Monomial zero locus in an affine toric variety ($V({\chi }^{\mu })={\bigcup }_{⟨\mu ,\tau ⟩\ne 0}V(\tau )$)

On the orbit of a face $\tau ⪯\sigma$, the monomial ${\chi }^{\mu }$, with $\mu \in {\sigma }^{\vee }\cap M$, vanishes exactly when $\mu \notin {\tau }^{\perp }$. Its reduced zero locus is consequently the union of those orbit closures. The principal subscheme need not be reduced: $V(x^2)$ and $V(x)$ have the same support but different ideals.

// Target: geometry-and-topology.bigb