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\title[Fans, Orbits, and Divisors on Toric Varieties]
{Fans, Orbits, and Divisors on 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{Fans, cones, and orbits}

\begin{frame}
{\bf Fans}

\begin{defn}
A \textbf{fan} is a collection $\Sigma$ of SCRPC's in $N_\R$ such that
\begin{itemize}
\item If $\tau$ is a face of $\sigma \in \Sigma$, then $\tau \in \Sigma$
\item if $\sigma_1,\sigma_2 \in \Sigma$, then $\sigma_1 \cap \sigma_2 \in \Sigma$ and
is a face of each.
\end{itemize}
\end{defn}

Example:  Our picture from Hal's last talk:
\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(10,10){\vector(1,0){10}}
\put(14,14){$\sigma_1$}
\put(10,10){\vector(0,1){10}}
\put(3,12){$\sigma_3$}
\put(10,10){\vector(-1,-1){6}}
\put(11,5){$\sigma_2$}
\put(1,1){$\sigma_{23}$}
\put(21,9){$\sigma_{12}$}
\put(9,21){$\sigma_{13}$}
\end{picture}
\end{figure}
shows a fan $\Sigma$ with three $2$-dimensional cones $\sigma_1,\sigma_2,\sigma_3$,
three $1$-dimensional cones (the rays $\sigma_{ij} = \sigma_i \cap \sigma_j$), and
one $0$-dimensional cone $\{0\}$.

\end{frame}

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

We will also need to consider the duals of these cones in $M$:

\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(70,20)(0,0)
\put(10,10){\vector(1,0){10}}
\put(10,10){\vector(0,1){10}}
\put(10,10){\vector(-1,0){10}}
\put(10,10){\vector(0,-1){10}}
\put(10,10){\vector(-1,1){6}}
\put(10,10){\vector(1,-1){6}}
\qbezier(12.1,10.1)(12.1,12.1)(10.1,12.1)
\put(14,14){$\sigma_1^\vee$}
\qbezier(7.9,10.1)(8.0,11.0)(8.5,11.5)
\put(0,12){$\sigma_3^\vee$}
\qbezier(11.5,8.5)(11.0,8.0)(10.1,7.9)
\put(12,0){$\sigma_2^\vee$}
\put(50,10){\vector(1,0){10}}
\put(50,10){\vector(0,1){10}}
\put(50,10){\vector(0,-1){10}}
\qbezier(50.1,12.1)(53,10)(50.1,7.9)
\put(55,15){$\sigma_{12}^\vee$}
\end{picture}
\end{figure}

Notice:  $U_{\sigma_1} = \mathrm{Spec}\ \C[x,y]$ and $U_{\sigma_2} = \mathrm{Spec}\ \C[xy^{-1},y^{-1}]$
Both are \emph{contained in} $U_{\sigma_{12}} = \mathrm{Spec}\ \C[x,y,y^{-1}]$.  By localization:
\begin{align*}
\C[x,y] &\longrightarrow \C[x,y]_{\langle 1/y\rangle} = \C[x,y,y^{-1}]\\
\C[xy^{-1},y^{-1}] & \longrightarrow \C[xy^{-1},y^{-1}]_{\langle 1/y^{-1}\rangle} = \C[x,y,y^{-1}]
\end{align*}
(This is the reason for the slightly weird $\sigma^\vee$ construction: smaller faces $\leftrightarrow$ smaller affines.)
\end{frame}

\begin{frame}
{\bf Exercises}

\begin{itemize}
\item Show that the above example gives a collection of affines that glue together to give $\P^2$
\item Identify the cones and the gluing for the complete fans with $1$-dimensional cones generated
by
\begin{enumerate}
\item $\pm e_1, \pm e_2$ (four 2-dimensional cones, four 1-dimensional cones, one 0-dimensional cone).
\item $e_1, e_2, -e_2, -e_1 + ae_2$, where $a$ is any integer $\ge 1$
(four 2-dimensional cones, four 1-dimensional cones, one 0-dimensional cone).
\end{enumerate}
\end{itemize}

\end{frame}

\begin{frame}
{\bf Abstract toric varieties $X_\Sigma$}

\vskip 10pt
As in these examples, any fan $\Sigma$ gives the combinatorial information to produce affine
torics gluing together to form an \emph{abstract
toric variety} $X_\Sigma$. 

\vskip 10pt
Most work on toric varieties in algebraic geometry uses this definition.

\vskip 10pt
All $X_\Sigma$ are \emph{normal} varieties (all affines $U_\sigma$ come from saturated semigroups
so coordinate rings are integrally closed). So not as
general as some examples $Y_{\cal A}$ we saw in first talks.
\end{frame}

\begin{frame}

\begin{thm}[Orbit-cone correspondence]
In the toric variety $X_\Sigma$, there is a 1-1 correspondence
\begin{align*}
\{\text{cones } \sigma \in \Sigma\} &\leftrightarrow \{ T_N-\text{orbits in } X_\Sigma\}\\
\sigma &\leftrightarrow O(\sigma) = \mathrm{Hom}_\Z(\sigma^\perp \cap M, \C^*).
\end{align*}
Under this correspondence,
\begin{enumerate}
\item $\dim O(\sigma) = \dim N - \dim \sigma$
\item $U_\sigma = \cup_{\tau \text{ face of }\sigma} O(\tau)$.
\item If $\tau$ is a face of $\sigma$, then $O(\sigma) \subset \overline{O(\tau)}$ and
$\overline{O(\tau)} = \cup_{\tau \text{face of }\sigma} O(\sigma)$.
\end{enumerate}
\end{thm}

Key technical point:  Uses identification $p \in U_\sigma$ as semigroup homomorphisms $S_\sigma \to \C$
via $p \mapsto (m \mapsto \chi^m(p))$.

\end{frame}

\begin{frame}
{\bf If that didn't make sense, try this example, ... }

Let $\Sigma$ be the following (non-complete) fan in $N = \R^2$:
\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(5,5){\vector(1,0){10}}
\put(5,5){\vector(0,1){10}}
\qbezier(7,5.1)(7,7)(5.1,7)
\put(11,11){$\sigma$}
\put(5,5){\circle*{1}}
\put(17,4){$\tau_1$}
\put(4,17){$\tau_2$}
\end{picture}
\end{figure}

The corresponding toric variety $X_\Sigma$ is $\C^2$; covering by affines
\begin{align*}
U_\sigma &= \mathrm{Spec}\ \C[x,y] = \C^2\\
U_{\tau_1} &= \mathrm{Spec}\ \C[x,y,y^{-1}] = \C \times \C^*\\
U_{\tau_2} &= \mathrm{Spec}\ \C[x,x^{-1},y] = \C^* \times \C\\
U_{\{0\}} &= \mathrm{Spec}\ \C[x,x^{-1},y,y^{-1}] = \C^* \times \C^*
\end{align*}
\end{frame}


\begin{frame}
{\bf The $(\C^*)^2$-orbits and how they correspond with the cones}

$$
\begin{array}{c|c|c}
\text{Cone} & \text{Orbit} & \text{Dimension} \\ \hline
\sigma &  \{ (0,0) \} & 0\\
\tau_1 &  \C^* \times \{0\} & 1\\
\tau_2 &  \{0\} \times \C^* & 1\\
\{0\} & (\C^*)^2 & 2
\end{array}
$$

Note: For instance for the cone $\tau_1$, $\tau_1^\perp \cap M \simeq \Z$
is the subgroup generated by $e_2$.  For the $0$-dimensional cone $\{0\}$,
we have $\{0\}^\perp \cap M \simeq \Z^2$.

\vskip 10pt
Check:  $$U_{\tau_1} = O(\tau_1) \cup O(\{0\}) = (\C^*\times\{0\}) \cup (\C^*\times\C^*) = \C^* \times \C$$

\end{frame}

\begin{frame}
{\bf Toric morphisms, II}

\vskip 10pt
If $\Sigma_1 \subset N_1$ and $\Sigma_2 \subset N_2$ are fans, then we say
a $\Z$-linear $\varphi: N_1 \to N_2$ is \emph{compatible} with the fans
if it maps cones to cones:
$$\sigma_1 \in \Sigma_1 \Rightarrow \varphi(\sigma_1) \subset \sigma_2 \in \Sigma_2$$
(for some $\sigma_2$).

\begin{thm}
``Toric morphisms are the same as in the affine case'' -- that is, any toric
morphism between $X_{\Sigma_1}$ and $X_{\Sigma_2}$ is given by
a mapping of this form.
\end{thm}

\end{frame}

\section{Divisors}

\begin{frame}
{\bf Weil Divisors}

\begin{defn}  Let $X$ be an irreducible variety.  $\mathrm{Div}(X)$
is the free abelian group generated by the irreducible codimension-1
subvarieties of $X$. The elements of $\mathrm{Div}(X)$ are
called \textbf{Weil divisors} on $X$.
\end{defn}

\vskip 10pt
Key: If $R$ is a normal domain and $P$ is a codimension-1 prime ideal
of $R$, then the localization  $R_P$ is a DVR.  This gives a way to talk about
orders of zeros and poles along $V(P)$.

\vskip 10pt
Example:  Let $P = \langle x \rangle$ in $\C[x,y]$.  If $f = \alpha/\beta \in \C(x,y)$,
can find $n_i \in \Z$ such that $x^{n_1} | \alpha$ and $x^{n_1} | \beta$ (maximal).
Then $\mathrm{ord}_{\langle x\rangle}(f) = n_1 - n_2$.
\end{frame}

\begin{frame}
{\bf Divisors on curves}

\begin{itemize}
\item<1->  The following special case may be familiar:  Say $X$ is a smooth, compact
curve over $\C$, $=$ compact Riemann surface
\item<2-> There are no nonconstant holomorphic functions on $X$
\item<3-> But can consider the meromorphic functions on $X$ with poles
bounded by a divisor $D = \sum a_i p_i$, $a_i\in \N$: $H^0(D) = \{f \mid (f) + D \ge 0\}$,
a finite-dimensional $\C$-vector space
\item<4-> For any holomorphic $1$-form $\omega$ on $X$, $\sum_{p \in X} \mathrm{res}_p(f\omega) = 0$.
Analysis of local Laurent series of $f$'s in $H^0(D)$ gives Riemann's inequality:
$\dim H^0(D) \ge  \deg(D) + 1 - g(X)$, and Riemann-Roch identifies the difference between left
and right sides.
\end{itemize}
\end{frame}

\begin{frame}
{\bf The divisor class group}

\begin{defn}  The divisor class group is
$$\mathrm{Cl}(X) = \mathrm{Div}(X)/\sim,$$
where $D_1 \sim D_2$ if $D_1 - D_2 = (f)$ for some
$f \in \C(X)$ (``linear equivalence'').
\end{defn}

\vskip 10pt
Exercise:  $\mathrm{Cl}(\P^1) = \Z$.  (Hartshorne says:  ``$\mathrm{Cl}(X)$  is a very important invariant, in
general not easy to calculate.''  This case is feasible, though!)

\end{frame}

\begin{frame}
{\bf $\mathrm{Cl}(X_\Sigma)$ for a toric variety}

\begin{thm}  There is an exact sequence
$$M \xrightarrow{A} \Z^{|\Sigma(1)|} \longrightarrow \mathrm{Cl}(X_\Sigma) \longrightarrow 0$$
where $A$ is the matrix whose rows are the primitive lattice generators of the $1$-dimensional
cones of $\Sigma$.
\end{thm}

\vskip 10pt
Sketch of proof:  First $\Z^{|\Sigma(1)|} \to \mathrm{Cl}(X_\Sigma) \to \mathrm{Cl}(T_N) \to 0$
is exact since $X_\Sigma$ is the union of the ``big torus'' isomorphic to $T_N$ and the
closures of the orbits corresponding to the 1-dimensional cones.  Since $\C[x_1^{\pm 1},\ldots,x_n^{\pm 1}]$
is a UFD, $\mathrm{Cl}(T_N) = \{0\}$.  For any $m \in M$, $(\chi^m) = 0 $ in $\mathrm{Cl}(X_\Sigma)$.
Hence $M \subseteq \ker(\Z^{|\Sigma(1)|} \to \mathrm{Cl}(X_\Sigma))$.  Exercise:  finish this proof.
\end{frame}

\begin{frame}
{\bf An example}

\begin{itemize}
\item<1->  For $X_\Sigma = \P^2$ from the fan $\Sigma$ considered earlier we have 
$$\Z^2 \xrightarrow{\begin{pmatrix} 1 & 0\\ 0 & 1\\ -1 & -1\end{pmatrix}} \Z^3 \longrightarrow
\mathrm{Cl}(\P^2) \longrightarrow 0$$
Hence ${\rm Cl}(\P^1) \simeq \Z$.  
\item<2-> Similarly, we have $\mathrm{Cl}(\P^n) \simeq \Z$ for all $n\ge 1$.
\end{itemize}
\end{frame}

\begin{frame}
{\bf Another example}

\begin{itemize}
\item<1->  For $X_\Sigma = \P^1\times\P^1$ from the fan $\Sigma$ with 1-dimensional
cones spanned by $\pm e_1, \pm e_2$ we have
$$\Z^2 \xrightarrow{\begin{pmatrix} 1 & 0\\ 0 & 1\\-1 & 0\\ 0 & -1 \end{pmatrix}} \Z^4 \longrightarrow
\mathrm{Cl}(\P^1\times\P^1) \longrightarrow 0$$
Hence ${\rm Cl}(\P^1\times\P^1) \simeq \Z^2$.
\item<2->  So now, when you see ${\cal O}_{\P^1\times\P^1}(a,b)$, you know what it means(!)  (See Hartshorne,
V, III.)
\end{itemize}
\end{frame}

\begin{frame}
{\bf Divisors}

\begin{thm}
Let $\rho \in \Sigma(1)$, and $D_\rho$ the corresponding divisor.  
Then $v_{D_\rho}(\chi^m) = \langle m, u_\rho\rangle$, where $u_\rho$
is the primitive lattice vector in that cone (the ``first lattice point''
on the ray).
\end{thm}

Example:  For $\Sigma$ below
\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(5,5){\vector(1,0){10}}
\put(5,5){\vector(0,1){10}}
\qbezier(7,5.1)(7,7)(5.1,7)
\put(11,11){$\sigma$}
\put(5,5){\circle*{1}}
\put(17,4){$\tau_1$}
\put(4,17){$\tau_2$}
\end{picture}
\end{figure}
take $\tau_1 = \mathrm{cone}(e_1)$.  Then $v_{\tau_1}(x^2y) = \langle (2,1),(1,0)\rangle = 2$.  
\end{frame}

\begin{frame}
{\bf Divisors give maps}

\begin{itemize}
\item<1-> Also \emph{Cartier} = ``locally principal'' divisors.  It is these that
correspond to invertible sheaves = line bundles.  If $X_\Sigma$ is smooth, though, the two 
notions \emph{coincide}, so we'll blur the distinction(!)
\item<2-> Given $D = \sum_{\rho\in\Sigma(1)} a_\rho D_\rho$,   
a basis of $H^0({\cal O}_{X_\Sigma}(D))$ can be used to define a map $X_\Sigma \to \P^n$ for $n = \dim H^0(D) + 1$.
\item<3->  We can ask whether the map is an isomorphism.
\item<4->  We investigate methods for finding a basis of $H^0({\cal O}_{X_\Sigma}(D))$ next.
\end{itemize}
\end{frame}

\begin{frame}
{\bf The global sections}

\begin{thm}
$$H^0\left({\cal O}_{X_\Sigma}\left(\sum_\rho a_\rho D_\rho\right)\right) = \mathrm{Span}\{\chi^m \mid \langle m,u_\rho\rangle \ge -a_\rho \text{ for all } \rho\}.$$
\end{thm}

Example:  For $\P^2$ with $\Sigma$: 
\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){6}}
\put(1,1){$D_3$}
\put(21,9){$D_2$}
\put(9,21){$D_1$}
\end{picture}
\end{figure}
Let $D = 2D_1$.  $H^0(2D_1)$ is spanned by the $\chi^m$ for $m = (a,b)$ satisfying
$\langle (a,b), (0,1)\rangle \ge -2$, $\langle (a,b),(1,0)\rangle \ge 0$ and $\langle (a,b),(-1,-1)\rangle \ge 0$.
\end{frame}

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

The inequalities $b \ge -2$, $a\ge 0$ and $a + b \le 0$ define a polytope as follows

\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(0,5){\line(1,0){20}}
\put(5,0){\line(0,1){20}}
\put(0,15){\line(1,0){20}}
\put(0,20){\line(1,-1){20}}
\end{picture}
\end{figure}

The lattice points are a translate of the lattice points in $\mathrm{conv}\{(0,0),(2,0),(0,2)\}$.
Hence we get a map to $\P^5$ whose image is the degree 2 Veronese image of $\P^2$ (same as
that from ${\cal O}_{\P^2}(2H)$). 
\end{frame}

\begin{frame}
{\bf An exercise}

\vskip 10pt
For $\Sigma$ as below
\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,20)(0,0)
\put(10,10){\line(1,0){10}}
\put(21,9){$D_2$}
\put(10,10){\line(0,1){10}}
\put(9,21){$D_1$}
\put(10,10){\line(0,-1){10}}
\put(9,-4){$D_3$}
\put(10,10){\line(-1,2){5}}
\put(2,21){$D_4$}
\end{picture}
\end{figure}
with the ray corresponding to $D_4$ equal to $\mathrm{cone}(-e_1 + 2e_2)$.
Compute $H^0(D_1 + D_2)$.  
\end{frame}

\begin{frame}
{\bf Support functions, ampleness}

\begin{defn}
A \textbf{support function} for $D = \sum_\rho a_\rho D_\rho$ is a function 
$\varphi_D : |\Sigma| \to \R$ such that
\begin{itemize}
\item $\varphi_D$ is linear on each cone in $\Sigma$
\item $\varphi_D(u_\rho) = -a_\rho$ for all $\rho \in \Sigma(1)$
\end{itemize}
\end{defn}

\begin{thm}
\begin{itemize}
\item If $\Sigma$ has convex support, and is full-dimensional, then $D$ is basepoint-free $\Leftrightarrow$ $\varphi_D$ is convex
\item If $\Sigma$ is complete, then $D$ is ample $\Leftrightarrow$ $\varphi_D$ is strictly convex
\end{itemize}
\end{thm}
\end{frame}

\begin{frame}
{\bf An Example, Demazure's theorem}

\vskip 10pt
Consider $2D_1$ on $\P^2$.  

\vskip 10pt
[Draw graph of a support function on board(!)]

\begin{thm}[Demazure]
If $\Sigma$ has convex support and $D$ is basepoint-free, the $H^i(D) = 0$ for $i > 0$.
\end{thm}
\end{frame}

\section{Toric varieties via quotients}

\begin{frame}
{\bf The quotient construction}

\begin{itemize}
\item<1-> Instead of patching affines to get $\P^n$, recall can build this
toric variety globally as $(\C^{n+1} \setminus \{0\})/\sim$ 
\item<2-> Cox:  There is an analog of this for any toric variety(!)
\item<3-> Define $\C[x_\rho \mid \rho \in \Sigma(1)]$ and an ideal
$$B_\Sigma = \langle x^{\widehat{\sigma}} = \prod_{\rho \notin \sigma(1)} x_\rho \mid \sigma \in \Sigma\rangle $$
\item<4-> Analog of $\C^{n+1} \setminus \{0\}$ is  $\C^{|\Sigma(1)|} \setminus V(B_{\Sigma})$
\end{itemize}
\end{frame}

\begin{frame}
{\bf Sanity check}

\begin{itemize}
\item<1->  For the standard fan $\Sigma$ defining $\P^2$, labeled like this:
\setlength{\unitlength}{.1cm}
\begin{figure}
\begin{picture}(20,25)(0,0)
\put(10,10){\vector(1,0){10}}
\put(14,14){$\sigma_1$}
\put(10,10){\vector(0,1){10}}
\put(3,12){$\sigma_3$}
\put(10,10){\vector(-1,-1){6}}
\put(11,5){$\sigma_2$}
\put(1,1){$\rho_3$}
\put(21,9){$\rho_1$}
\put(9,21){$\rho_2$}
\end{picture}
\end{figure}
we have $x^{\widehat{\sigma_1}} = x_3$, $x^{\widehat{\sigma_2}} = x_2$, $x^{\widehat{\sigma_3}} = x_1$.
So $V(B_\Sigma) = V(x_1,x_2,x_3)$ as expected.
\end{itemize}

\end{frame}

\begin{frame}
{\bf The quotient map}

\begin{itemize}
\item<1-> If $N_\R$ is spanned by the $u_\rho$ for $\rho \in \Sigma(1)$, then 
$$0 \longrightarrow M \longrightarrow \Z^{|\Sigma(1)|} \longrightarrow \mathrm{Cl}(X_\Sigma) \longrightarrow 0 $$
is exact.  
\item<2-> Hit this with $\mathrm{Hom}_\Z(-,\C^*)$ get
$$0 \longrightarrow G \longrightarrow (\C^*)^{|\Sigma(1)|} \longrightarrow T_N$$
\item<3->  Since $G \subseteq (\C^*)^{|\Sigma(1)|}$, $G$ acts on $\C^{|\Sigma(1)|}$
\item<4->  Modulo some (hard) technical details about quotients by algebraic group actions (GIT)
we get $X_\Sigma$ as the quotient modulo this action
\end{itemize}
\end{frame}

\begin{frame}
{\bf Sanity check, II}

\vskip 10pt
For $\P^2$, the first exact sequence above is
$$\Z^2 \xrightarrow{\begin{pmatrix} 1 & 0\\ 0 & 1\\ -1 & -1\end{pmatrix}} \longrightarrow \mathrm{Cl}(\P^2) 
\longrightarrow 0$$
After taking $\mathrm{Hom}_\Z(-,\C^*)$, we have
$$(\C^*)^2 \xleftarrow{\begin{pmatrix} 1 & 0 & -1\\ 0 & 1 & -1\end{pmatrix}} (\C^*)^3 \xleftarrow{
\begin{pmatrix} 1 \\ 1\\ 1\end{pmatrix}}  \C^* \longleftarrow 0$$
And indeed $\C^*$ acts on $\C^3$ via $a \sim \lambda a$.
\end{frame}

\begin{frame}
{\bf Why is this important?}

\begin{thm} Any coherent sheaf on a toric variety $X_\Sigma$ comes from a 
finitely-generated $\mathrm{Cl}(X_\Sigma)$-graded module over the ``Cox ring''
$$S = \C[x_\rho \mid \rho \in \Sigma(1)]$$
\end{thm}

(Generalizes a result of Serre in the case of $X_\Sigma = \P^n$, where 
the ``Cox ring'' is the same as the homogeneous coordinate ring 
$\C[x_0,\ldots,x_n]$.)

\vskip 10pt
Translates abstract questions about sheaf cohomology into \emph{concrete, computable}
questions in multigraded commutative algebra(!)
\end{frame}
\end{document}

