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\title[Affine and Projective Toric Varieties]
{Affine and Projective Toric Varieties\\
Math in the Mountains Tutorial}

\author
{Henry K. Schenck}

\institute[University of Illinois]
{
  Department of Mathematics\\
  University of Illinois, Urbana-Champaign\\
}

\date[July 29-31, 2013]
{July 29-31, 2013}

\begin{document}

\begin{frame}
  \titlepage
\end{frame}

\begin{frame}
{\bf Outline}

\tableofcontents

\end{frame}

\section{Affine toric varieties}

\begin{frame}
{\bf The torus, characters}

\begin{itemize}
\item<1->  $T = (\C^*)^n$ is an abelian group under coordinatewise multiplication,
\emph{and} an algebraic variety.  Ex.  $\C^* \leftrightarrow V(xy - 1) \subset \C^2$
\item<2->  A \emph{character} of $T$ is morphism of varieties and group homomorphism
$\chi : T \to \C^*$
\item<3->  Any character of $T$ has the form
$$(t_1,\ldots,t_n) \mapsto \prod_i t_i^{a_i}$$
for some $a_i \in \Z$.
\item<4->  Write $M = \{(a_1,\ldots,a_n) \mid a_i \in \Z\}\simeq \Z^n$,
the \emph{character lattice}
of $T$.
\item<5-> \emph{Mnemonic: $M$ for ``monomial map''}
\end{itemize}
\end{frame}

\begin{frame}
{\bf One-parameter subgroups}

\begin{itemize}
\item<1-> A one-parameter subgroup of $T$ is a morphism of varieties and group
homomorphism $\lambda : \C^* \to T$
\item<2-> Any such mapping has the form
$$t \mapsto (t^{a_1}, \ldots, t^{a_n})
\text{ for some } a_i \in \Z.$$
\item<3-> So we have a second lattice $N \simeq \Z^n$, the lattice of 1-parameter subgroups,
and $T_N = N \otimes_\Z \C^*$ is the associated torus
\item<4-> Given $m \in M$ and $n \in N$ we get $\chi^m : T \to \C^*$ and $\lambda^n : \C^* \to T$
\item<5-> Gives a pairing $\langle\ ,\ \rangle : M \times N \to Z$ via
\begin{align*} \chi^m \circ \lambda^n : \C^* &\to \C^*\\
                                           t &\mapsto t^{\langle m,n\rangle}
\end{align*}
\end{itemize}
\end{frame}

\begin{frame}
{\bf A theorem and a definition}

\begin{thm}[Sumihiro]
\begin{enumerate}
\item If $T_1$ and $T_2$ are tori and $\varphi: T_1 \to T_2$ is a morphism of varieties
and group homomorphism, then $\mathrm{im}(\varphi)$ is a closed torus in $T_2$
\item If $H$ is an irreducible subvariety of a torus $T$ that is a subgroup
of $T$, then $H$ is a torus.
\end{enumerate}
\end{thm}

\begin{defn}
An \textbf{affine toric variety} is an irreducible variety containing $T$ as
a Zariski open subset, and such that the action of $T$ on itself extends
to a morphism $T \times V \to V$.
\end{defn}

\end{frame}

\begin{frame}
{\bf First Examples}

\begin{itemize}
\item<1->  $T = (\C^*)^n$ itself
\item<2->  $\C^n$, $\P^n$, via the ``obvious'' inclusions $T \subset \C^n \subset \P^n$
\item<3->  The cuspidal cubic $V(y^2 - x^3) \subset \C^2$. The torus $T$ in this case is
$T = \{(t^2,t^3) \mid t\in \C^*\}$
\end{itemize}
\end{frame}

\begin{frame}
{\bf ``Why do we care?''}

\vskip 10pt
The action of $T$ lets us break things into simple bits --

\begin{itemize}
\item<1->  The ``three-fold path:"
\begin{itemize}
\item<2-> Lattice points (free abelian groups of finite rank)
\item<3->  Toric ideals
\item<4->  Affine semigroups
\end{itemize}
\item<5-> If $T_N$ has character lattice $M$ and ${\cal A} = \{a_1,\ldots,a_n\} \subset M$, let
\begin{align*} \varphi_{\cal A} : T &\to \C^n\\
     t &\mapsto (\chi^{a_1}(t),\ldots,\chi^{a_n}(t))
\end{align*}
\end{itemize}
\end{frame}

\begin{frame}
{\bf Lattice points and the variety $Y_{\cal A}$}

\begin{itemize}
\item<1->  Here is the formal definition of the variety defined by a monomial parametrization:
\begin{defn} $Y_{\cal A} = \overline{\varphi_{\cal A}(T)}$, where the bar is Zariski closure. \end{defn}
\item<2->
\begin{prop} $Y_{\cal A}$ is an affine toric variety with character lattice $\Z({\cal A})$.
The dimension of $Y_{\cal A}$ is the rank of the lattice $\Z({\cal A})$.
\end{prop}
\item<3->  Example:  $M = \Z$, ${\cal A} = \{2,3\}$ gives $Y_{\cal A} = V(y^2 - x^3)$
\end{itemize}
\end{frame}

\begin{frame}
{\bf The ideal of $Y_{\cal A}$}

\begin{itemize}
\item<1-> For ${\cal A}$ as above, consider $\Z^n \to M$ defined by $e_i \mapsto a_i$.
\item<2-> Let $L$ be the kernel of this map, so $0 \to L \to \Z^n \to M$ is exact
\item<3-> Write each $\ell \in L$ as $\ell = \ell_+ - \ell_-$ (that is, $\ell_+ = \sum_{\ell_i > 0} \ell_i e_i$
and $\ell_- = -\sum_{\ell_i < 0} \ell_i e_i$)
\item<4->  Then $x^{\ell_+} - x^{\ell_-}$ vanishes on the image of $\varphi_{\cal A}$, hence
is in $I(Y_{\cal A})$.
\item<5->  In fact we have,
\begin{thm}
$I(Y_{\cal A}) = \langle x^\alpha - x^\beta \mid \alpha,\beta \in \N^n, \alpha - \beta \in L\rangle.$
\end{thm}
\end{itemize}
\end{frame}

\begin{frame}
{\bf Toric ideals}

\begin{itemize}
\item<1-> As we saw in the statistics talk, ideals of this form have a name:
\begin{defn}  A \textbf{toric ideal} is a prime lattice ideal
$$\langle x^\alpha - x^\beta \mid \alpha,\beta \in \N^n, \alpha - \beta \in L\rangle$$
for $L \subset \Z^n$.
\end{defn}
\item<2-> In fact, the binomial form of the generators is enough:
\begin{thm} $I \subset \C[x_1,\ldots,x_n]$ is toric $\Leftrightarrow$ $I$ is prime and
generated by binomials.
\end{thm}
\end{itemize}
\end{frame}

\begin{frame}
{\bf Two Examples}

\begin{itemize}
\item<1->  Consider $\Z^2 \to \Z$ defined by columns of ${\cal A} = (2\ \ 3)$.  (Note: we'll sometimes
be sloppy about what ${\cal A}$ represents -- it could be the set $\{2,3\}$ or the $1 \times 2$
integer matrix with those as the columns(!))
\item<2->  The kernel is $L = \langle (3\ -2)\rangle$
\item<3->  Gives binomial $x^3 - y^2$, which generates the toric ideal
\item<4->  Similarly if
$${\cal A} = \begin{pmatrix} 2 & 1 & 0\\
0 & 1 & 2\end{pmatrix}$$ then
$L = \langle (1\ -2\ 1)\rangle$ and we get $I = \langle xz - y^2\rangle$, so $V(I)$ is a quadric cone.
\end{itemize}
\end{frame}

\begin{frame}
{\bf Affine semigroups}

\begin{itemize}
\item<1-> A semigroup is a set with a binary operation satisfying all of the group axioms
except for the existence of inverses.
\item<2-> Affine semigroups are abelian, finitely generated, contained in a lattice, operation
written as $+$
\item<3->  If $S$ is a semigroup, we can define a \emph{semigroup algebra}
$$\C[S] = \mathrm{Span}\{x^\a : \a \in S\}
\ \mathrm{with}\  x^\alpha \cdot x^\beta = x^{\alpha + \beta}$$
\item<4->  It follows that
\begin{thm}  $\C[S]$ is an integral domain, finitely generated as a $\C$-algebra, hence
corresponds to an affine variety $\mathrm{Spec}\ \C[S]$. This variety is toric; if
$S = \N {\cal A}$ for $A \subset M$, then it is isomorphic to $Y_{\cal A}$.
\end{thm}
\end{itemize}
\end{frame}

\begin{frame}
{\bf An example}

\vskip 10pt
Let $S$ be the affine semigroup in $\Z^2$ generated by $(2,0), (1,1), (0,2)$.

\vskip 10pt
The semigroup algebra $\C[S] = \C[s^2, st, t^2]$

\vskip 10pt
The corresponding variety ${\rm Spec}\ \C[S]$ is the affine toric variety
$Y_{\cal A}$ for ${\cal A} = \{(2,0),(1,1),(0,2)\}$.

\vskip 10pt
This is the quadric cone $V(xz - y^2)$ from before.
\end{frame}

\begin{frame}
{\bf Three ways to describe affine toric varieties}

\vskip 10pt
From the last example, the next statement should
be plausible at least:

\vskip 10pt
\begin{thm}  The following are equivalent:
\begin{itemize}
\item $V$ is an affine toric variety
\item $V = Y_{\cal A}$ for some ${\cal A} \subset M$
\item $V = V(I)$ for a toric ideal
\item $V = {\rm Spec}\ \C[S]$ for an affine semigroup $S$.
\end{itemize}
\end{thm}
\end{frame}

\begin{frame}
{\bf Cones}

\begin{itemize}
\item<1->  Let $F \subset N_\R = N\otimes_\Z \R$ be a finite set.  We define the \emph{cone spanned
by $F$} to be:
$$\mathrm{cone}(F) = \left\{\sum_{u\in F} \lambda_u u \mid \lambda_u \ge 0\right\}$$
\item<2->  The \emph{convex hull} of $S$ is
$$\mathrm{conv}(F) = \left\{\sum_{u\in F} \lambda_u u \mid \lambda_u \ge 0, \sum_{u\in F} \lambda_u = 1\right\}$$
\end{itemize}
\end{frame}

\begin{frame}
{\bf Example}

\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(5,5){\vector(1,0){10}}
\put(5,5){\vector(1,2){10}}
\put(10,5){\circle*{2}}
\put(10,15){\circle*{2}}
\put(10,10){\circle{2}}
\linethickness{0.4mm}
\put(10,5){\line(0,1){10}}
\end{picture}
\caption{$F = \{(1,0),(1,2)\}$ in $\Z^2$}
\end{figure}

$\mathrm{cone}(F)$ is the region between the two rays; $\mathrm{conv}(F)$
is the thicker vertical line segment.
\end{frame}

\begin{frame}
{\bf The dual cone}

\vskip 10pt
Given a cone $\sigma$ in $N_\R$, the dual cone is
$$\sigma^\vee = \{n \in M_\R \mid \langle m, n\rangle \ge 0 \forall n \in \sigma\}$$

\vskip 10pt
For $\sigma = \mathrm{cone}(\{(1,0),(1,2)\})$ as on the previous slide, the dual
cone is
$$\sigma^\vee = \mathrm{cone}(\{(0,1),(2,-1)\})$$

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\begin{figure}
\begin{picture}(20,20)(0,0)
\put(5,5){\vector(0,1){10}}
\put(5,5){\vector(2,-1){10}}
\end{picture}
\end{figure}

\end{frame}

\begin{frame}
{\bf Some terminology}

\begin{itemize}
\item<1-> A cone $\sigma = \mathrm{cone}(F)$ is \emph{rational} if $F \subset N$; \emph{polyhedral} if $F$
is finite (no ``ice cream cones!''); \emph{strongly convex} if the largest linear
subspace contained in $\sigma$ is $\{0\}$
\item<2->  RPC = rational polyhedral cone, SCRPC = strongly convex, rational, polyhedral cone
\item<3->  A SCRPC is \emph{smooth} if the minimal generators of the rays of the
cone are a subset of a $\Z$-basis of $N$
\item<4-> A SCRPC is \emph{simplicial} if the minimal generators are a subset
of a basis of $N_\Q = N \otimes_\Z \Q$
\end{itemize}
\end{frame}

\begin{frame}
{\bf An important fact}

\begin{thm}[Gordan's Lemma]  If $\sigma$ is a RPC in $N$, then $S_\sigma = \sigma^\vee \cap M$
is a finitely-generated semigroup.
\end{thm}

\vskip 10pt
Proof:  $\sigma^\vee$ is also an RPC, so $\sigma^\vee = \mathrm{cone}(T)$
for $T$ a finite subset of $M$.  The parallelotope
$K = \{\sum_{m\in T} \lambda_m m \mid 0 \le \lambda_m < 1\}$
is bounded, so $|K\cap M|$ is finite and $T \cup (K \cap M) \subset S_\sigma$.
The key point is $T \cup (K\cap M)$ generates $S_\sigma$ since if $w \in S_\sigma$,
$$w = \sum_m \lambda_m m = \sum \lfloor \lambda_m \rfloor m + \sum (\lambda_m - \lfloor \lambda_m\rfloor) m$$
The first term is in $T$, the second is in $K$. //
\end{frame}

\begin{frame}
{\bf A corollary, saturated semigroups}

\begin{itemize}
\item<1->  Let $S_\sigma = \sigma^\vee \cap M$; by the previous result:
\begin{cor}  Let $\sigma \subset N_\R$ be an RPC.  Then $\mathrm{Spec}\ \C[S_\sigma]$ is an affine
toric variety, denoted $U_\sigma$.  It has dimension $n = \mathrm{rank}(N)$ if and only if $\sigma$ is
a SCRPC.
\end{cor}
\item<2-> An affine semigroup $S \subset M$ is said to be \emph{saturated} if 
$km \in S \Rightarrow m \in S$ for $k \in \N$.
\item<3->  Example:  $S = \langle (4,0),(3,1),(1,3),(0,4)\rangle$ is \emph{not saturated} (why not?)
\end{itemize}
\end{frame}

\begin{frame}
{\bf Normality, smoothness, etc.}

\begin{thm}  Let $V$ be an affine toric variety with torus $T_N$.  The following are equivalent:
\begin{itemize}
\item $V$ is normal (coordinate ring is integrally closed)
\item $V = \mathrm{Spec}\ \C[S]$ for a saturated affine semigroup $S$
\item $V = \mathrm{Spec}\ \C[S_\sigma]$ for a SCRPC $\sigma \subset N_\R$
\end{itemize}
\end{thm}

\begin{thm}  $U_\sigma$ is a smooth variety if and only if $\sigma$ is a smooth
cone.
\end{thm}

\end{frame}

\begin{frame}
{\bf Points, morphisms}

\vskip 10pt
A point $p \in \mathrm{Spec}\ \C[S]$ $\leftrightarrow$ a semigroup
homomorphism $\gamma$ in $\mathrm{Hom}_{sg}(S,\C)$ via
$$m \in S \mapsto \chi^m(p) \in \C.$$
Intrinsically, if $p = m \mapsto \gamma(m)$, then for all $t \in T$,
$t\cdot p : m \mapsto \chi^m(t)\gamma(m)$.

\begin{defn} A morphism $V_1 \to V_2$ is \textbf{toric} if
$\C[S_2] \to \C[S_1]$ is induced by a semigroup homomorphism $S_2 \to S_1$.
\end{defn}

\begin{thm}  $\varphi : V_1 \to V_2$ is toric if and only if
$\varphi(T_{N_1}) \subseteq T_{N_2}$ and $\varphi$ restricts
to a group homomorphism on $T_{N_1}$.  Toric morphisms are
$T$-equivariant:  $\varphi(t\cdot p) = \varphi(t) \varphi(p)$.
\end{thm}

\end{frame}

\section{Projective toric varieties}

\begin{frame}
{\bf The basics}

\begin{itemize}
\item<1->  $\P^n = (\C^{n+1}\setminus \{0\})/\sim$ where $(x_0,\ldots,x_n) \sim \lambda\cdot(x_0,\ldots,x_n)$
for all $\lambda \in \C^*$
\item<2->  The torus in $\P^n$ is the complement of $V(x_0\cdots x_n)$
\item<3->  Projective parametric toric varieties:
\begin{defn}  If ${\cal A} = \{m_1, \ldots, m_{n+1}\} \subset M$, we have
$$T_N \xrightarrow{\varphi_{\cal A}} (\C^*)^{n+1} \longrightarrow \C^{n+1} \longrightarrow \P^n$$
The image is the projective toric variety $X_{\cal A} = \overline{\varphi_{\cal A}(T_N)}$.
\end{defn}
\end{itemize}
\end{frame}

\begin{frame}
{\bf Two subtle points}

\begin{itemize}
\item<1->  Since $1 \to \C^* \to (\C^*)^{n+1} \to T_N \to 1$ is exact,
the character lattice $M_{\P^n}$ is
$\{(a_0,\ldots,a_n) \mid \sum_i a_i = 0\} = \Z^{n+1}/\Z(1,\ldots,1)$
\item<2-> Two rather different parametrizations can give the same projective
toric variety:
\item<3-> Example A:  $M = \{0,1,\ldots,n\}$ in $\Z$ gives
$\C^* \to (\C^*)^{n+1} \to \P^n$ defined by $t \mapsto (1,t,\ldots,t^n)$
\item<4->  Example B:  $M = \{(n,0),(n-1,1),\ldots,(0,n)\} \subset \Z^2$
gives $(s,t) \mapsto (s^n,s^{n-1}t,\ldots,t^n)$
\item<5-> The projective varieties in A and B are the same, but the
affine varieties are not (parametrization of $Y_{\cal A}$ is not homogeneous;
for $Y_{\cal B}$, it is)
\end{itemize}
\end{frame}

\begin{frame}
{\bf Connection with affine torics}

\begin{defn}
Given ${\cal A} \subset M$, let $\Z^1({\cal A}) = \{\sum_i a_i m_i \mid \sum_i a_i = 0\}$
\end{defn}

\begin{thm}  The following are equivalent
\begin{itemize}
\item $Y_{\cal A}$ is the affine cone over the projective variety $X_{\cal A}$
\item $I(Y_{\cal A})$ is a homogeneous ideal
\item There exists a $u \in N$ and $k \in \N$ such that $\langle m_i, u\rangle=k$
for all $i$.  (Note: this says the $m_i$ all lie on a hyperplane in $M$.
\end{itemize}
\end{thm}
\end{frame}

\begin{frame}
{\bf Consequences}

\begin{thm}  The character lattice of $X_{\cal A}$ is $\Z^1({\cal A})$.
This has rank $\mathrm{rank}\ \Z({\cal A}) - 1$ if the $m_i$ all lie
on a hyperplane and $\mathrm{rank}\ \Z({\cal A})$ otherwise.
\end{thm}

\vskip 10pt
{\bf Example}:  If ${\cal A}$ consists of the columns of $$\begin{pmatrix} 4 & 3 & 1 & 0\\
                                                                    0 & 1 & 3 & 4\end{pmatrix},$$
                                                                    then $\mathrm{rank} \Z^1({\cal A}) = 1$
and $X_{\cal A}$ is a curve.
\end{frame}

\begin{frame}
{\bf Affine charts}

\begin{itemize}
\item<1-> In $\P^n$, let $U_i = \{x_i \ne 0\}$ -- a covering of $\P^n$ by affines isomorphic
to $\C^n$.
\item<2->  If $X \subset \P^n$ is a variety, let $X_i = X \cap U_i$.
\item<3->  In toric case, since $x_i = \chi^{m_i}$, passing to $U_i = \mathrm{Spec}\ \C[x_0/x_i,\ldots,x_n/x_i]$
is equivalent to considering $A_i = \{m_j - m_i \mid j \ne i\}$
\item<4->  Have
\begin{thm}  $X_{\cal A} \cap U_i = Y_{{\cal A}_i} = \mathrm{Spec}\ \C[S_i]$, where $S_i = \N {\cal A}_i$.
\end{thm}
\item<5-> In fact, can do better:
$$X_{\cal A} = \bigcup_{m_i \text{ a vertex of } \mathrm{conv}({\cal A})} X_{\cal A} \cap U_i$$
\end{itemize}

\end{frame}

\begin{frame}
{\bf An example, projective normality}

\begin{itemize}
\item<1-> For the rational quartic from ${\cal A} = \{(4,0),(3,1),(1,3),(0,4)\}$, we have
$$X_{\cal A} \cap U_0 = \mathrm{Spec}\ \C[1,t/s,t^3/s^3,t^4/s^4] = \C[t/s]$$
\item<2-> So $X_{\cal A} \cap U_0$ is $\C$.
\item<3-> Similarly for $U_3$ $\Rightarrow$ a smooth, projective, rational curve.
\item<4-> Applying a standard notion from algebraic geometry:
\begin{defn}  $X_{\cal A}$ is said to be \textbf{projectively normal} if the affine
cone $Y_{\cal A}$ is normal (for toric, $\Leftrightarrow$ saturated)
\end{defn}
\item<5-> Exercise: the rational quartic above is not projectively normal
\end{itemize}
\end{frame}

\begin{frame}
{\bf Polytopes}

\begin{itemize}
\item<1->  A polytope $P$ can be defined either as the convex hull of a finite
set of points, \emph{or} as a compact intersection of finitely many half-spaces
\item<2-> The lattice polytope
\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(5,5){\circle*{2}}
\put(5,10){\circle*{2}}
\put(5,15){\circle*{2}}
\put(10,5){\circle*{2}}
\put(10,10){\circle*{2}}
\put(15,5){\circle*{2}}
\put(5,5){\line(1,0){10}}
\put(5,5){\line(0,1){10}}
\put(15,5){\line(-1,1){10}}
\end{picture}
\end{figure}
is
$$\mathrm{conv}(\{(0,0),(2,0),(0,2)\}$$
and
$$\{x \ge 0\} \cap \{y \ge 0\} \cap \{x + y \le 2\}.$$

\end{itemize}
\end{frame}

\begin{frame}
{\bf Polytope terminology}

\begin{defn} Let $P$ be a lattice polytope in $M$
\begin{itemize}
\item<1-> A hyperplane $H = \{m \mid \langle m, n_F\rangle = -a_F\}$ is a \textbf{supporting hyperplane}
if $P \subset \{m \mid \langle m, n_F\rangle \ge -a_F\}$
\item<2-> $P \cap H$ for a supporting hyperplane is a \textbf{face} of $P$; dimension
of a face is dimension of the affine span
\item<3-> $P$ is a \textbf{simplex} if $P$ has $\dim(P) + 1$ vertices
\item<4-> $P$ is \textbf{simplicial} if each facet (codim 1 face) is a simplex
\item<5-> $P$ is \textbf{simple} if each vertex lies in $\dim(P)$ facets.
\end{itemize}
\end{defn}
\end{frame}

\begin{frame}
{\bf Two examples}

\begin{figure}
\centering
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=.7\linewidth]{Oct2.jpg}
  \caption{simplicial $6,8,12$}
  \label{fig: massplot1}
\end{minipage}%
\begin{minipage}{.5\textwidth}
  \centering
  \includegraphics[width=.7\linewidth]{cube.jpg}
  \caption{simple $8,12,6$}
  \label{fig: massplot2}
\end{minipage}
\end{figure}
\end{frame}

\begin{frame}
{\bf Normal polytopes}

\begin{itemize}
\item<1->  The definition is:
\begin{defn}  A lattice polytope $P \subset M_\R$ is \textbf{normal} if
$$(k P \cap M) + (\ell P \cap M) = (k + \ell)P \cap M$$
for all $k,\ell \ge 1$. (Note: The $\subseteq$ here is ``automatic.'')
\end{defn}
\item<2->  Exercise: $P = \mathrm{conv}(\{0,e_1,e_2,e_1+e_2+2e_3\})$ is not normal.
\item<3->  A nice fact:
\begin{thm}
If $P$ is a lattice polytope with $\dim(P) = n \ge 2$, then $kP$ is normal for all $k \ge n -1$.
\end{thm}
\end{itemize}
\end{frame}

\begin{frame}
{\bf The toric variety of a polytope}

\vskip 10pt
The key idea here is that charts $\Leftrightarrow$ vertices of $P$.
Given a lattice polytope $P \subset M$ we get a cone and a semigroup
from each vertex $m_i$ of $P$ as follows:

\begin{defn}
\begin{itemize}
\item The vertex cone is
$$\sigma_i = \mathrm{cone}(P\cap M - m_i)^\vee \subseteq N$$
(Note:  The notation $P\cap M - m_i$ means: translate $P\cap M$ to place $m_i$ at the origin!)
\item The semigroup is
$$S_{P,v} = \N(P \cap M - v) \subseteq M$$
\end{itemize}
\end{defn}
\end{frame}

\begin{frame}
{\bf An example will make all clear, we hope ... }

\vskip 10pt
Let $P$ be the following polytope from before:

\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(5,5){\circle*{2}}
\put(5,10){\circle*{2}}
\put(5,15){\circle*{2}}
\put(10,5){\circle*{2}}
\put(10,10){\circle*{2}}
\put(15,5){\circle*{2}}
\put(5,5){\line(1,0){10}}
\put(5,5){\line(0,1){10}}
\put(15,5){\line(-1,1){10}}
\end{picture}
\end{figure}

\vskip 10pt
and take $v = 2e_2 = (0,2)$.  The cone $\sigma_i$ is $\mathrm{cone}(\{e_1,-e_1-e_2\})$:

\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(10,15){\vector(-1,-1){7}}
\put(10,15){\vector(1,0){10}}
\end{picture}
\end{figure}

\end{frame}

\begin{frame}
{\bf Example, continued}

\vskip 10pt
The lattice points in $M$ making up the semigroup $S_{P,v}$
are shown below:  top vertex is translated to the origin:

\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(25,25)(0,0)
\put(5,5){\circle*{2}}
\put(5,10){\circle*{2}}
\put(5,15){\circle*{2}}
\put(5,20){\circle*{2}}
\put(10,5){\circle*{2}}
\put(10,10){\circle*{2}}
\put(10,15){\circle*{2}}
\put(15,5){\circle*{2}}
\put(15,10){\circle*{2}}
\put(20,5){\circle*{2}}
\put(5,20){\vector(0,-1){20}}
\put(5,20){\vector(1,-1){20}}
\end{picture}
\end{figure}
\end{frame}

\begin{frame}
{\bf The affine charts}

\begin{thm} $X_{P\cap M} \cap U_i = \mathrm{Spec}\ \C[\sigma_i^\vee \cap M]$ \end{thm}

\begin{defn} $P$ is \textbf{very ample} if for all vertices $v$, $S_{P,v}$ is saturated. \end{defn}

\vskip 10pt
Exercise:
Let $P = \mathrm{conv}\{0,e_1,e_2,e_1+e_2+2e_3\}$ from before.  Show $|P\cap M| = 4$ so
$X_{P\cap M} \to \P^3$.  Show that $X_{P \cap M}$ is not smooth. Show that $P$ is
not very ample.
\end{frame}

\begin{frame}
{\bf How it all fits together, ... }

\vskip 10pt
Let's draw in the vertex cones at each vertex of the lattice polytope
in the previous example:

\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(5,5){\circle*{1}}
\put(5,10){\circle*{1}}
\put(5,15){\circle*{1}}
\put(10,5){\circle*{1}}
\put(10,10){\circle*{1}}
\put(15,5){\circle*{1}}
\put(5,5){\line(1,0){10}}
\put(5,5){\line(0,1){10}}
\put(15,5){\line(-1,1){10}}
\put(5,5){\vector(1,0){4}}
\put(5,5){\vector(0,1){4}}
\put(5,15){\vector(-1,-1){4}}
\put(5,15){\vector(1,0){4}}
\put(15,5){\vector(0,1){4}}
\put(15,5){\vector(-1,-1){4}}
\linethickness{.001mm}
\qbezier(7.1,5.1)(6.4,6.4)(5.1,7.1)
\qbezier(3.1,13.1)(5,12.8)(7.1,14.9)
\qbezier(13.1,3.1)(12.8,5)(14.9,7.1)
\end{picture}
\end{figure}

If we ``reassemble'' these cones with vertices at the origin, we see they fit together to cover $N_\R = \R^2$.
Will return to this in the next talk(!)

\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(10,10){\vector(1,0){10}}
\put(10,10){\vector(0,1){10}}
\put(10,10){\vector(-1,-1){8}}
\end{picture}
\end{figure}
\end{frame}

\begin{frame}
{\bf The toric variety of a polytope $P$}

\vskip 10pt
Note that we have only defined $X_{P \cap M}$ so far.

\begin{defn}
Let $P$ be a lattice polytope in $M$.  The toric variety of $P$
$X_P = X_{kP \cap M}$ where $kP$ is very ample.
\end{defn}

\begin{cor} For an $n$-dimensional polytope, $(n-1)P$ always
gives an embedding of $X_P$ in $\P(V)$.
\end{cor}
\end{frame}

\begin{frame}
{\bf Final comments}

\begin{itemize}
\item<1->  Warning:  There are smooth, complete varieties that are not projective
(Shafarevich example for surfaces).  Such examples exist for toric varieties, too,
but only in dimensions $\ge 3$.
\item<2-> Motivated by the last example,
\begin{defn}  The \textbf{normal fan} of a polytope $P$ is the collection of cones
glued like a simplicial complex, generated by the inward normals to the faces.
\end{defn}
\item<3-> The point is that $P$ and $kP$ have the same normal fan, so they
give, intrinsically the same (abstract) toric variety, to be described in
the next talk.
\end{itemize}
\end{frame}

\end{document}
