%\documentclass{amsart}
\documentclass[11pt]{amsart}
\usepackage[hmargin=1 in, vmargin = 1 in]{geometry}
\usepackage{hyperref}

\parindent=0 in
\parskip=0.05 in



\newcommand{\isom}{\cong} %The isomorphism symbol
\newcommand{\union}{\cup}
\newcommand{\intersection}{\cap}
\newcommand{\bigunion}{\bigcup}
\newcommand{\bigintersection}{\bigcap}
\newcommand{\disjointunion}{\sqcup}
\newcommand{\bigdisjointunion}{\bigsqcup}

\newcommand{\join}{\vee}
\newcommand{\meet}{\wedge}
\newcommand{\bigjoin}{\bigvee}
\newcommand{\bigmeet}{\bigwedge}


%This makes alternating tensors look right in displayed equations
\newcommand{\Alt}{\bigwedge\nolimits}


%Blackboard bold letters.
\renewcommand{\AA}{\mathbb{A}}
\newcommand{\BB}{\mathbb{B}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\DD}{\mathbb{D}}
\newcommand{\EE}{\mathbb{E}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\GG}{\mathbb{G}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\II}{\mathbb{I}}
\newcommand{\JJ}{\mathbb{J}}
\newcommand{\KK}{\mathbb{K}}
\newcommand{\LL}{\mathbb{L}}
\newcommand{\MM}{\mathbb{M}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\renewcommand{\SS}{\mathbb{S}}
\newcommand{\TT}{\mathbb{T}}
\newcommand{\UU}{\mathbb{U}}
\newcommand{\VV}{\mathbb{V}}
\newcommand{\WW}{\mathbb{W}}
\newcommand{\XX}{\mathbb{X}}
\newcommand{\YY}{\mathbb{Y}}
\newcommand{\ZZ}{\mathbb{Z}}

%Calligraphic letters

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}

\newcommand{\fm}{\mathfrak{m}}
\newcommand{\GL}{\text{GL}}
\newcommand{\Mat}{\text{Mat}}

\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\Hom}{Hom}


\theoremstyle{definition}
\newtheorem{Problem}{Problem}
\newtheorem{problem}{Problem}





\newcommand{\into}{\hookrightarrow}
\newcommand{\onto}{\twoheadrightarrow} 
\newcommand{\from}{\leftarrow}

\newenvironment{sbm}
    {\left[ \begin{smallmatrix}
    }
    { 
     \end{smallmatrix} \right]
    }
    
\newcommand{\newword}[1]{\textbf{\emph{#1}}}

     
\pagestyle{empty}
\begin{document}

\thispagestyle{empty}
\centerline{\sc Problem Set Three: Due Thursday, February 8}

See the course website for policy on collaboration. 
All references to FOAG refer to the July 31, 2023 edition of \href{http://math.stanford.edu/~vakil/216blog/FOAGjul3123public.pdf}{\emph{Foundations of Algebraic Geometry}}, by Ravi Vakil.

\textbf{Problems from the textbook:} For all of these problems, feel  free to assume that your scheme is noetherian, so that you don't have to think about the distinction between ``finitely generated", ``finitely presented" and ``coherent".


\textbf{Exercise 14.4.F}: This shows that ``locally free" is equivalent to ``free stalks".\\
\textbf{Exercise 14.4.K} This shows that dimension of fibers only jumps up, not down, when we approach special points.\\
\textbf{Exercise 14.4.L} This justifies an assertion in class; that vector bundles are, in a certain sense, characterized by having fibers of constant dimension.\\

\textbf{Exercise 14.7.C} This is an excellent exercise to understand the sheaf corresponding to a graded module.\\
\textbf{Exercise 14.7.D} Compare the result of this exercise to problem~\ref{ExtraAndMissingSections}.\\

\textbf{Exercise 15.1.C} This computes that the sections of $\cO(d)$ on $\PP^{n-1}$ are precisely the degree $d$ polynomials in $n$ variables; in class we saw why such polynomials give sections but didn't see why all sections were of this form.\\
\textbf{Exercise 15.1.E}  This exercise classifies line bundles on $\PP^1$. If you want to do $\PP^{n-1}$ by brute force, this is also possible, but you'll have better tools for this problem after you've read Section 15.4.\\


\begin{Problem} \label{ExtraAndMissingSections}
This Problem gives you practice in computing the global sections of a sheaf on a projective scheme. Throughout, let $k$ be a field. We identify $\PP^{n-1}$ with $\Proj k[x_1, x_2, \ldots, x_n]$.
\begin{enumerate}
\item Let $X$ be the subscheme of $\PP^1$ given by the homogenous ideal $\langle x_1 x_2 \rangle$. What are the global sections of $\cO$ on $X$? 
\item Let $Y$ be the subscheme of $\PP^3$ given by the 	homogenous ideal $\langle x_1 x_3, \ x_1 x_4,\  x_2 x_3,\ x_2 x_4 \rangle$. What are the global sections of $\cO$ on $X$?
\item Let $Z$ be the subscheme of $\PP^3$ given by the homogenous ideal $\langle x_1^2 x_3 - x_2^3,\ x_2 x_4^2 - x_3^3,\ x_1 x_4 - x_2 x_3 \rangle$. What are the global sections of $\cO(1)$ on $X$? Hint: First that $Z$ is covered by the affine opens $Z \cap D_+(x_1)$ and $Z \cap D_+(x_4)$.
\end{enumerate}
\end{Problem}

\end{document}


