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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.

Let $u,v,w$ be the consecutive primitive toric ray vectors around the compact invariant curve $F=F_v$. Smoothness gives oriented determinants $\det (u,v)=\det (v,w)=1$, hence $u+w=bv$ for an integer $b$. Choose an algebraic torus character $m$ with $⟨m,u⟩=0$ and $⟨m,v⟩=1$. The principal divisor on a toric variety has coefficient one at $F$, coefficient $b$ at $F_w$, and zero at $F_u$. The other invariant curves do not meet $F$, while each adjacent one meets it transversely once. Intersecting this principal divisor with the proper curve $F$ gives $0=F^2+b$. This proves the self-intersection formula for a toric surface divisor even when the ambient surface is not complete.

If $F^2=-1$, then $b=1$ and $v=u+w$. The vectors $u,w$ form a lattice basis since $\det (u,w)=1$, and their toric cone is the union of the two toric cones adjacent to $v$. Deleting $v$ is consequently a smooth toric fan coarsening. Its inverse is precisely the toric blowup at a torus-fixed point, obtained by inserting the sum of the two primitive basis vectors. Conversely, that insertion gives $u+w=v$, and hence algebraic self-intersection $-1$. Thus the toric contraction of a minus-one curve satisfiesFor the cyclic quotient description, first take a strongly convex toric cone of dimension two. Write its primitive boundary vectors as $a,b$ and put $N_0=\mathbb{Z}a+\mathbb{Z}b$, $r=[N:N_0]$. Since $a$ is primitive, extend it to a basis of $N$ and write $b=ua+rc$ with $\gcd (u,r)=1$. Thus $N/N_0$ is cyclic of order $r$. In the lattice $N_0$, the toric cone is smooth and its toric variety is ${\mathbb{A}}^2$. Passing to the larger lattice $N$ takes the finite quotient whose invariant monomials are precisely those in the smaller dual lattice $M=N^{\ast }\subset N_0^{\ast }$. Relative to the two boundary coordinates, $c=(b-ua)/r$, so a generator acts with weights $(-u,1)$ modulo $r$. Replacing that generator by a suitable power makes the first weight one and the second some $q$ with $0<q<r$ and $\gcd (q,r)=1$. Therefore, for the singular case,The cyclic quotient surface singularity can equivalently be described by the first-quadrant toric cone inIndeed its coordinate ring is $k[x^i{y}^j:i,j\ge 0,i+qj\equiv 0(\mathrm{mod}r)]=k[x,y{]}^{{\mu }_r}$. Characteristic zero ensures the stated ordinary finite-group quotient interpretation.

The restriction to singular toric cones of dimension two is necessary for $r>1$. A smooth toric cone has $r=1$ and gives ${\mathbb{A}}^2$ with the trivial quotient. A toric cone of dimension one gives ${\mathbb{A}}^1\times k^{\ast }$, and the zero toric cone gives $(k^{\ast }{)}^2$. These latter surfaces cannot be a quotient of the displayed kind, since they have nonconstant invertible regular functions, whereas $k[x,y{]}^{{\mu }_r}$ has only constant units. Thus the blanket affine-surface wording requires these separate cases.

For the singular case, compute the negative continued fraction by the ceiling form of the Euclidean algorithm. Start with $a_0=r$, $a_1=q$, and, while $a_i>0$, setThen $0\le a_{i+1}<a_i$, so the process terminates at $a_{s+1}=0$. The gcd is unchanged at each step, hence $a_s=1$. Since $a_{i-1}>a_i>0$, every $b_i\ge 2$. Rearranging the recurrence givesNow define lattice vectorsTo verify this Hirzebruch–Jung resolution algorithm, write $v_i=(p_i,a_i)/r$ with $p_0=0$, $p_1=1$ and $p_{i+1}=b_i{p}_i-p_{i-1}$. Induction gives $a_i\equiv q{p}_i(\mathrm{mod}r)$, so $v_i\in L$. It also gives $p_i{a}_{i+1}-a_i{p}_{i+1}=-r$. Thus every adjacent pair has absolute determinant $1/r$, the covolume of $L$, and is a basis of $L$. The $p_i$ increase strictly, while the $a_i$ decrease to zero, so these toric rays occur in order inside the first quadrant. At termination, $a_s=1$ and the determinant identity gives $p_{s+1}=r$, hence $v_{s+1}=(1,0)$. All boundary and inserted vectors are primitive because they occur in lattice bases.

Subdivide the quadrant by $v_1,\dots ,v_s$. Every resulting toric cone is smooth by the smoothness criterion for a toric variety, and the fan subdivision gives a proper birational toric morphism, hence a resolution. Each interior toric ray has a complete one-dimensional star, so its exceptional curve of a surface resolution is ${\mathbb{P}}^1$. The recurrence yieldsThis proves minimality: no exceptional curve of a surface resolution can be contracted to a smooth point by the Castelnuovo contraction criterion. There is also a direct toric fan check. For fixed $i$, the determinants of $v_i$ with $v_{i+h}$, measured in units of the lattice covolume, satisfy the same recurrence with coefficients at least two and initial values zero and one. They are at least $h$. Omitting any intervening toric ray therefore leaves a toric cone of determinant at least two, so no nontrivial coarsening is smooth. Thus this is the minimal resolution of a cyclic quotient surface singularity.

If all exceptional algebraic self-intersections are $-2$, all $b_i=2$ and induction gives $[2,\dots ,2{]}^{-}=(s+1)/s$. Since $r,q$ are coprime, this forces $r=s+1$, $q=s$. Conversely $r/(r-1)$ has exactly $r-1$ coefficients equal to two. ThereforeFor the final toric fan, the minimal smooth fan subdivision of the toric cone from $(0,1)$ to $(r,1)$ inserts the toric rays $(j,1)$ for $1\le j<r$. Adjacent determinants have absolute value one, and $(j-1,1)+(j+1,1)=2(j,1)$, so its inserted curves have algebraic self-intersection $-2$ and it is the minimal fan subdivision just described. The second original toric cone, from $(r,1)$ to $(1,0)$, is already smooth.

Write $v_j=(j,1)$ and $e=(1,0)$. In the resolved toric fan, the rightmost interior toric ray obeys $v_{r-1}+e=v_r$, so its curve has algebraic self-intersection $-1$ and can be blown down to a smooth point. After deleting it, the same relation $v_{r-2}+e=v_{r-1}$ applies to the new rightmost interior toric ray. Continue deleting $v_r,v_{r-1},\dots ,v_1$ in that order. Each deletion is the inverse of a toric blowup at a torus-fixed point. The last toric fan has only the quadrant toric cone and defines ${\mathbb{A}}^2$. ConsequentlyAll maps induce the identity on the dense algebraic torus, so this composite is the same morphism as $X_{{\Sigma }^{\prime }}\to X_{\Sigma }\to {\mathbb{A}}^2$. The intermediate smooth toric fans need not include the singular toric fan $\Sigma$.

For the rays $v_1=(1,0)$, $v_2=(0,1)$, and $v_3=(-a,-b)$, the determinants of consecutive pairs have absolute valuesBy the smoothness criterion for a toric variety, $\mathbb{P}(1,a,b)$ is smooth exactly when $a=b=1$.

Let $v_1,\dots ,v_r$ be the primitive lattice generators of the rays of the strongly convex rational cone $\sigma \subseteq N_{\mathbb{R}}$. The smoothness criterion for a toric variety says that the affine toric variety $U_{\sigma }$ is smooth exactly when $v_1,\dots ,v_r$ form part of a $\mathbb{Z}$-basis of the cocharacter lattice of an algebraic torus $N$. Equivalently, there is a basis $e_1,\dots ,e_n$ of $N$ such thatIn particular, the cone must be a simplicial polyhedral cone and its ray generators must be primitive; for a full-dimensional cone the criterion says that those generators form a lattice basis.

The primitive ray generators of $\sigma$ areand $|\det (v_0,v_3)|=3$. They do not form a lattice basis, so the smoothness criterion for a toric variety shows that $U_{\sigma }$ is singular. The presentation from part (i) gives the same conclusion by the Jacobian criterion: all derivatives of $B^3-AC$ vanish at $A=B=C=0$.

Insert the primitive raysLet $\Sigma$ consist of the conesand all their faces. Sinceevery cone of $\Sigma$ is smooth. Its support is $\sigma$, so this fan subdivision induces a proper birational toric morphismThe source is smooth, and $p$ is an isomorphism over the dense algebraic torus; it is therefore a toric resolution of singularities.

Take the weighted projective plane $X=\mathbb{P}(1,1,2)$. Its fan in $N={\mathbb{Z}}^2$ has rays generated byand all three two-dimensional cones spanned by adjacent rays. Their union is $N_{\mathbb{R}}$, so the fan is complete and the toric variety $X$ is proper.

The determinants of the cones $⟨v_1,v_2⟩$ and $⟨v_2,v_0⟩$ have absolute value one, whileThe smoothness criterion for a toric variety therefore shows that the affine chart for $⟨v_0,v_1⟩$ is singular; it is the cyclic quotient singularity of type $\frac{1}{2}(1,1)$. Thus $X$ is a singular proper toric surface.