# Friday, October 06

:::{.remark}
Recall that $\OO_X(D)$ is bpf iff $\phi$ is convex and ample iff $\phi$ is *strictly* convex.
If $\phi_0, \phi_1$ have graphs differing by a linear function, then $D_1$ is linearly equivalent to $D_0$, i.e. $D_1 - D_2 = \div(f)$ for some rational functional $f$.
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## Polytopes and Projective Toric Varieties

:::{.remark}
Goal: complete toric varieties with ample line bundles correspond to $n\dash$dimensional lattice polytopes \( \Delta \subseteq M_\RR \).
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:::{.example title="?"}
Consider $\PP^2$ with $D = D_1 + D_2 + D_3$, where the $D_i$ correspond to $\inp{\wait}{-e_1-e_2}, \inp{\wait}{2e_1 - e_2}, \inp{\wait}{-e_1 + 2e_2}$ respectively.
These three vectors determine a simplex-shaped polytope in $M \da \Hom(N, \ZZ)$.
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:::{.lemma title="?"}
\[
H^0(X_ \Sigma; \OO_{X_ \Sigma}(D) ) = \bigoplus_{m\in P_D \intersect M} \CC z^m
.\]
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:::{.proof title="?"}
$\impliedby$: if $m\in P_D \intersect M$, note $z^m$ is a rational function. 
Recall that $v_{z^m}(D_ \rho) = \inp{m}{u_{ \rho}}$. 
Let $f = z^m$, we want to show $\div(f) + D$ is effective.
Check $\sum _{\rho} v_{z^m} D_{ \rho} + \sum_{ \rho} a_ \rho D_{ \rho} \geq 0$ using that $\inp{m}{u_ \rho} + a _{{\rho}} \geq 0$, which is true since $\inp{m}{u_ \rho} \geq - a _{\rho}$ by definition of $P_D$.
So $D$ is Cartier and bpf.

$\implies$: let $f$ be a global section, we want to show $f \in \bigoplus \CC z^m$.
Note $D = 0$ on $X_{\Sigma} \sm D \cong (\cstar)^n$, and so $f$ is regular on $(\cstar)^n$.
So it suffices to show this holds on $(\cstar)^n$.
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:::{.theorem title="?"}
If $D$ is ample then $P_D$ is maximal dimensional.
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:::{.exercise title="?"}
Find the polytopes for $\OO_{\PP^2}(n)$ for every $n$.
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