\chapter{Homogeneous Coordinates}
\label{homogeneous}

\setcounter{section}{-1}

\section{Background:\ Quotients in Algebraic Geometry}
 
\begin{itemize}
    \item{$\P^n = \C^{n+1} \setminus 0 / C^*$}
    \item{$G \subseteq (C^*)^n$ closed subgroup 
     (possibly disconnected, with torsion)}
    \item{Good categorical quotient(GCQ), Good geometric quotient (GGQ)}
    \item{$\Spec(A) = X \rightarrow X/G = \Spec(A^G)$ is GCQ}
    \item{$A$ normal implies $A^G$ normal}
    \item{$\phi(x) = \phi(y)$ iff $\overline{Gx} \cap \overline{Gy} \ne
    \emptyset$} 
    \item{$X \rightarrow Y$ normal, onto, $\pi^{-1}(y) = \mbox{orbit}
    \forall y$ implies GGQ [dolgachev p.95]}
    \item{$V(xy-zw)$ GCQ but not GGQ}
\end{itemize}

\section{Total Coordinate Rings of Toric Varieties}
 
\begin{itemize}
    \item{define $\Cl(X)$}
    \item{ $M \rightarrow \Z^{\Sigma(1)} \rightarrow \Cl(X) \rightarrow 0$}
    \item{$S = \C[X_\rho \mid \rho \in \Sigma(1)]$, grading}
    \item{Examples: $\P^n$, weighted projective space, $\P^n \times
       \P^m$, $Bl_0(C^m)$, Hirzebruchs} 
    \item{$\mathrm{V}(xy-zw)$, mirror symmetry (MS)}
\end{itemize}

\section{Quotient Constructions of Toric Varieties}
 
\begin{itemize}
    \item{Assume $\Sigma(1)$ spans $N_{\R}$}
    \item{$0 \rightarrow M \rightarrow \Z^{\Sigma(1)}\rightarrow
    \Cl(X)\rightarrow 0$} 
    \item{$G = Hom_{\Z}(\Cl(X),{\C}^*)$}
    \item{concrete description of action on $\C^{\Sigma(1)}$}
    \item{$B = \langle X^{\hat{\sigma}} \mid \sigma \in \Sigma_{max}
       \rangle$, example of $\P^n$ with $B = \langle x_0,\ldots, x_n
       \rangle$, $G = \C^*$ acts as usual} 
    \item{$\C^{\Sigma(1)} \setminus V(B) / G = X_{\Sigma}$ GCQ, GGQ
       iff $\Sigma$ simplicial -- maybe use Batyrev-Cox proof} 
\end{itemize}

\section{Examples}

\begin{itemize}
    \item{Examples from section 1}
    \item{MS case}
    \item{relate to change of lattice}
\end{itemize}

\section{Polytopes and Homogenization}
 
\begin{itemize}
    \item{$P$ corresponds to a degree in $\Cl(X)$, so $S_{m\,deg}
       \leftrightarrow |mP \cap M| $}
    \item{pictures similar to ``What is paper'' of how to homogenize}
    \item{$\C^{\Sigma(1)} \setminus V(B) / G  \leftrightarrow \C^*
       \rightarrow \P^n$}
    \item{Mirror constructions}
\end{itemize}