= Solution
Write $C=\check\sigma=\sigma^\vee$, $S=C\cap M$, and $U_\sigma=X_{\check\sigma}=\operatorname{Spec}k[S]$. The <semigroup algebra> has basis $\chi^m=x^m$, $m\in S$, with $\chi^m\chi^{m'}=\chi^{m+m'}$. 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 $k[z_1^{\pm1},\ldots,z_n^{\pm1}]$ after a basis of $M$ is chosen. This proves the <dense torus of an affine toric variety> assertion:
$$
\boxed{X_M=\operatorname{Spec}k[M]\cong(k^*)^n\text{ is a dense open subset of }U_\sigma.}
$$
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> $\nu\preceq C$, 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 $X_\nu=\operatorname{Spec}k[\nu\cap M]$ 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 $S_p=\{m\in S:\chi^m(p)\ne0\}$. Multiplicativity gives $m+m'\in S_p$ if and only if both $m,m'\in S_p$. 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 $C$. Therefore
$$
\boxed{U_\sigma\setminus X_M=\bigcup_{\nu\prec C}X_\nu.}
$$
One 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> is
$$
\boxed{V(\chi^\mu)_{\mathrm{red}}=\bigcup_{\substack{\nu\preceq C\\\mu\notin\nu}}X_\nu.}
$$
This 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 maps
$$
\boxed{\tau\preceq\sigma\ \longmapsto\ C\cap\tau^\perp,\qquad \nu\preceq C\ \longmapsto\ \sigma\cap\nu^\perp.}
$$
Their 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 $\tau\preceq\sigma$, 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 $\operatorname{Spec}k[L]=X_{\tau^\perp}$. 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 $\operatorname{Hom}(M,k^*)\to\operatorname{Hom}(L,k^*)$ is surjective. The extended <algebraic torus> action is multiplication on this open <algebraic torus> and hence transitive. All other <monomials> vanish there. It follows that
$$
\boxed{O(\tau)=X_{\tau^\perp},\qquad F_\tau=\overline{O(\tau)}=X_{C\cap\tau^\perp}.}
$$
This is the <torus orbit-cone correspondence> and its <orbit closure in a toric variety> description. Under <convex face> duality, $F_\gamma=\bigcup_{\eta\succeq\gamma}O(\eta)$.
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 $\gamma\preceq\tau$. The requested <complement of an affine toric face chart> is therefore
$$
\boxed{X_{\check\sigma}\setminus X_{\check\tau}=\bigcup_{\substack{\gamma\preceq\sigma\\\gamma\npreceq\tau}}F_\gamma.}
$$
All 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.
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