\documentclass[12pt]{amsart}
\usepackage{amsthm}
\usepackage{amsmath,amssymb}
\usepackage[margin=1in]{geometry}
\usepackage{enumerate}

\usepackage[all]{xy}
\usepackage{footmisc}

\renewcommand{\thefootnote}{\fnsymbol{footnote}}
\usepackage{nopageno}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amscd}
\usepackage{amsfonts}
\usepackage[latin1]{inputenc}
%\usepackage{pgfpages}
%\usepackage{pgfplots}
\usepackage[mathscr]{euscript}
%\usepackage{bm}
%\usepackage{tikz}
\usepackage{enumerate}


\theoremstyle{definition} % amsthm only
\newtheorem*{theorem*}{Theorem}



\newcommand{\ds}{\displaystyle}

\newcommand{\pro}[1]{\noindent {\textbf{Problem #1.}}}


\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\KK}{\mathbb{K}}
\newcommand{\Aff}{\mathbb{A}}




\newcommand{\Hom}{\ensuremath{\operatorname{Hom}}}
\newcommand{\Aut}{\ensuremath{\operatorname{Aut}}}
\newcommand{\End}{\ensuremath{\operatorname{End}}}


\newcommand{\gps}{\scriptstyle{\operatorname{Gps}}}
\newcommand{\Ker}{\operatorname{Ker}}
\newcommand{\Image}{\operatorname{Im}}
\newcommand{\Zentrum}{\mathsf{Z}}
\newcommand{\quat}{\mathsf{Q}}


\newcommand{\Sn}[1]{\ensuremath{S_{#1}}}
\newcommand{\An}[1]{\ensuremath{A_{#1}}}

\newcommand{\PGL}{\operatorname{PGL}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\pr}{\ensuremath{\mathbb{P}}}
\newcommand{\diag}{\ensuremath{\operatorname{diag}}}
\newcommand{\Stab}{\ensuremath{\operatorname{Stab}}}
\newcommand{\from}{\colon}
\newcommand{\sign}{\ensuremath{\operatorname{sign}}}

\newcommand{\Autg}[1]{\operatorname{Aut}_{\scriptscriptstyle{gp}}\left(#1\right)}
\newcommand{\mult}[2]{\operatorname{Mult}(#1; #2)}
\newcommand{\gr}{\ensuremath{\operatorname{gr}}}

\newcommand{\supp}{\ensuremath{\operatorname{Supp}}}
\newcommand{\Min}{\ensuremath{\operatorname{Min}}}

\begin{document}



\begin{center}
  \Large{\textsf{\textbf{MATH 7141 - Algebraic Geometry I}}}\\
\vspace{1ex}
  \Large{\textsf{\textbf{Homework 1}}}\\
\vspace{2ex}

\large{\textbf{\textsf{
Zariski topology, basic properties of affine varieties and defining ideals
}}}\normalsize

\vspace{1ex}

\end{center}
\normalsize

Please indicate any source in the literature used in finding the solution to a given problem. You are encouraged to work in teams. If you do so, please indicate the name of your collaborator(s) on your submission.
\vspace{1ex}

Solutions to each problem can be uploaded (on Carmen) by at most one student. There is no deadline, so work at your own pace.
\vspace{1ex}

Please use the following name for the file you upload \texttt{HW\#\_Problem\#.pdf}.


\vspace{3ex}




\hrule
\vspace{1ex}

\begin{center}
In all problems below, we assume $\KK$ is an arbitrary field.
\end{center}

\vspace{1ex}

\hrule

\vspace{3ex}

\noindent \emph{Definition:} A topological space $X$ is called \textbf{Noetherian} if there is no infinite strictly decreasing chain $X_0\supsetneq X_1\supsetneq X_2 \supsetneq \ldots$ of closed subsets of $X$.\\

\pro{1} Show that any affine variety (with the Zariski topology) is a Noetherian topological space. In particular, show  that the affine space $\Aff^n_{\KK}$ is Noetherian.\\

\pro{2} Show that every Noetherian topological space is compact (i.e., every open cover of such a space has a finite subcover). Conclude from Problem 1 that every open subset of an affine variety is compact in the Zariski topology.\\
 

\pro{3} Let $X$ be an irreducible affine subvariety of $\Aff^n_{\KK}$ and $U$ be any non-empty open subset of $X$ (with respect to the Zariski topology). Show that $U$ is dense in $X$. Show that the irreducibility of $X$ is necessary by providing a counter-example to this density statement.\\



\pro{4} Show that a subvariety $V$ of $\Aff^n_{\KK}$ is irreducible if, and only if, for every variety $W\subsetneq V$, the difference $V\smallsetminus W$ is Zariski dense in $V$.\\

\pro{5} Find the irreducible components of the affine variety $V(x-yz, xz-y^2)\subset \Aff^3_{\CC}$.\\

\pro{6} Let $X\subset \Aff^n$ be an arbitrary subset. Prove that $V(I(X)) =\overline{X}\subset \Aff^n$.\\

\vspace{1ex}
\hrule
\vspace{2ex}
\noindent Recall that for two ideals $\mathfrak{a}_1, \mathfrak{a}_2$ of a ring $R$ the \textbf{ideal quotient} (or \textbf{colon ideal}) equals
\[
(\mathfrak{a}_1: \mathfrak{a}_2) = \{f\in R: f\mathfrak{a}_2\subset \mathfrak{a}_1\}.
\]
\vspace{1ex}

\pro{7} Let $X_1$ and $X_2$ be affine subvarieties of $\Aff^n_{\KK}$. Show that
\[I(\overline{X_1\smallsetminus X_2}) = (I(X_1): I(X_2)).\]
\vspace{1ex}



\pro{8} Consider two ideals $\mathfrak{a}_1, \mathfrak{a}_2\subset \KK[x_1,\ldots, x_n]$. Show that $V(\mathfrak{a}_1) = V(\mathfrak{a}_1+\mathfrak{a}_2) \cup V((\mathfrak{a}_1:\mathfrak{a}_2))$.\\

\pro{9} Consider two ideals  $\mathfrak{a}_1$ and $\mathfrak{a}_2$ of $\KK[x_1,\ldots, x_n]$.
\begin{enumerate}[(i)]
  \item Prove that
    $\overline{V(\mathfrak{a}_1)\smallsetminus V(\mathfrak{a}_2)} \subseteq V((\mathfrak{a}_1:\mathfrak{a}_2))$.    
\item Show that the inclusion in the previous item can be strict by considering the ideals $\mathfrak{a}_1 = (x^2(y-1))$ and $\mathfrak{a}_2=(y)$ of $\CC[x,y]$.\\
\end{enumerate}

\pro{10} Consider two affine subvarieties $V,W$ of $\Aff^n_{\KK}$. Show that $V = (V\cap W) \cup \,\overline{V\smallsetminus W}$.\\

\vspace{1ex}
\hrule
\vspace{2ex}
%\pagebreak
\pro{11} Let $X\subset \Aff^n_{\KK}$ and $Y\subset \Aff^m_{\KK}$ be two affine varieties.

\begin{enumerate}[(i)]
\item Prove that their product $X\times Y\subset \Aff^{n+m}_{\KK}$ is also an affine variety.
\item Show that $X\times Y$ is irreducible if both $X$ and $Y$ are irreducible.\\
\end{enumerate}


\pro{12} Consider the following parameterization of a subset $X$ of $\Aff_{\CC}^2$, via $x=t/(1+t)$, and $y=1-1/t^2$ for $t\neq -1, 0$ in $\CC$.
\begin{enumerate}[(i)]
\item Determine $I(\overline{X})$.
\item Compute $\overline{X}\smallsetminus X$.\\
\end{enumerate}

\pro{13}\textbf{(The Twisted Cubic)} Assume $\KK$ is infinite and consider $X=V(y-x^2, z-x^3)\subseteq\Aff^3_{\KK}$. This set is a curve in $\Aff^3$, known as the twisted cubic.
\begin{enumerate}[(i)]
\item Show that $I(X) = (y-x^2, z-x^3)\subset \KK[x,y,z]$.
  
  \item  Show that $y^2-xz\in I(X)$ by writing it explicitly as a polynomial combination of the generators of $I(X)$.\\
\end{enumerate}

\pro{14} Assume $\KK$ is infinite, and consider the subset $X$ of $\Aff^3_{\KK}$ parameterized by $t\mapsto (t, t^3, t^4)$.
\begin{enumerate}[(i)]
\item Show that $X$ is an affine subvariety of $\Aff^3_{\KK}$.
  \item Determine generators for $I(X)$.\\
\end{enumerate}

\end{document}
