= Lattice polytope of a toric divisor
{title2=$P_D$}
= Toric divisor polyhedron
{synonym}
For a <toric divisor> $D=\sum_\rho a_\rho D_\rho$ on a <normal variety>[normal] split <toric variety>, define the rational <polyhedron>
$$
P_D=\{m\in M_{\mathbb R}:\langle m,v_\rho\rangle\geq-a_\rho\text{ for all }\rho\}.
$$
Its integral <points> index the <torus character> basis of <global sections> of the divisorial <sheaf> $\mathcal O_X(D)$. If $D$ is a <Cartier divisor>, this is an <invertible sheaf>. For a general <Weil divisor> it is a rank-one reflexive <sheaf>.
The <polyhedron> can be unbounded. For <affine space> $\mathbb A^1$ and $D=0$, its inequality is $m\geq0$, and the integral <points> give the basis $1,t,t^2,\ldots$ of $k[t]$. If the <toric fan> is <complete>, the recession <toric cone> is zero, so a nonempty $P_D$ is bounded. Even then its vertices need not be integral for a non-Cartier <Weil divisor>. <Toric Varieties by David A. Cox and others>[chapter4, sections of invariant divisors]{ref}
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