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Write $C={\sigma }^{\vee }$ for the dual cone, $S=C\cap M$, and $U_{\sigma }=\mathrm{Spec}\mathbb{C}[S]$ 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 $\tau ⪯\sigma$, define ${\tau }^{\ast }=C\cap {\tau }^{\perp }$. 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 ${\tau }^{\ast }=C\cap v^{\perp }$ is a face of $C$. Conversely a face $\gamma$ of $C$ gives ${\gamma }^{\ast }=\sigma \cap {\gamma }^{\perp }$. 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 $\mu \in {\tau }^{\ast }$, soThe same argument applied to $C$ proves $({\gamma }^{\ast }{)}^{\ast }=\gamma$, 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 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 claimOne 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 isThus the face chart is an open subvariety.

For a face $\gamma ⪯C$, 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 ${\chi }^m(p)\ne 0$ form a semigroup face: a product is nonzero exactly when both factors are. These exponents are precisely the point of the toric lattices 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,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 givesThis 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 $\nu ⪯\sigma$. 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 isIndeed, 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 the global complement is exactly

The smoothness criterion for a toric variety says that every cone must be regular: its primitive ray generators must be part of a lattice basis of $N$. Such a $d$-dimensional cone gives the chart ${\mathbb{A}}^d\times ({\mathbb{C}}^{\ast }{)}^{n-d}$; all charts are smooth precisely under this criterion.

Here is a terminating construction of a smooth fan subdivision. First subdivide compatibly into simplicial polyhedral cones. One method is the barycentric subdivision: choose a rational interior ray in each nonzero cone and perform the star subdivisions in decreasing dimension. Its cones correspond to chains of faces, hence are simplicial. For a simplicial cone with primitive generators $v_i$, let its multiplicity be the index of their integer span in the saturated lattice $N\cap {\mathrm{span}}_{\mathbb{R}}\sigma$.

If a cone is not regular, choose a nonregular face $\tau$ minimal under inclusion. Its half-open fundamental parallelepiped contains a nonzero point of the toric lattice $w={\sum }_i{\alpha }_i{v}_i$, $0\le {\alpha }_i<1$. Every coefficient is positive: otherwise $w$ lies in the span of a regular proper face, whose generators extend to a lattice basis, forcing all the remaining coefficients to be integers and hence zero. Replace $w$ by the primitive vector $w^{\prime }$ on its ray; its coefficients ${\beta }_i$ still satisfy $0<{\beta }_i<1$. Star-subdivide the entire fan at this ray. In each cone containing $\tau$, the new maximal cones replace one of these $v_i$ by $w^{\prime }$, so multilinearity of the determinant makes the new multiplicity ${\beta }_i$ times the old one. It is a positive integer strictly smaller than the old multiplicity. Other cones are unchanged. Select $\tau$ inside a maximal cone of largest multiplicity. The number of maximal cones having that largest multiplicity decreases, and once it reaches zero the largest multiplicity decreases. This lexicographic integer descent terminates at multiplicity one. It proves every finite rational fan has a smooth subdivision, not merely a simplicial one; this is multiplicity descent in toric desingularization.

For the specified threefold, abbreviate the primitive rays byIts boundary two-dimensional faces are $ab,bc,cd,da$. For example their supporting normals, chosen inward, are $(0,1,-1),(1,1,-2),(1,-1,2),(0,0,1)$; each is zero on the named pair and positive on the other two rays. These normals are primitive. Equivalently the greatest common divisor of the two-by-two minors for each ray pair is one, so these faces are regular; the rays themselves are primitive as well. The full cone is three-dimensional with four extremal rays, hence not simplicial and not regular. The orbit-cone correspondence now shows the closed torus-fixed point is its only singular point: every other orbit belongs to the smooth chart of a proper face.

Define the first subdivision by its maximal conesHere a string of letters denotes their nonnegative span; all faces are included. This is the triangulation along diagonal $ac$ of a positive cross-section. Both determinants have absolute value one, so it is smooth. The subdivision has the same support as the original cone, hence its toric morphism is proper and birational and is a resolution. No new rays were added, so it has no exceptional surfaces. Its internal wall $ac$ gives a compact exceptional curve isomorphic to ${\mathbb{P}}^1$, by the two-ray quotient fan at that wall. The only proper coarsening is the original nonregular cone, so this resolution is minimal in the stated fan sense.

For a second resolution, setStart with the other diagonal $bd$. The cone $abd$ is regular, but $bcd$ has determinant of absolute value two. Its primitive interior point $e$ divides it into the three regular cones shown, each with determinant of absolute value one. Therefore ${\Sigma }_2$ is again a smooth resolution and is different from ${\Sigma }_1$.

This fan has exactly one new ray, so its only exceptional surface is $E_e\cong {\mathbb{P}}^2$. To identify it, quotient $N$ by $\mathbb{Z}e$. The three neighboring rays are $\bar{b},\bar{c},\bar{d}$, their sum is zero, and consecutive pairs are bases because $bce,cde,dbe$ are regular. Their quotient fan is the simplex fan of the projective plane. In fact its normal bundle is ${\mathcal{O}}_{{\mathbb{P}}^2}(-2)$: the chart $U_{bcd}$ before subdivision is ${\mathbb{A}}^3/\{\pm 1\}$ since its index-two lattice extension is generated by $(b+c+d)/2$, and the new radial parameter is the square of the ordinary radial parameter of $\mathcal{O}(-1)$. It has the transition factors of $\mathcal{O}(-2)$.

For minimality, the ray $e$ cannot be deleted in a smooth coarsening. Deleting an interior ray in this triangulation merges all three incident cones into $bcd$, which is nonregular, or merges further into the original nonsimplicial cone. If $e$ is retained, deleting a wall between its three incident cones gives a nonconvex union unless all three are merged, again giving $bcd$. The remaining internal wall $bd$ separates $abd$ and $bde$; their union has four extremal rays $a,b,d,e$, since $a+e=b+d$, and is not regular. Thus no proper coarsening is smooth. These are the two minimal toric resolutions of a four-ray threefold cone.

The factorization can be made completely explicit. Starting with ${\Sigma }_1$, add $e=a+c$ in the wall $ac$, producing a smooth fan ${\Sigma }_A$ with maximal cones $abe,bce,ade,cde$. Next addin the wall $ae$. The resulting common fan has maximal conesStarting instead with ${\Sigma }_2$, the star subdivision at $w=b+d$ in the wall $bd$ gives exactly the same six cones. Each insertion is the sum of the generators of a smooth two-dimensional cone. Such a star subdivision is the toric blowup along an orbit closure of its smooth invariant curve: in local coordinates the center has ideal $(x_1,x_2)$, and the two blowup charts with ratios $x_2/x_1$ and $x_1/x_2$ give the two new cones. We have the zigzag of blowup morphismsThe required birational map is the composite of the inverses of the first two morphisms with the last. This supplies individual smooth toric blowups, rather than appealing to an abstract factorization theorem.

The orbit-cone correspondence is an inclusion-reversing bijectionIf $n=\mathrm{rank}N$, thenThe affine chart and orbit closure in a toric variety areIn particular, the zero cone corresponds to the dense algebraic torus, while maximal cones correspond to torus-fixed points.

For every $t\in T_N$, the torus action mapis an automorphism of $X_{\Sigma }$. It induces isomorphisms of local rings, so it carries smooth points to smooth points and singular points to singular points. The singular locus is consequently invariant under $T_N$. If it contains one point $x$, it contains the entire orbit $T_N\cdot x$; hence the torus-invariance of the singular locus of a toric variety proves that $X_{\Sigma }^{\mathrm{sing}}$ is a union of the orbits in the orbit-cone correspondence.

Write the invariant prime divisors in the order of the raysas $D_1,\dots ,D_4$. The toric divisor class sequence isand the two characters in the standard basis of $M$ giveThuswhere $S=D_2$, $F=D_1=D_3$, and $D_4=S+kF$. Since the fan is smooth, this is also the Picard group.

The orbit-cone correspondence decomposes ${\mathbb{F}}_k$ into one two-dimensional torus, four one-dimensional torus orbits, and four fixed points. Only the zero-dimensional orbits contribute to the compactly supported Euler characteristic, so