%\documentclass{amsart}
\documentclass[11pt]{amsart}
\usepackage[hmargin=1 in, vmargin = 1 in]{geometry}
\usepackage{hyperref}
\usepackage{enumitem}
\usepackage[normalem]{ulem}
\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 Five: Due Thursday, February 22}

See the course website for policy on collaboration. 
All references to FOAG refer to the February 6, 2024 edition of \href{http://math.stanford.edu/~vakil/216blog/FOAGfeb0624public.pdf}{\emph{Foundations of Algebraic Geometry}}, by Ravi Vakil.

\textbf{Problems from the textbook:} \\
\textbf{15.4.J} We carry out the verification, promised in class, of a divisor which is not locally principal. \\
\textbf{15.4.M} Here is a nice example of an explicit class group which is easy to compute by algebraic geometry, but which I think would be very hard by pure algebra.\\
\textbf{15.4.Q} The hints for this exercise aren't exactly the same as the way I'd do this. Feel free to prove that $\text{Cl}(\PP^1 \times \PP^1) \cong \ZZ^2$ any way that you like.\\
\textbf{16.1.C } As the text says, this is ``easy but important". It is a good example of the use of Nakayama's lemma.\\



\begin{Problem}
Here are two problems on pulling back line bundles. 
\begin{enumerate}
\item Let $\alpha : \PP^{n-1} \to \PP^{n}$ be the inclusion of the hyperplane $x_{n+1}=0$. Show that $\alpha^{\ast} \cO_{\PP^n}(d) \cong \cO_{\PP^{n-1}}(d)$. 
\item Let $\beta : \PP^1 \to \PP^r$ be the $r$-fold Veronese map; we can realize this in in homogeneous coordinates as $\beta(x:y) = (x^d : x^{d-1} y : \cdots : y^d)$. Show that $\beta^{\ast} \cO_{\PP^d}(1) \cong \cO_{\PP^1}(d)$. Hint: Writing $(z_0 : z_1 : \cdots : z_d)$ for the coordinates on $\PP^d$, it may help to show that $\alpha(\PP^1) \subset D_+(z_0) \cup D_+(z_d)$.
\end{enumerate}
\end{Problem}

\begin{Problem}
This problem gives an example where $D$ is an effective locally principle divisor, but the restriction of $\cO(D)$ to $D$ is not effective.

Let $k$ be a field and let $K = k(x,y)$. Let $A$ and $B$ be the subrings $k[x, x^{-1} y]$ and $k[x y^{-1} , y]$ of $K$.
\begin{enumerate}
\item Verify that $A_{x^{-1} y} = B_{xy^{-1}}$ as subrings of $K$.
\end{enumerate}
We define the scheme $X$ to be the gluing of $\Spec(A)$ to $\Spec(B)$ along $\Spec(A_{x^{-1} y}) = \Spec(B_{x y^{-1}})$. We define the divisor $D$ to be the closed subscheme where $D \cap \Spec(A) = V(x)$ and $D \cap \Spec(B) = V(y)$. 
\begin{enumerate}[resume]
\item Verify that $D \cong \PP^1$.
\item Let $\phi : D \to X$ be the inclusion of $D$ into $X$. Verify that $\phi^{\ast} \cO(D)$ is the line bundle $\cO(-1)$ on $D$.
\end{enumerate}
The variety $X$ is called ``the blowup of the affine plane at the origin." It is also (fun fact!) the total space of the line bundle $\cO(-1)$.
\end{Problem}




\end{document}


