\documentclass[12pt,letterpaper]{article}
\usepackage{multicol}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{mathrsfs}
\usepackage{enumerate}
\usepackage{hyperref}
%\usepackage{pstricks, pst-node, pst-plot}

\newcommand{\ds}{\displaystyle}
\newcommand{\R}{\mathbb{R}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\C}{\mathbb{C}}
\renewcommand{\P}{\mathbb{P}}

\begin{document}


\begin{center}\begin{large}\textbf{Math 137 - Problem Set 7\\
Due Friday, Mar 27}\end{large}\end{center}



\pagestyle{empty}
All rings are commutative, and $k$ is an algebraically closed field.

\begin{enumerate} 

%\item Let $f$ be an affine plane curve. Let $L$ be a line that is not a component of $f$ (i.e. $L$ doesn't divide $f$). Suppose $L = \{(a+tb, c+td) \mid t \in k\}$. Define $g(t) = f(a+tb, c+td)$. Factor $$g(t) = \prod (t-\lambda_i)^{e_i},$$ $\lambda_i$ distinct.

%\begin{enumerate}
%\item Show that there is a natural one-to-one correspondence between the $\lambda_i$ and the points $P_i \in L \cap f$.

%\item Show that $I_{P_i}( L, f) = e_i$. In particular, $\sum_i I_{P_i}( L , f) \leq \deg(f)$.
%\end{enumerate}

\item Suppose $P=(0,0)$ is a point of multiplicity two on an affine plane curve $f$, and suppose $f$ has only one unique tangent $L$ at $P$. 

\begin{enumerate}
\item Show that $I_P(f, L) \geq 3$. The curve $f$ is said to have a \textit{\textbf{cusp}} at $P$ if equality holds.

\item Suppose $L = y$. Show that $P$ is a cusp if and only if $f_{xxx}(P)\neq0$, where $f_{xxx}$ is the third partial derivative of $f$ with respect to $x$.

\item Show that if $P$ is a cusp on $f$, then $f$ has only one component passing through $P$.

\end{enumerate}

\item Let $f \in k[x_1, \ldots, x_{n+1}]$. Write $f = \sum_i f_i$, where $f_i$ is a form of degree $i$. Let $P \in \P^n$ and suppose $f(a_1, \ldots, a_{n+1}) = 0$ for every choice of homogeneous coordinates $[a_1: \ldots: a_{n+1}]$ for $P$. Show that for all $i$, $f_i(a_1, \ldots, a_{n+1}) = 0$ for every choice of homogeneous coordinates for $P$.
\end{enumerate}

\noindent You might want to wait until after class on Monday 3/23 to work on the remaining problems.

\begin{enumerate}
 \setcounter{enumi}{2}

\item Let $V \subset \P^n$ be an algebraic set and $I$ a homogeneous ideal. Verify the following claims from class:

\begin{enumerate}
\item  If $V \neq \emptyset$, then $I_a(C(V)) = I_p(V)$.
\item if $V_p (I) \neq \emptyset$, then $C(V_p(I)) = V_a(I)$.
\end{enumerate}

\item Let $I \subset k[x_1, \ldots, x_{n+1}]$ be a homogeneous ideal. 

\begin{enumerate}
\item Show that $I$ is prime if and only if for any forms (i.e. homogeneous polynomials) $F, G \in k[x_1, \ldots, x_{n+1}]$, such that $FG \in I$, we have $F \in I$ or $G\in I$. (One direction should be obvious.)

\item Show $\sqrt{I}$ is homogeneous.

\end{enumerate}

\item Let $S =  k[x_1, \ldots, x_{n+1}]$,  $I \subset S$ a homogeneous ideal, and $R = S/I.$

\begin{enumerate}
\item Let $d \geq 0$ and $S_d \subset S$ the set of forms of degree $d$. Show that $S_d$ is a $k$-vector space and calculate its dimension.

\item Let $R_d$ be the image of $S_d$ in the quotient. Show $R_d$ is also a $k$-vector space of finite dimension.
\end{enumerate}

\item Let $S = k[x,y,z]$, and $f \in S$ an irreducible form of degree $n$. Let $C = V(f) \subset \P^2$, and $R = S/(f)$.

\begin{enumerate}
\item Show that there is a short exact sequence $$0 \to S \stackrel{\cdot f}{\to} S \to R \to 0,$$ where the map from $S$ to $S$ is multiplication by $f$, and the second map is the natural quotient. (i.e. this amounts to showing that the first map is injective, and the second map is the  quotient of the first.)

\item For each $d$, calculate $\dim_k ({R_d})$. Keep in mind that your answer may depend on whether $d>n$, $d=n$, or $d<n$.


\end{enumerate}


\end{enumerate}

\end{document}