%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%% Math 273 Homework \#2, Fall 2010 %%%%%%
%%%%%% %%%%%%
%%%%%% Instructor: Ezra Miller %%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[11pt]{article}
\oddsidemargin=17pt \evensidemargin=17pt
\headheight=9pt \topmargin=26pt
\textheight=564pt \textwidth=433.8pt
\usepackage{amsmath,amssymb,graphicx}
\newcommand{\excise}[1]{}
\newcommand{\comment}[1]{{$\star$\sf\textbf{#1}$\star$}}
\leftmargini=5.5ex
\leftmarginii=3.5ex
%new math symbols taking no arguments
\newcommand\CC{\mathbb{C}}
\newcommand\cF{\mathcal{F}}
\newcommand\cG{\mathcal{G}}
\newcommand\NN{\mathbb{N}}
\newcommand\OO{\mathcal{O}}
\newcommand\QQ{\mathbb{Q}}
\newcommand\RR{\mathbb{R}}
\newcommand\ZZ{\mathbb{Z}}
\newcommand\bb{\mathbf{b}}
\newcommand\kk{\Bbbk}
\newcommand\mm{\mathfrak{m}}
\newcommand\pp{\mathfrak{p}}
\newcommand\minus{\smallsetminus}
\newcommand\goesto{\rightsquigarrow}
%redefined math symbols taking no arguments
\newcommand\<{\langle}
\renewcommand\>{\rangle}
\renewcommand\iff{\Leftrightarrow}
\renewcommand\implies{\Rightarrow}
%new math operators
\DeclareMathOperator{\HHom}{\ensuremath{\mathcal{H}\hspace{-.25ex}\mathit{om}}}
\DeclareMathOperator{\Hom}{Hom}
%new math symbols taking arguments
\newcommand\ol[1]{{\overline{#1}}}
%redefined math symbols taking arguments
\renewcommand\mod[1]{\ (\mathrm{mod}\ #1)}
%roman font math operators
\DeclareMathOperator\aut{Aut}
%for easy 2 x 2 matrices
\newcommand\twobytwo[1]{\left[\begin{array}{@{}cc@{}}#1\end{array}\right]}
%for easy column vectors of size 2
\newcommand\tworow[1]{\left[\begin{array}{@{}c@{}}#1\end{array}\right]}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{\mbox{}\\[-8ex]Math 273 Homework \#2, Fall 2010\\\normalsize
Instructor: Ezra Miller}
\author{Solutions by: ...your name...\\[2ex]
Collaborators: ...list those with whom you worked on this assignment...\\[1ex]}
\date{Due: Tuesday 28 September 2010}
\maketitle
\vspace{-3ex}%
\noindent
\textsc{Reading assignments} in \cite{vakil}\vspace{-.5ex}
\begin{itemize}
\item%
by Tuesday 21 September: Chapter~4; all but \S4.5--\S4.6 should be review\vspace{-1ex}
\item%
by Thursday 23 September: \S5.1--\S5.4; note that \S5.2 should be review\vspace{-1ex}
\item%
by Tuesday 28 September: \S5.5, Chapter~6 (should be mostly review)\vspace{-1ex}
\item%
by Thursday 30 September: Chapter~12, \S13.1--\S13.3; this plus
Chapter~6 is a lot of material, but most of it is review (skip any
item mentioning morphisms of schemes)
\end{itemize}
\noindent
\textsc{Exercises}:
In \cite{vakil}, exercises have labels C.S.N, for ``Chapter~C,
Section~S, Exercise~N'', where $\text{C},\text{S} \in \ZZ_+$ and
$\text{N} \in \text{A},\ldots,\text{Z}$. It is not expected that
everyone will complete all of the assigned exercises, but those marked
``[required]'' are essential.\vspace{-1ex}%
\begin{enumerate}
\item[3.5.H]%
\item[3.6.C]%
\item[3.6.G][required]%
\begin{enumerate}
\item%
\item%
\end{enumerate}
\item[3.7.D][required]%
\item[4.2.M][required]%
\item[4.4.C]%
\begin{enumerate}
\item%
\item%
\item%
\end{enumerate}
\item[4.4.G][required]%
%\item[4.5.C][required, unless Prof.\ Saper covered it]%
%\item[4.5.E][required, unless Prof.\ Saper covered it]%
%\item[4.6.D][required, unless Prof.\ Saper covered it]%
%\begin{enumerate}
%\item%
%
%\item%
%
%\item%
%
%\end{enumerate}
\item[4.6.F]%
\item[4.6.H]%
\begin{enumerate}
\item%
\item%
\end{enumerate}
\item[4.6.M]%
\item[4.6.O][required]%
\item[4.6.S]%
\item[4.7.E]%
\item[5.1.A]%
\item[5.3.F][required]%
\item[14.1.A]%
\item[14.1.C][required]%
\item[14.1.E][required]%
\end{enumerate}
Additional exercise.
\begin{enumerate}
\item%
Fix a coherent sheaf $\cF$ on a scheme $(X,\OO_X)$. Prove that the
set of points $\pp \in X$ where $\cF(\pp)$ has dimension at least~$r$
is closed in~$X$, for each $r \geq 0$. Hint: what condition on an $m
\times n$ matrix with entries in a field guarantees that it has rank
at most $n - r$? [You need only what we did in class concerning
coherent sheaves for this.]
\end{enumerate}
%raggedbottom%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{Artin}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Vakil]{vakil}
Ravi Vakil, \emph{Foundations of algebraic geometry}, notes dated
August 26, 2010.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{thebibliography}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%