\chapter{Sheaves on Toric Varieties}
\label{invertible}

\section{Invertible and Reflexive Sheaves}
 
\begin{itemize}
    \item{$\cO_X$, sheaf of a Weil divisor $\cO_X(D)$}
    \item{formal definition of sheaf}
    \item{invertible sheaves (when is $\cO_X(D)$ invertible)}
    \item{sheaf of modules (in general, $\cO_X(D)$ need not be locally
       free, define reflexive)} 
    \item{Invertible iff Cartier, every invertible is sheaf of a
       Cartier divisor}
    \item{linear equivalence of divisors corresponds to iso of sheaves 
       (rmk: in appendix, defns of $\otimes$, reflexive module)}
\end{itemize}

\section{Connection to Polytopes}
 
\begin{itemize}
    \item{Global sections, definition of $P_D$}
    \item{maps to $\P^n$}
    \item{support functions}
    \item{$S_\alpha = H^0(O_X(D)) \simeq \oplus_{m \in P_D} \chi^m$}
    \item{general defs for v.ample, ample, gen'd by sections}
\end{itemize}

\section{Ampleness Criteria on Toric Varieties}
 
\begin{itemize}
    \item{toric criteria for nef, ample (example of not nef divisor on
       Hirzebruch } 
    \item{Smooth then ample iff v.amp?}
    \item{toric Nakai}
    \item{nef cone, Mori cone}
    \item{primitive collections}
    \item{nef and not big implies mod by subspace get smaller fan}
\end{itemize}

\section{The Canonical Sheaf}
\begin{itemize}
    \item{$\Omega^1$, $K$, Gorenstein, Fano}
    \item{Cotangent s.e.s, residue sequence}
    \item{Graded $S$-module gives sheaf on $X_{\Sigma}$; examples for
    $\Omega_X$, $T_X$}
\end{itemize}