\magnification=\magstep1
\centerline{Math in the Mountains Toric Varieties Tutorial}
\centerline{July 29 - 31, 2013}
\bigskip
\noindent
{\it Exercises on Toric Varieties}
\bigskip
\item{1.}  Show that the rational quartic in ${\bf P}^3$
 corresponding to ${\cal A} = \{(4,0),(3,1),(1,3),(0,4)\}$
is not projectively normal (Harshorne, I.3.18)
\item{2.}  Show that the tetrahedron
$P = {\rm conv}\{0,e_1,e_2,e_1 + e_2 + 2e_3\} \subset {\bf R}^3$
is not normal.  (A lattice polytope $P \subset M_{\bf R}$ is normal
if $(kP \cap M) + (\ell P\cap M) = (k+\ell)P \cap M$ for all $k,\ell \in {\bf N}$.)
\item{3.}  Show that the complete fan $\Sigma$ in $N_{\bf R} \simeq {\bf R}^2$ with rays ${\bf R}_{\ge 0} e_1$,
${\bf R}_{\ge 0} e_2$, and ${\bf R}_{\ge 0} (-e_1-e_2)$ gives $X_\Sigma = {\bf P}^2$. 
\item{4.}  Identify the toric surface given by the complete fans with rays $\pm {\bf R}_{\ge 0} e_1$, 
$\pm {\bf R}_{\ge 0} e_2$ (four 2-dimensional cones)
\item{5.}  Identify the toric surface given by the complete fans with rays ${\bf R}_{\ge 0} e_1$, 
$\pm {\bf R}_{\ge 0} e_2$, and 
$\pm {\bf R}_{\ge 0} (-e_1 + a e_2)$ where $a \ge 1$  (four 2-dimensional cones)
\item{6.}  Show using the toric description that ${\rm Cl}({\bf P}^1) = {\bf Z}$.
\item{7.}  Complete the proof that we have an exact sequence for the class group of $X_\Sigma$:
$$M \to {\bf Z}^{|\Sigma(1)|} \to {\rm Cl}(X_\Sigma) \to 0,$$
where the first map is given by the integer matrix whose rows are the 
primitive generators of the rays of $\Sigma$.
\item{8.}  (Toric surfaces, Riemann-Roch, etc.)  The Riemann-Roch theorem on a smooth
algebraic surface $X$ can be written as
$$\chi({\cal O}_X(D)) = {D(D - K_X)\over 2} + \chi({\cal O}_X),\leqno(1)$$
where $D$ is a divisor on $X$, $K_X$ is the canonical divisor, and 
$$\chi({\cal F}) = h^0({\cal F}) - h^1({\cal F}) + h^2({\cal F})$$
is the Euler characteristic (there would be additional terms if $\dim X > 2$). 
\medskip
\itemitem{a.}  Given a lattice polytope $P$ in ${\bf Z}^2$, show how to use
the facet description of $P$ to find $D$ and $\Sigma$ such that a basis for
$H^0({\cal O}_{X_\Sigma}(D))$ is given by $P \cap {\bf Z}^2$.  
\itemitem{b.}  Show that in this case $\chi({\cal O}_{X_\Sigma}(D)) = |P\cap {\bf Z}^2|$
and $\chi({\cal O}_{X_\Sigma}) = 1$ (Hint:  Demazure + support function.)
\itemitem{c.}  Replace $D$ with $nD$ in the formula (1) above to get 
$$h^0({\cal O}_{X_\Sigma}(nD)) = {D^2\over 2} n^2 - {DK\over 2} n + 1$$
\itemitem{d.}  Pick's formula for a lattice polygon implies 
$$|nP \cap {\bf Z}^2| = {\rm Area}(P) n^2 + {|\partial P \cap {\bf Z}^2|\over 2} n + 1.$$
Check this for the triangle $P = {\rm conv}\{(0,0), (2,0), (0,2)\}$.  
\itemitem{e.}  Conclude:  We can compute $D^2$ and $D K$ from purely combinatorial 
data from $P$ for a toric surface.  In fact $K_{X_\Sigma} = - \sum_{\rho \in \Sigma(1)} D_\rho$
when $X_\Sigma$ is smooth.
\bigskip   
\noindent
{\it Exercises on Algebraic Statistics Topics}
\bigskip
\item{1.}  Give an integer matrix ${\cal A}$ 
such that the corresponding toric model coincides with 
the $3\times 3$ independence model.  Generalize to
obtain a matrix corresponding to the $k \times \ell$
independence model for all $k,\ell \ge 2$.  
\bigskip
\item{2.}  For the Jukes-Cantor model, 
\itemitem{a.}  Compute $P(ACG)$, 
$P(AAC)$, $P(ACA)$ and $P(CAA)$.  Explain where the 
$p_{dis}, p_{ij}$ polynomials come from and why 
every component of the full Jukes-Cantor model
can be expressed using just these 5 distinct polynomials.
\itemitem{b.}  Verify that $p_{123} + p_{dis} + p_{12} + p_{13} + p_{23} = 1$.
\itemitem{c.}  Show that 
$$\eqalign{
q_{111} &= (\theta_1 - \pi_1)(\theta_2 - \pi_2)(\theta_3 - \pi_3)\cr
q_{110} &= (\theta_1 - \pi_1)(\theta_2 - \pi_2)(\theta_3 + 3\pi_3)\cr
q_{101} &= (\theta_1 - \pi_1)(\theta_2 + 3\pi_2)(\theta_3 - \pi_3)\cr
q_{011} &= (\theta_1 + 3\pi_1)(\theta_2 - \pi_2)(\theta_3 - \pi_3)\cr
q_{000} &= (\theta_1 + 3\pi_1)(\theta_2 + 3\pi_2)(\theta_3 + 3\pi_3)\cr}$$
are linear combinations of $p_{123},p_{dis},p_{ij}$ (and
monomials in linear combinations of the original parameters).
\itemitem{d.}  What is the corresponding toric variety for this
reparametrized model?
\bigskip
\item{3.}  Let ${\cal A}$ be any integer matrix with equal column sums.
Let $\varphi$ be the parametrization of the corresponding toric model.  Write
$\widehat{p}$ for $\varphi(\widehat{\theta})$, where
$\widehat{\theta}$ is the MLE for $\theta$.  Show using the method
of Lagrange multipliers that if $L$ is the likelihood function for 
given data $u$, then
$$L \cdot b = L \cdot A u = \lambda \cdot A \widehat{p}$$
\bigskip
\noindent
{\it Exercises on Geometric Modeling Topics}
\bigskip
\item{1.}  Show that for all choices of the control points
$P_i \in {\bf R}^2$, the B\'ezier cubic curve
$$b(t) = (x(t),y(t)) = \sum_{i=0}^3 {3\choose i} t^i (1-t)^{3-i} P_i$$
lies on an algebraic curve defined by an equation $F(x,y) = 0$, 
where $F$ has total degree 3 in $x,y$.  The curve $C = V(F)$ is 
automatically rational ($g(C) = 0$) because of this.  Explain why this
implies $C$ must have a singular point (an ordinary node or cusp), possibly
with coordinates in ${\bf C}$.   
\bigskip
\item{2.}  Some questions about toric surface patches. 
\medskip
\itemitem{a.} Determine the Krasauskas toric blending functions
for the triangle 
$$\Delta_k = {\rm conv}\{(0,0),(k,0),(0,k)\}$$
and show that the corresponding surface patch is a reparametrization
of the B\'ezier triangle patch of order $k$. 
\itemitem{b.} Now consider the Krasauskas toric blending functions
for the triangle 
$$\Delta = {\rm conv}\{(0,0),(2,0),(1,1)\}$$   
What happens on the image of a toric surface patch
for this $\Delta$ at the corner $(1,1)$?  
\itemitem{c.}  Let $P$ be a vertex of $\Delta$.  The ``corner triangle''
of $P$ is the triangle consisting of $P$ and the nearest
two lattice points on the edges adjacent to $P$. One of the results of [K]
is that the image of $P$ on the toric surface patch is nonsingular
if and only if the corner triangle has area $1/2$ (equivalently, 
the two vectors from $P$ to the nearest lattice points on the adjacent
edges form a lattice basis for ${\bf Z}^2$).  Prove this statement and
identify the singularity if the corner triangle has area $\ell/2$ for $\ell > 1$.
\bigskip
\item{3.}  Let ${\cal P}$ be a set of control points in ${\bf R}^3$.
Show that the degree of the implicit equation defining an 
algebraic surface containing the image of $b_{{\cal A},w,{\cal P}}$
is at most 
$$2\cdot {\rm Area}({\rm conv}({\cal A}))$$
and equals this for generic
choices of ${\cal P}$.   (Hint: Use the BKK bound.)
\bigskip
\item{4.}  What is the relation between the image of $b_{{\cal A},w,{\cal P}}$
and the image of $b_{{\cal A}',w,{\cal P}'}$ if ${\cal A}' \subset {\cal A}$ and 
${\cal P}'$ is the subset of ${\cal P}$ corresponding to ${\cal A}'$?
\bigskip
\item{5.}  Let $\Delta$ be a lattice polygon in the plane, let
$h_m$ be the equations of the edges using inward normals, and 
let $H : \Delta \to {\bf R}^\ell$ be the map defined by the $h_i$ 
The toric blending functions
come from composing this $H$ with
$\chi : {\bf R}^\ell \to {\bf R}^{\ell}$ defined by 
$$y \longmapsto (y_1^{h_1(m)}\cdots y_\ell^{h_\ell(m)} : m \in {\cal A})$$
The toric surface
patch is the composition $\pi_{P} \circ \chi \circ H$, where
$\pi_{\cal P} : {\bf R}^\ell \to {\bf R}^3$ is the affine form of a 
linear projection defined by the set of control points
Show that the
$m$-component of $\chi(y)$ is just $y^a z^m$, 
where $a = (a_1,\ldots,a_\ell)$ comes from the constant terms, and
$z = (z_j)$ where  
$$z_j = \prod_{i=1}^\ell y_i^{ \langle {\bf v}_i,{\bf e}_j \rangle }, 
\quad j = 1,2$$
\bigskip
\item{6.}  Show that the toric suface patch for the triangle 
${\rm conv}\{(0,0),(k,0),(0,k)\}$ has linear precision. 
\bigskip
\item{7.}  Exactly how does the set-up for Birch's Theorem relate
to the algebraic moment map for $Y_{\cal A}$?
\bye
