Projective toric variety of a lattice polytope (${\mathbb{P}}_{\Delta }$)

For a lattice polytope $\Delta$, form the semigroup of lattice points $(m,l)$ with $l\ge 0$ and $m\in l\Delta$, using only $(0,0)$ in degree zero. Its semigroup algebra is graded by $l$, and its Proj construction gives ${\mathbb{P}}_{\Delta }$. The chart at a lattice vertex $v$ is $\mathrm{Spec}k[\mathrm{cone}(\Delta -v)\cap M]$; these charts give the normal fan of a polytope construction. For a polytope of smaller dimension, use its intrinsic affine lattice after translation.

// Target: geometry-and-topology.bigb