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For a normal complex toric variety with no torus factor, its Cox ring isThere is one variable per toric ray, and the grading is by the divisor class group, rather than necessarily by ordinary integers. The degree-$[D]$ part corresponds to sections of the divisorial sheaf ${\mathcal{O}}_X(D)$ after choosing a representative toric divisor.

For ${\mathbb{P}}^n$, this is the usual polynomial ring with every variable in degree one. For a well-formed weighted projective space, the class group is $\mathbb{Z}$ and the variable degrees are its weights. When an algebraic torus factor is present, its invertible Laurent coordinates must be included separately; the ray-variable ring alone cannot recover a positive-dimensional algebraic torus.^[ref]

For a toric divisor $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$ on a normal split toric variety, define the rational polyhedronIts integral points index the torus character basis of global sections of the divisorial sheaf ${\mathcal{O}}_X(D)$. If $D$ is a Cartier divisor, this is an invertible sheaf. For a general Weil divisor it is a rank-one reflexive sheaf.

The polyhedron can be unbounded. For affine space ${\mathbb{A}}^1$ and $D=0$, its inequality is $m\ge 0$, and the integral points give the basis $1,t,t^2,\dots$ of $k[t]$. If the toric fan is complete, the recession toric cone is zero, so a nonempty $P_D$ is bounded. Even then its vertices need not be integral for a non-Cartier Weil divisor. ^[ref]

Use the toric scheme over a base ring construction over $\mathbb{Z}$. Take the cocharacter lattice of an algebraic torus $N={\mathbb{Z}}^2$ and dual character lattice of an algebraic torus $M=\mathrm{Hom}(N,\mathbb{Z})$. For every $i\in \mathbb{Z}$, setThese cones in toric geometry, together with their rays and the zero cone, form an infinite fan in toric geometry. Adjacent two-dimensional cones meet in their common ray; all other pairs meet only at zero. Thus the face and intersection axioms hold even though there are infinitely many charts.

The consecutive vectors $v_i,v_{i+1}$ are a basis of $N$, since their determinant is $-1$. The dual cone has integral generatorswith $⟨a_i,v_i⟩=0$, $⟨a_i,v_{i+1}⟩=1$, $⟨b_i,v_i⟩=1$, $⟨b_i,v_{i+1}⟩=0$. Its integral points form the free commutative monoid $\mathbb{N}{a}_i+\mathbb{N}{b}_i$. Hence the associated semigroup algebra chart is simplyOn the dense algebraic torus, write $t={\chi }^{(1,0)}$ and $q={\chi }^{(0,1)}$. The chart coordinates areAlthough these expressions use inverses on the torus, $x_i,y_i,q$ are regular on the indicated charts.

The common-ray chart of $U_i$ and $U_{i+1}$ is $D(y_i)\subset U_i$, identified with $D(x_{i+1})\subset U_{i+1}$ byThe inverse is $y_i=x_{i+1}^{-1}$, $x_i=y_{i+1}{x}_{i+1}^2$. Nonadjacent charts meet in the common dense torus, and all transition maps agree there with the same characters of an algebraic torus $t,q$, so their cocycle identities follow. By gluing of schemes along open subschemes, these charts construct a scheme $X$.

The lattice projection $(a,b)\mapsto b$ carries every ${\sigma }_i$ into the nonnegative ray of the fan of the affine line. It therefore defines a toric morphismExplicitly its pullback on each chart sends $q$ to $x_i{y}_i$. The adjacent-chart formula preserves that product, so the local morphisms glue.

Compute its scheme-theoretic fibre over $q=0$:This is the union of the two coordinate affine lines, each with multiplicity one. Let $D_i$ denote the toric divisor corresponding to ${\rho }_i$. Its part in $U_i$ is $y_i=0$, with coordinate $x_i$; its part in $U_{i-1}$ is $x_{i-1}=0$, with coordinate $y_{i-1}$. On the overlap these coordinates satisfy $x_i=y_{i-1}^{-1}$. Consequently these two affine lines glue toThe neighbouring components $D_i,D_{i+1}$ meet at the section $p_i=(x_i,y_i)=(0,0)$, isomorphic to $\mathrm{Spec}\mathbb{Z}$. No other components meet. Every special-fibre chart has exactly these two branches and no embedded or extra components. ThusThe union is an infinite chain, not a finite-type curve disguised as one: its infinitely many components account for the infinitely many toric charts.

Over the principal open subset $D(q)$, both $x_i$ and $y_i$ are units because their product is $q$. Every chart then becomes the same torus chart:The transition maps identify the coordinate $t$ exactly, soThis is the generic region described in the question; over the actual generic point of $\mathrm{Spec}\mathbb{Z}[q]$ its fibre is ${\mathbb{G}}_{m,\mathbb{Q}(q)}$.

The construction is also flat. Indeed, each node chart has a free $\mathbb{Z}[q]$-module basisEvery monomial $x_i^u{y}_i^v$ reduces uniquely by taking out $q^{\min (u,v)}$, and the resulting monomials are independent. Thus the chart morphisms are flat morphisms. This verifies directly that the semistable infinite chain from a toric fan is a degeneration of the multiplicative group scheme rather than an unrelated special fibre.

Let $D_1$ and $D_2$ be the toric divisors corresponding respectively to the primitive ray generators $v_1=(3,-2)$ and $v_2=(0,1)$ of the original cone. The principal divisor on a toric variety formula applied to the characters $x={\chi }^{e_1^{\ast }}$ and $y={\chi }^{e_2^{\ast }}$ givesThe toric divisor class sequence therefore presents the divisor class group asThe class of $D_1$ is a generator and $[D_2]=2[D_1]$.

A primitive lattice vector is a nonzero element $u$ of a finite-rank free abelian group $N$ that cannot be written $u=kv$ with $v\in N$ and an integer $|k|>1$. In an integral basis, this is equivalent to the greatest common divisor of its coordinates being one. The property is independent of the chosen lattice basis.

For example, $(2,3)$ is primitive although neither coordinate is one: the integer functional $\ell (x,y)=-x+y$ satisfies $\ell (2,3)=1$. Hence $N=\mathbb{Z}u\oplus \ker \ell$, showing that $u$ extends to a lattice basis. In contrast, $(2,4)=2(1,2)$ is not primitive.

Every rational ray of a fan contains a unique primitive generator pointing along it. This generator gives the integral normalization used in toric divisors and cocharacters; replacing it by a larger positive multiple leaves the geometric ray unchanged but changes its lattice data.

Every toric ray $\rho$ of a normal toric variety determines a torus-invariant prime Weil divisor $D_{\rho }$. An invariant toric toric divisor is an integer combination $D={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$.

Such a toric divisor is Cartier exactly when each toric cone $\sigma$ admits a algebraic torus character $m_{\sigma }\in M$ satisfyingThen the corresponding local algebraic torus character sections trivialize its divisor line bundle. Without this integral local condition, the toric divisor gives a divisorial rank-one reflexive sheaf and need not define a line bundle. The positive class generator of $\mathbb{P}(1,1,2)$, for example, has Cartier index two. Thus an arbitrary combination of invariant prime toric divisors must not be silently identified with invertible-sheaf data.^[ref]