 Lattice polytope of a toric divisor

For a toric divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$ on a normal split toric variety, define the rational polyhedronIts 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 $m\ge 0$, and the integral points give the basis $1,t,t^2,\dots$ 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. ^[ref]