we drop the coordinate $u_0$, we obtain a map
	$\varphi :  \P^2 \setminus \{q\} \longrightarrow \P^4$
	that is defined at all points except $q = (0, 0, 1)$ by
	$(x,y,z) \mapsto (y^2, x^2, yz, xz, xy)$.
	Prove that the inverse map $\varphi^{-1}$ is a morphism, and that the fibre over $q$ is a projective line.


\item Let $C$ be the projective plane curve $x^3 - y^2 z = 0$.
	\begin{enumerate}
		\item Show that the function field $K(C)$ of $C$ is the field $C(t)$ of rational functions in $t = y/x$.
		\item Show that the point $(t^2 - 1, t^3 - 1)$ of $\P^1(K(C))$ defines a morphism $C \to \P^1$.
	\end{enumerate}

\item Show that the conic $C$ in $\P^2$ defined by the polynomial $x^2 + y^2 + z^2 = 0$ and the twisted cubic $V = V(v_0 v_2 -v_1^2, v_0 v_3 - v_1 v_2, v_1 v_3 - v_2^2)$ in $\P^3$ are isomorphic, by exhibiting a inverse morphisms between them.


\item Show that the morphism $\varphi: \A^1 \to \A^1$ given by $\varphi(t) = t^2$ is an integral morphism.

\end{enumerate}


\end{document}
