= Solution
Write $C=\sigma^\vee$ for the <dual cone>, $S=C\cap M$, and $U_\sigma=\operatorname{Spec}\mathbb C[S]$ for the <affine toric variety>. This is the variety denoted $X_{\check\sigma}$ in the source; the subscripts there refer to the cone of exponents, whereas $U_\sigma$ refers to the cone in $N_{\mathbb R}$.
For a <face of a polyhedral cone> $\tau\preceq\sigma$, define $\tau^*=C\cap\tau^\perp$. If $v$ is in the relative interior of $\tau$, nonnegativity on $\sigma$ shows that a functional in $C$ vanishes on $v$ exactly when it vanishes on all of $\tau$. Hence $\tau^*=C\cap v^\perp$ is a face of $C$. Conversely a face $\gamma$ of $C$ gives $\gamma^*=\sigma\cap\gamma^\perp$. Every polyhedral face is exposed: the supporting functionals for its containing facets can be added, since they are nonnegative. If $\tau=\sigma\cap\mu^\perp$, then $\mu\in\tau^*$, so
$$
\tau\subseteq\sigma\cap(\tau^*)^\perp\subseteq\sigma\cap\mu^\perp=\tau.
$$
The same argument applied to $C$ proves $(\gamma^*)^*=\gamma$, using the closed-cone identity $C^\vee=\sigma$. That identity follows from separation of a point outside a closed convex cone. Finally, enlarging $\tau$ makes its annihilator smaller. Thus \b[these maps are inverse order-reversing bijections], proving <face duality for polyhedral cones> even when $\sigma$ is not full-dimensional.
For the given integral exposing functional $\mu$, we claim
$$
\tau^\vee\cap M=S+\mathbb Z_{\geq0}(-\mu).
$$
One inclusion is immediate since $\mu$ vanishes on $\tau$. For the other, let $m$ be nonnegative on $\tau$ and test it on finitely many generating rays of $\sigma$. Where $\mu$ vanishes, those rays belong to $\tau$, so $m$ is already nonnegative. On every other ray $\mu$ is positive, so choosing a sufficiently large integer $k$ makes $m+k\mu$ nonnegative on all the rays. This proves the claim. The corresponding <semigroup algebra> identity is
$$
\mathbb C[\tau^\vee\cap M]=\mathbb C[S]_{\chi^\mu},\qquad \boxed{U_\tau=D(\chi^\mu)\subset U_\sigma.}
$$
Thus the face chart is an open subvariety.
For a face $\gamma\preceq C$, the span of monomials $\chi^m$ with $m\notin\gamma$ is an ideal: if $m+n\in\gamma$ with $m,n\in C$, the face property forces both into $\gamma$. Its quotient is $\mathbb C[\gamma\cap M]$, defining the closed subvariety $X_\gamma$. If $\mu\notin\gamma$, this subvariety lies in $V(\chi^\mu)$.
Conversely, at a point $p\in U_\sigma$, the exponents with $\chi^m(p)\ne0$ form a <semigroup face>: a product is nonzero exactly when both factors are. These exponents are precisely the point of the <toric lattice>s of a face $\gamma$ of $C$. To see the geometric assertion, choose finitely many monoid generators by the <Gordan lemma>. The nonzero generators span a cone. If that cone were not a face, a rational nonnegative relation would express a sum of its generators using an omitted generator with a positive coefficient. Clearing denominators and evaluating the corresponding monomials would equate a nonzero product to zero. Conversely, any point of the <toric lattice> in their cone has a positive integral multiple expressible using them, so its monomial must also be nonzero. This proves the <monomial support face of a toric point> assertion. If $\chi^\mu(p)=0$, then $\mu\notin\gamma$ and every monomial outside $\gamma$ vanishes at $p$, placing $p$ in $X_\gamma$. Therefore, as reduced closed subvarieties,
$$
\boxed{U_\sigma\setminus U_\tau=\bigcup_{\substack{\gamma\preceq C\\\mu\notin\gamma}}X_\gamma.}
$$
The assertion concerns the closed set with its reduced structure; a monomial's principal zero subscheme can have multiplicities.
For the intersection assertion, choose integral exposing functionals $\mu_i$ for $\tau_i$. Both are nonnegative on $\sigma$, so $\tau_1\cap\tau_2=\sigma\cap(\mu_1+\mu_2)^\perp$. Localization then gives
$$
\boxed{U_{\tau_1}\cap U_{\tau_2}=D(\chi^{\mu_1})\cap D(\chi^{\mu_2})=D(\chi^{\mu_1+\mu_2})=U_{\tau_1\cap\tau_2}.}
$$
This uses the dual of the intersection, as printed in the PDF; the TeX's perpendicular symbol in that position is not the correct transcription.
For a general <toric fan>, the points with a fixed monomial support face form a torus orbit. The nonzero evaluations extend to a character of the lattice generated by that face, and restriction from the ambient <algebraic torus> is surjective over $\mathbb C$. Thus the affine calculation yields the <orbit-cone correspondence>: $U_\sigma$ contains exactly the orbits $O(\nu)$ for faces $\nu\preceq\sigma$. The orbit associated to $\tau$ has character lattice $M\cap\tau^\perp$, so it is the source's $X_{\tau^\perp}$. The same monomial ideal calculation, chart by chart, shows that its closure is
$$
F_\tau=\bigcup_{\nu\in\Sigma,\ \tau\preceq\nu}O(\nu).
$$
Indeed, inside a chart containing $\tau$, the closure has exponent face $\sigma^\vee\cap\tau^\perp$, and its smaller support faces correspond exactly to cones containing $\tau$.
Fix $\sigma\in\Sigma$, as required for its affine chart. If $\tau$ is not contained in $\sigma$, none of the larger cones $\nu\supseteq\tau$ can be contained in $\sigma$, so $F_\tau$ is disjoint from $U_\sigma$. Conversely every orbit outside $U_\sigma$ has a cone $\nu$ not contained in $\sigma$, and it lies in $F_\nu$. Consequently \b[the global complement is exactly]
$$
\boxed{X_\Sigma\setminus U_\sigma=\bigcup_{\tau\in\Sigma,\ \tau\not\subseteq\sigma}F_\tau.}
$$
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