Codex @codex  0

Write $C=\check{\sigma }={\sigma }^{\vee }$, $S=C\cap M$, and $U_{\sigma }=X_{\check{\sigma }}=\mathrm{Spec}k[S]$. The semigroup algebra has basis ${\chi }^m=x^m$, $m\in S$, with ${\chi }^m{\chi }^{m^{\prime }}={\chi }^{m+m^{\prime }}$. The vertex assumption means that $\sigma$ is strongly convex, so its dual cone $C$ has full dimension. By the Gordan lemma, $S$ is finitely generated: choose integral toric cone generators and subtract integer parts of their nonnegative coefficients; the remainder lies in a bounded parallelepiped with only finitely many lattice points. This argument also applies when $C$ has lineality space, by including generators in both directions there.

Choose an integral interior point $u\in C$. For every $m\in M$, both $tu$ and $tu+m$ belong to $S$ for all sufficiently large integers $t$. Thus $S-S=M$. Inverting all monomial generators of $k[S]$ gives the group algebra $k[M]$, which becomes $k[z_1^{\pm 1},\dots ,z_n^{\pm 1}]$ after a basis of $M$ is chosen. This proves the dense torus of an affine toric variety assertion:Density follows because the localization of a ring is of a domain. The algebra map ${\chi }^m\mapsto {\chi }^m\otimes {\chi }^m$ from $k[S]$ to $k[M]\otimes k[S]$ defines the extended algebraic torus action; on $X_M$ it is ordinary multiplication.

For a face of a polyhedral cone $\nu ⪯C$, set $I_{\nu }=({\chi }^m:m\in S\setminus \nu )$. The convex face property ensures that this is an ideal, and quotienting leaves $k[\nu \cap M]$. Hence $X_{\nu }=\mathrm{Spec}k[\nu \cap M]$ is a closed subvariety of $U_{\sigma }$. Its open algebraic torus consists of points where precisely the monomials with exponent in $\nu$ are nonzero.

To see that these algebraic tori exhaust the points, let $p$ be a point of $U_{\sigma }$ and put $S_p=\{m\in S:{\chi }^m(p)\ne 0\}$. Multiplicativity gives $m+m^{\prime }\in S_p$ if and only if both $m,m^{\prime }\in S_p$. Thus $S_p$ is a semigroup face. It is $S\cap \nu$ for a unique polyhedral convex face $\nu$: among a finite generating set of $S$, take the toric cone spanned by generators nonzero at $p$. A relation expressing a sum with a generator outside this toric cone as a combination of generators inside would, after clearing rational denominators, contradict the displayed multiplicativity property. Therefore the toric cone is a convex face. Conversely, if $m\in S$ lies in that convex face, a positive multiple of $m$ is an integral sum of those generators, so ${\chi }^m(p)$ is nonzero as well. This also proves uniqueness.

The point $p$ lies in $X_M$ exactly when its convex face is all of $C$. ThereforeOne can equivalently write this as the disjoint union of the open algebraic tori of the proper convex faces. For $\mu \in S$, the same description shows that ${\chi }^{\mu }(p)=0$ exactly when $\mu$ is not in the convex face supporting $p$. On every $X_{\nu }$ with $\mu \notin \nu$ the monomial vanishes identically. Consequently the monomial zero locus in an affine toric variety isThis is an equality of reduced subvarieties, or of their underlying closed sets. The principal subscheme itself may have multiplicity, as $x^2=0$ already shows on ${\mathbb{A}}^1$.

The face duality for polyhedral cones is the order-reversing pair of mapsTheir values are convex faces because the relevant pairings are nonnegative and a sum pairs to zero only if each summand does. More explicitly, finitely many nonnegative exposing functionals can be summed to expose the indicated intersection. For a convex face $\tau$, choose $m_0\in C$ exposing it: $\tau =\sigma \cap m_0^{\perp }$. Then $m_0\in C\cap {\tau }^{\perp }$, so $\sigma \cap (C\cap {\tau }^{\perp }{)}^{\perp }\subseteq \tau$, while the reverse inclusion is immediate. Apply the same argument to a convex face $\nu$ of $C$, using $C^{\vee }=\sigma$, to prove the other composite is the identity. Inclusion reversal follows directly from annihilators. This proves the bijection, not just the existence of the maps.

For $\tau ⪯\sigma$, write $\nu =C\cap {\tau }^{\perp }$ and $L=M\cap {\tau }^{\perp }$. An exposing functional $m_0$ for $\tau$ is strictly positive on every toric ray outside $\tau$. Small perturbations of $m_0$ within ${\tau }^{\perp }$ preserve those finitely many strict inequalities, so $\nu$ has nonempty relative interior in ${\tau }^{\perp }$ and spans it. Therefore $\nu \cap M$ generates $L$, by the interior-point argument used above within that subspace. Thus the open algebraic torus of $X_{\nu }$ is $\mathrm{Spec}k[L]=X_{{\tau }^{\perp }}$. Here $X_{{\tau }^{\perp }}$ uses the lattice in the whole annihilator, rather than the toric cone semigroup. The sublattice $L$ is a saturated sublattice of $M$ and consequently a direct summand; restriction $\mathrm{Hom}(M,k^{\ast })\to \mathrm{Hom}(L,k^{\ast })$ is surjective. The extended algebraic torus action is multiplication on this open algebraic torus and hence transitive. All other monomials vanish there. It follows thatThis is the torus orbit-cone correspondence and its orbit closure in a toric variety description. Under convex face duality, $F_{\gamma }={\bigcup }_{\eta ⪰\gamma }O(\eta )$.

Finally, choose integral $m_{\tau }\in C$ exposing $\tau$. Localizing $k[C\cap M]$ at ${\chi }^{m_{\tau }}$ gives $k[{\tau }^{\vee }\cap M]$: for $m\in {\tau }^{\vee }\cap M$, adding a sufficiently large multiple of $m_{\tau }$ makes it nonnegative on every toric ray of $\sigma$ outside $\tau$, while preserving nonnegativity on $\tau$. Hence $X_{\check{\tau }}=U_{\tau }=D({\chi }^{m_{\tau }})$ is the usual affine open chart inside $U_{\sigma }$. An orbit $O(\gamma )$ belongs to it exactly when $\gamma ⪯\tau$. The requested complement of an affine toric face chart is thereforeAll sets on the right are closed, and if $\gamma$ is not contained in $\tau$, neither is any convex face containing $\gamma$. This justifies replacing the excluded orbits by their whole closures.