Monomial support face of a toric point

At a point $p$ of $\mathrm{Spec}k[C\cap M]$, the exponents $m$ with ${\chi }^m(p)\ne 0$ form a semigroup face. They are the lattice points of one face of $C$. Points with that same support face form a single algebraic torus orbit: their nonzero monomial evaluations extend to homomorphisms from the lattice generated by the face into $k^{\ast }$, and restriction from the full torus is surjective.

// Target: mathematical-optimization.bigb