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Let $X_{\Sigma }$ be a normal complex toric variety whose toric rays span $N_{\mathbb{R}}$, so it has no torus factor. Form the Cox ring $S=\mathbb{C}[x_{\rho }]$, its toric irrelevant ideal $B_{\Sigma }$, and the diagonalizable groupThe divisor class group grading defines the action of $G$. The Cox construction recovers $X_{\Sigma }$ as the categorical quotient of $\mathrm{Spec}S\setminus V(B_{\Sigma })$ by $G$. It is a geometric quotient exactly when the toric fan is simplicial.

For the projective line, $S=\mathbb{C}[x_0,x_1]$, $B=(x_0,x_1)$ and $G={\mathbb{C}}^{\ast }$ acts by simultaneous scaling, giving $({\mathbb{A}}^2\setminus \{0\})/{\mathbb{G}}_m={\mathbb{P}}^1$. The ray-span condition matters: a positive-dimensional algebraic torus has a zero toric fan and no toric ray variables, so its construction needs the omitted torus factor as well.^[ref]

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 an algebraic group $G$ acting on a algebraic variety $X$ over an algebraically closed field, a geometric quotient is a surjective invariant morphism $q:X\to Y$ whose fibers are exactly the group orbits, whose topology is the quotient topology, and whose regular functions are the invariant regular functions locally:The orbit condition gives actual orbit parameters; the function-sheaf condition specifies the algebraic structure on the orbit space. Both matter beyond a set-theoretic quotient.

For scalar multiplication on ${\mathbb{A}}^{n+1}\setminus \{0\}$, the map $x\mapsto [x]$ is the geometric quotient by ${\mathbb{G}}_m$, giving projective space. On a standard chart, coordinate ratios are the invariant functions. In the Cox construction, the quotient is geometric for a simplicial toric fan; in general the categorical quotient can identify more than a single orbit.^[ref]

For the polynomial Cox ring $S=k[z_{\rho }]$ of a normal split toric variety whose toric rays span the ambient real toric lattice space, the toric irrelevant ideal isThe Cox construction uses the open set $\mathrm{Spec}S\setminus V(B_{\Sigma })$. For the projective line, the two maximal toric cones give generators $z_0,z_1$, so the excluded locus is the origin. For affine space, one maximal toric cone contains every toric ray; the empty product is one, hence $B=(1)$ and nothing is excluded.

The ray-span/no-torus-factor hypothesis belongs to this polynomial presentation. A pure algebraic torus has no toric rays and requires its algebraic torus coordinates in addition to the displayed polynomial data.^[ref]

// Target: geometry-and-topology.bigb