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First suppose $\Delta$ has full dimension in $M_{\mathbb{R}}$. Form the toric cone over it in $M_{\mathbb{R}}\oplus \mathbb{R}$ and the graded semigroup algebrawhere degree zero contains only $k$. The lattice semigroup is finitely generated by the Gordan lemma. The projective toric variety of a lattice polytope is ${\mathbb{P}}_{\Delta }=\mathrm{Proj}{R}_{\Delta }$. Using all lattice points in all dilates avoids any assumption that the degree-one lattice points generate the ring.

For each vertex $v$ of $\Delta$, the degree-zero part after inverting ${\chi }^{v}t$ isIndeed its exponents are $m-lv$ with $m\in l\Delta$. Conversely, every lattice point of the tangent toric cone has this form for sufficiently large integral $l$. These charts cover the Proj construction: each point of a dilate is a rational convex combination of vertices, so some power of its monomial is a product of vertex monomials. Their dual cones are the vertex toric cones of the inward normal fan of a polytope. The normal toric fan is complete because every linear functional on $\Delta$ attains its minimum on a convex face. This describes ${\mathbb{P}}_{\Delta }$ as a complete toric variety. Translating $\Delta$ by $m_0\in M$ gives the graded isomorphism $(m,l)\mapsto (m+l{m}_0,l)$, so containing the origin is unnecessary. If $\Delta$ is not full-dimensional, translate a lattice vertex to zero and use $M\cap {\mathrm{span}}_{\mathbb{R}}(\Delta -\Delta )$; the construction is then complete of dimension $\dim \Delta$.

For the simplex in question put $Q={\prod }_i{q}_i$. Its inequalities are $x_i\ge 0$ and ${\sum }_i{q}_i{x}_i\le Q$, so its inward normal toric rays are ${\mathbb{R}}_{+}{e}_i$ and ${\mathbb{R}}_{+}(-q_1,\dots ,-q_n)$. Here the $e_i$ in the dual space denote the standard dual basis. Apply the real linear map $A(e_i)=e_i/q_i$. It carries these toric rays and their convex face incidences to the standard simplex fan, with toric rays $e_1,\dots ,e_n,-{\sum }_i{e}_i$. The transported lattice isWith $\gcd (q_1,\dots ,q_n)=1$, the primitive vectors of this toric fan in $N^{\prime }$ are $v_i=e_i/q_i$ and $v_0=-{\sum }_i{e}_i$. The latter is primitive because $\lambda {\sum }_i{e}_i\in N^{\prime }$ requires each $\lambda q_i$ to be integral, whose smallest positive solution is $\lambda =1$. Their unique primitive relation isThus mutual coprimality beyond the displayed common gcd condition is not needed.

For a general complete toric variety, introduce one variable $z_{\rho }$ for each toric ray. Its toric homogeneous coordinate ring, or Cox ring, isThe toric divisor class sequence is $0\to M\to {\mathbb{Z}}^{\Sigma (1)}\to \mathrm{Cl}(X_{\Sigma })\to 0$, where the first map is $m\mapsto (⟨m,v_{\rho }⟩{)}_{\rho }$. Thus the graded piece $S_{\alpha }$ has basis the monomials $\prod z_{\rho }^{b_{\rho }}$ with $b_{\rho }\ge 0$ and $\sum b_{\rho }[F_{\rho }]=\alpha$. For an invariant divisor $D=\sum a_{\rho }{F}_{\rho }$, monomials in degree $[D]$ correspond to algebraic torus characters ${\chi }^m$ through $b_{\rho }=a_{\rho }+⟨m,v_{\rho }⟩\ge 0$. This also identifies that graded piece with the global sections of $\mathcal{O}(D)$.

In our example $M^{\prime }=(N^{\prime }{)}^{\ast }={⨁}_i{q}_i\mathbb{Z}{e}_i^{\ast }$, and its map to ${\mathbb{Z}}^{n+1}$ sends ${\sum }_i{q}_i{b}_i{e}_i^{\ast }$ to $(-{\sum }_i{q}_i{b}_i,b_1,\dots ,b_n)$. The quotient is therefore $\mathbb{Z}$, via $(c_0,\dots ,c_n)\mapsto c_0+{\sum }_i{q}_i{c}_i$. In particular $\mathrm{deg}{z}_0=1$ and $\mathrm{deg}{z}_i=q_i$, giving the explicit graded isomorphismThe target has the grading inherited from ordinary total degree in $k[Y_0,\dots ,Y_n]$. Its displayed generators are algebraically independent, which proves injectivity as well as surjectivity onto that subring.

To obtain the quotient, the toric irrelevant ideal is generated by ${\prod }_{\rho \notin \sigma (1)}{z}_{\rho }$. Here each maximal toric cone omits exactly one toric ray, so it is $(z_0,\dots ,z_n)$, and the open set is ${\mathbb{A}}^{n+1}\setminus \{0\}$. The diagonalizable algebraic group with algebraic torus character group $\mathrm{Cl}(X)=\mathbb{Z}$ is $k^{\ast }$. Its action on the Cox coordinates is dictated by their degrees, namelyHere the $y_i$ name the independent Cox coordinates; the graded-subring realization above does not change their weights. More concretely, the quotient chart where $y_i\ne 0$ has coordinate ring $(S[y_i^{-1}]{)}_0$, the invariant ring for this action. These charts glue to $\mathrm{Proj}S$ and to the specified toric fan charts. Positive weights make each orbit closed inside the punctured affine space, and in characteristic zero the finite stabilizers cause no failure of this geometric quotient. This is the weighted projective space construction.

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 $\Sigma$.

The dense algebraic torus has coordinate ring $k[M]$, a Laurent polynomial ring and hence a unique factorization domain. Its divisor class group is therefore zero. Restrict an arbitrary Weil divisor $D$ to the algebraic torus and choose a rational function on an algebraic variety $f$ whose divisor there is $D{|}_T$. Then $D-\mathrm{div}(f)$ is supported on the algebraic torus boundary. The prime divisors in that boundary are precisely the $F_j$ corresponding to the toric rays, by the torus orbit-cone correspondence. ConsequentlySince the variety is smooth, every Weil divisor is Cartier. This gives invariant representatives for all divisor classes, without assuming that the original divisor was invariant.

Now use such a representative. The associated vector space isThe principal divisor on a toric variety formula gives $\mathrm{div}({\chi }^m)={\sum }_j⟨m,e_j⟩F_j$. Define the lattice polytope of a toric divisorFor a maximal toric cone $\sigma$, its basic toric ray vectors are a basis of $N$, so there is a unique integral $m_{\sigma }$ with $⟨m_{\sigma },e_j⟩=-a_j$ on its toric rays. The algebraic torus character ${\chi }^{m_{\sigma }}$ is a local frame for $\mathcal{O}(D)$ on $U_{\sigma }$. Every global section restricts on the algebraic torus to a finite Laurent polynomial ${\sum }_m{c}_m{\chi }^m$. Relative to this frame, regularity on $U_{\sigma }$ means ${\sum }_m{c}_m{\chi }^{m-m_{\sigma }}\in k[{\sigma }^{\vee }\cap M]$. Since distinct algebraic torus characters are independent, this holds exactly when each exponent with nonzero coefficient satisfies the inequalities for the toric rays of $\sigma$. Taking all charts provesCompleteness makes $P_D$ bounded when nonempty: a nonzero recession vector would be nonnegative on every toric ray of a complete toric fan, and therefore on all of $N_{\mathbb{R}}$, which is impossible. The displayed space is consequently finite-dimensional.

At the torus-fixed point $p_{\sigma }$ in the smooth chart $U_{\sigma }\cong {\mathbb{A}}^n$, a section ${\chi }^m$ has local monomial ${\chi }^{m-m_{\sigma }}$. Its value is nonzero exactly when this monomial is constant, namely when $m=m_{\sigma }$. Thus if the complete linear system of a divisor has no base point, some section is nonzero at $p_{\sigma }$, forcing $m_{\sigma }\in P_D$. Conversely, if $m_{\sigma }\in P_D$ for every maximal toric cone, its algebraic torus character is a global section whose local expression on $U_{\sigma }$ is one. It therefore vanishes nowhere on that whole chart, and these charts cover $X$. This proves the toric basepoint-free criterion:The analogous toric ampleness criterion, stated without proof, replaces the inequalities off $\sigma$ by strict ones:Equivalently, the normal fan of a polytope $P_D$ is exactly $\Sigma$.

For the final fan subdivision, take $n\ge 2$ so that $\tilde{e}$ is a new toric ray, as the construction requires. The vectors $e_1,\dots ,e_n$ are a lattice basis, so their sum is primitive. Replacing any one basis vector by this sum gives another basis. Thus the star subdivision gives a smooth toric blowup at a torus-fixed point $\pi :X_{{\Sigma }^{\prime }}\to X_{\Sigma }$, with algebraic exceptional divisor $E=\tilde{F}$. On the old chart, a local equation of $D$ is ${\chi }^{-m_{\sigma }}$, whose order on the new toric ray is $-⟨m_{\sigma },\tilde{e}⟩={\sum }_{i=1}^n{a}_i=a$. All old toric ray coefficients are unchanged. ThereforeWe prove ampleness of $D_c=c{D}^{\prime }-E$ directly by inequalities, giving the ample divisor after blowing up a torus-fixed point result.

Let $u_1,\dots ,u_n$ be the basis of $M$ dual to $e_1,\dots ,e_n$. On the new maximal toric cone ${\sigma }_i$, which omits $e_i$, the required local algebraic torus character isIndeed its pairings on $e_j$, $j\ne i$, are $-c{a}_j$, while its pairing on $\tilde{e}$ is $-ca+1$, exactly the negative of the new coefficient $ca-1$. Its inequality on the omitted old toric ray has strict gap one:For any toric ray $e_j$ outside the original toric cone, put ${\delta }_j=⟨m_{\sigma },e_j⟩+a_j$. The original ampleness gives positive integers ${\delta }_j$, and the new strict gaps arefor all sufficiently large $c$.

For any unchanged maximal toric cone $\eta \ne \sigma$, the local algebraic torus character is $c{m}_{\eta }$. All strict inequalities on old toric rays follow from those for $D$. Its inequality on the new toric ray has gapEach summand is nonnegative, and at least one is positive because $\eta \ne \sigma$ cannot contain all its toric rays. Hence ${ϵ}_{\eta }$ is a positive integer, making this gap positive for $c\ge 2$. There are only finitely many remaining inequalities. For example, any integer satisfyingworks, with the inner maximum omitted when its index set is empty. The toric ampleness criterion now givesThe new-ray assumption is essential to the printed construction. In dimension one, $\tilde{e}=e_1$ and the fan subdivision is unchanged, so the two purported distinct divisors coincide. If the displayed definition of $D^{\prime }$ is applied literally on ${\mathbb{P}}^1$ with coefficients $a_1=-2$, $a_2=3$, then $D$ has degree one but $D^{\prime }$ has degree $-1$, contradicting the conclusion. The argument above applies to the intended blowup in dimension at least two.