The Cox homogeneous coordinate ring and the toric GIT quotient

Intuition Beginner

Projective space has a beautiful one-line description: a point of ${\mathbb{P}}^n$ is a list of homogeneous coordinates $[x_0:\dots :x_n]$, not all zero, where two lists count as the same point when one is a nonzero scalar multiple of the other. In other words ${\mathbb{P}}^n$ is the space ${\mathbb{A}}^{n+1}$ with the origin removed, then collapsed by the rule "rescale everything at once." A single polynomial ring in $n+1$ variables, plus one scaling rule, describes the whole variety.

David Cox discovered in 1995 that every toric variety has a description of exactly this shape. Recall from the fan unit that a toric variety $X_{\Sigma }$ is built from a fan, a fan-out of cones whose one-dimensional edges are called rays. Cox's idea: introduce one variable $x_{\rho }$ for each ray $\rho$. These variables generate a single polynomial ring $S$, the total coordinate ring (also called the Cox ring). It is the toric replacement for the homogeneous coordinate ring of projective space.

The scaling rule for ${\mathbb{P}}^n$ was one copy of ${\mathbb{C}}^{\ast }$ acting by simultaneous multiplication. For a general toric variety the rule is richer: a group $G$ acts on the variable space, and $G$ is read off from the geometry of the fan. The toric variety is then the space of variable-tuples, minus a forbidden set, collapsed by the action of $G$.

So the slogan is short. One polynomial ring, one clever grading, one group action: a whole toric variety presented as a single quotient, just like projective space.

Visual Beginner

A two-panel picture comparing the projective-space template with the general toric quotient. Left panel: the space ${\mathbb{A}}^{n+1}$ drawn as a large region with the origin marked as a small forbidden dot. Radial lines emanate from the origin; each radial line is one orbit of the scaling action of ${\mathbb{C}}^{\ast }$, and each line collapses to a single point of ${\mathbb{P}}^n$, drawn on a separate target line to the right of the panel. The caption reads "${\mathbb{P}}^n=({\mathbb{A}}^{n+1}\setminus 0)/{\mathbb{C}}^{\ast }$: remove the origin, collapse each ray to a point."

Right panel: the space ${\mathbb{A}}^{\Sigma (1)}$, one coordinate axis per ray of the fan, drawn as a box. A shaded region inside the box is labelled the forbidden set $Z(\Sigma )$ (the place where too many coordinates vanish at once). The group $G$ acts on the box, and its orbits are drawn as small curved sheets; collapsing each sheet to a point yields the toric variety $X_{\Sigma }$, drawn as a target region. The caption reads "$X_{\Sigma }=({\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma ))/G$: remove the forbidden set, collapse each $G$-orbit to a point."

The picture makes the analogy precise: the right panel is the left panel with $n+1$ replaced by "number of rays," the single ${\mathbb{C}}^{\ast }$ replaced by the group $G$, and the origin replaced by the forbidden set $Z(\Sigma )$.

Worked example Beginner

Build the projective plane ${\mathbb{P}}^2$ as a Cox quotient. The fan of ${\mathbb{P}}^2$ has three rays, with primitive generators $u_0=-e_1-e_2$, $u_1=e_1$, $u_2=e_2$, as recalled in the fan unit. One variable per ray gives three variables $x_0,x_1,x_2$, so the total coordinate ring is $S=\mathbb{C}[x_0,x_1,x_2]$.

The grading. The three rays satisfy one relation, $u_0+u_1+u_2=0$. This single relation says that the three variables share one common scaling degree. So $S$ is graded by $\mathbb{Z}$, with each variable in degree one, and a monomial $x_0^a{x}_1^b{x}_2^c$ sits in degree $a+b+c$. This is the ordinary grading of the homogeneous coordinate ring of ${\mathbb{P}}^2$.

The forbidden set. The rule is that we delete the locus where the coordinates fail to lie in any cone of the fan. For ${\mathbb{P}}^2$ this deleted locus is the single point $x_0=x_1=x_2=0$, the origin of ${\mathbb{A}}^3$. So we work on ${\mathbb{A}}^3\setminus 0$.

The group. One $\mathbb{Z}$ grading direction corresponds to one copy of ${\mathbb{C}}^{\ast }$. The group $G={\mathbb{C}}^{\ast }$ acts by $t\cdot (x_0,x_1,x_2)=(t{x}_0,t{x}_1,t{x}_2)$, simultaneous scaling.

The result. The quotient $({\mathbb{A}}^3\setminus 0)/{\mathbb{C}}^{\ast }$ is the standard construction of ${\mathbb{P}}^2$. The Cox machine has reproduced homogeneous coordinates on ${\mathbb{P}}^2$, with $x_0,x_1,x_2$ the familiar coordinates. What the machine adds is that the recipe — one variable per ray, grade by the relations among rays, delete a forbidden set, collapse by the matching group — works for any toric variety, not only projective space.

Check your understanding Beginner

Formal definition Intermediate+

Let $X_{\Sigma }$ be the toric variety of a fan $\Sigma$ in $N_{\mathbb{R}}$, as constructed in [04.11.04], with $\Sigma (1)$ the set of rays and $u_{\rho }\in N$ the primitive generator of $\rho$. Write $M=\mathrm{Hom}(N,\mathbb{Z})$ for the character lattice.

Definition (total coordinate ring). The total coordinate ring (or Cox ring) of $X_{\Sigma }$ is the polynomial ring $$ S = \mathbb{C}[x_\rho : \rho \in \Sigma(1)], $$ one variable per ray. A monomial $x^a={\prod }_{\rho }{x}_{\rho }^{a_{\rho }}$ corresponds to the divisor $D_a={\sum }_{\rho }{a}_{\rho }{D}_{\rho }$ in the group ${\mathbb{Z}}^{\Sigma (1)}={⨁}_{\rho }\mathbb{Z}{D}_{\rho }$ of $T$-invariant Weil divisors of [04.11.08].

Definition (the class-group grading). Assume $X_{\Sigma }$ has no torus factor, so that the rays span $N_{\mathbb{R}}$. The divisor-class exact sequence of [04.11.08], $$ 0 \to M \xrightarrow{\ \mathrm{div}\ } \mathbb{Z}^{\Sigma(1)} \xrightarrow{\ \pi\ } \mathrm{Cl}(X_\Sigma) \to 0, \qquad m \mapsto \sum_\rho \langle m, u_\rho\rangle D_\rho, $$ defines a grading of $S$ by the class group: the monomial $x^a$ has degree $\mathrm{deg}(x^a):=\pi (D_a)=[D_a]\in \mathrm{Cl}(X_{\Sigma })$. This makes $S={⨁}_{\beta \in \mathrm{Cl}(X_{\Sigma })}{S}_{\beta }$ a $\mathrm{Cl}(X_{\Sigma })$-graded ring, with $S_{\beta }={⨁}_{[D_a]=\beta }\mathbb{C}{x}^a$.

Definition (irrelevant ideal). For each cone $\sigma \in \Sigma$, write $x^{\hat{\sigma }}={\prod }_{\rho \notin \sigma (1)}{x}_{\rho }$ for the product of the variables of the rays not in $\sigma$. The irrelevant ideal is $$ B(\Sigma) = \big\langle x^{\hat\sigma} : \sigma \in \Sigma \big\rangle \subseteq S, $$ generated by these monomials (it suffices to range over maximal cones). Its vanishing locus $Z(\Sigma )=V(B(\Sigma ))\subseteq {\mathbb{A}}^{\Sigma (1)}=\mathrm{Spec}S$ is the exceptional set (or irrelevant locus). Concretely a point lies in $Z(\Sigma )$ when the set of rays $\{\rho :x_{\rho }=0\}$ fails to be contained in the rays of any single cone of $\Sigma$.

Definition (the quotient group). Applying $\mathrm{Hom}(-,{\mathbb{C}}^{\ast })$ to the exact sequence and using $\mathrm{Hom}(M,{\mathbb{C}}^{\ast })=T$ gives $$ 1 \to G \to (\mathbb{C}^)^{\Sigma(1)} \to T \to 1, \qquad G = \mathrm{Hom}(\mathrm{Cl}(X_\Sigma), \mathbb{C}^). $$ The group $G$ is a diagonalisable group (a product of a torus and a finite abelian group), and it acts on ${\mathbb{A}}^{\Sigma (1)}$ as the subgroup of the diagonal torus $({\mathbb{C}}^{\ast }{)}^{\Sigma (1)}$ that maps to the identity of $T$. Explicitly $g=(g_{\rho })\in G$ iff ${\prod }_{\rho }{g}_{\rho }^{⟨m,u_{\rho }⟩}=1$ for all $m\in M$.

Definition (the Cox quotient). The quotient presentation of $X_{\Sigma }$ is $$ X_\Sigma ;=; \big(\mathbb{A}^{\Sigma(1)} \setminus Z(\Sigma)\big) /!!/ G, $$ the categorical quotient of the open set ${\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma )$ by the action of $G$. When $\Sigma$ is simplicial this is a geometric quotient (the orbits are closed in the open set and the quotient is an orbit space); the quotient is identified with a GIT quotient in the sense of [04.10.02] for a suitable character linearisation. The variables $x_{\rho }$ are the homogeneous coordinates on $X_{\Sigma }$.

Counterexamples to common slips

• "The grading is always by $\mathbb{Z}$." Only when $\mathrm{Cl}(X_{\Sigma })\cong \mathbb{Z}$, as for ${\mathbb{P}}^n$. For ${\mathbb{P}}^1\times {\mathbb{P}}^1$ the class group is ${\mathbb{Z}}^2$, so the grading is bigraded; for a singular toric surface the class group can carry torsion, and the grading lands in a group with a finite part.

• "The forbidden set is always the origin." Only for fans like that of ${\mathbb{P}}^n$ where every proper coordinate subspace meets a cone. For ${\mathbb{P}}^1\times {\mathbb{P}}^1$ the exceptional set $Z(\Sigma )$ is the union of two coordinate planes, $\{x_1=x_2=0\}\cup \{x_3=x_4=0\}$, not a single point.

• "The quotient is always geometric." For non-simplicial $\Sigma$ the action of $G$ has non-closed orbits, and the construction yields only a categorical (good) quotient that identifies non-separated orbits; the geometric-quotient statement needs the simplicial hypothesis.

Key theorem with proof Intermediate+

Theorem (Cox quotient presentation; Cox 1995). Let $X_{\Sigma }$ be the toric variety of a fan $\Sigma$ whose rays span $N_{\mathbb{R}}$, with total coordinate ring $S$, irrelevant ideal $B(\Sigma )$, exceptional set $Z(\Sigma )=V(B(\Sigma ))$, and group $G = \mathrm{Hom}(\mathrm{Cl}(X_\Sigma), \mathbb{C}^)$.*

(a) Quotient. There is a canonical morphism $\pi :{\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma )\to X_{\Sigma }$ that is a good categorical quotient for the action of $G$. When $\Sigma$ is simplicial, $\pi$ is a geometric quotient, and $X_{\Sigma }$ is the orbit space.

(b) GIT. The quotient $\pi$ agrees with the GIT quotient $({\mathbb{A}}^{\Sigma (1)}{)}^{ss}(\mathcal{L})//G$ for the linearisation $\mathcal{L}$ given by an ample class $\beta \in \mathrm{Cl}(X_{\Sigma })$, with semistable locus $({\mathbb{A}}^{\Sigma (1)}{)}^{ss}(\mathcal{L})={\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma )$.

(c) Sheaves. For $X_{\Sigma }$ simplicial, the functor $\widetilde{(-)}$ sending a graded $S$-module $F$ to the sheaf $\tilde{F}$ on $X_{\Sigma }$ induces an equivalence between the category of finitely generated graded $S$-modules modulo those supported on $Z(\Sigma )$ and the category of coherent sheaves on $X_{\Sigma }$. In particular $\widetilde{S(\beta )}={\mathcal{O}}_{X_{\Sigma }}(\beta )$ for every $\beta \in \mathrm{Cl}(X_{\Sigma })$.

Proof.

(a) Construction of the map. For each maximal cone $\sigma \in \Sigma$ let $U_{\sigma }=\mathrm{Spec}\mathbb{C}[{\sigma }^{\vee }\cap M]$ be the affine chart of [04.11.04]. Consider the principal open set $W_{\sigma }=\{x^{\hat{\sigma }}\ne 0\}\subseteq {\mathbb{A}}^{\Sigma (1)}$, where every variable $x_{\rho }$ with $\rho \notin \sigma (1)$ is invertible. The ring of $G$-invariant functions on $W_{\sigma }$ is exactly the degree-zero part of the localisation $S_{x^{\hat{\sigma }}}$, namely $$ \big(S_{x^{\hat\sigma}}\big)0 = \mathbb{C}[x\rho^{\pm 1} (\rho \notin \sigma), x_\rho (\rho \in \sigma)]0 . $$ A Laurent monomial $\prod\rho x_\rho^{a_\rho}$hasdegreezeroin$\mathrm{Cl}(X_\Sigma)$iff$\sum_\rho a_\rho D_\rho = \mathrm{div}(\chi^m)$forsome$m \in M$,thatisiff$a_\rho = \langle m, u_\rho\rangle$.Restrictingtomonomialsregularon$W_\sigma$(so$a_\rho \geq 0$for$\rho \in \sigma(1)$)forces$\langle m, u_\rho\rangle \geq 0$for$\rho \in \sigma(1)$,equivalently$m \in \sigma^\vee \cap M$. Hence $$ \big(S_{x^{\hat\sigma}}\big)0 = \mathbb{C}[\sigma^\vee \cap M], $$ which is the coordinate ring of $U\sigma$.Taking$\mathrm{Spec}$producesamorphism$W_\sigma /!!/ G \to U_\sigma$thatisanisomorphism.Theseglueovertheraysintoaglobalmorphism$\pi : \mathbb{A}^{\Sigma(1)} \setminus Z(\Sigma) \to X_\Sigma$,since$\mathbb{A}^{\Sigma(1)} \setminus Z(\Sigma) = \bigcup_\sigma W_\sigma$bythedefinitionof$Z(\Sigma)$ as the locus lying in no cone.

(a) Quotient properties. On each $W_{\sigma }$ the map is the affine GIT quotient $\mathrm{Spec}(S_{x^{\hat{\sigma }}})\to \mathrm{Spec}((S_{x^{\hat{\sigma }}}{)}_0)$, which is a good categorical quotient by the basic theory of [04.10.02]. Categorical quotients glue, so $\pi$ is a good categorical quotient globally. When $\Sigma$ is simplicial, each $\sigma$ has linearly independent ray generators, so the isotropy of $G$ at a point of $W_{\sigma }$ is finite and the orbits are closed in $W_{\sigma }$; a good quotient with closed orbits is geometric. Hence $X_{\Sigma }$ is the orbit space $({\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma ))/G$.

(b) GIT identification. Choose $\beta \in \mathrm{Cl}(X_{\Sigma })$ ample, corresponding to an ample line bundle ${\mathcal{O}}_{X_{\Sigma }}(\beta )$. Linearise the structure-sheaf bundle on ${\mathbb{A}}^{\Sigma (1)}$ by the character of $G$ dual to $\beta$ under $\mathrm{Cl}(X_{\Sigma })=\hat{G}$. The Hilbert-Mumford numerical criterion of [04.10.02] selects as semistable exactly those points whose closure of the $G$-orbit avoids the locus where every section of the linearised bundle vanishes; for the toric character this locus is precisely $Z(\Sigma )$. Thus $({\mathbb{A}}^{\Sigma (1)}{)}^{ss}(\mathcal{L})={\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma )$, and the GIT quotient is the categorical quotient of part (a).

(c) Sheaves. The localisations $(S_{x^{\hat{\sigma }}}{)}_0=\mathbb{C}[{\sigma }^{\vee }\cap M]$ are the structure rings of the charts $U_{\sigma }$; a graded $S$-module $F$ localises to a module over each, gluing to a quasi-coherent sheaf $\tilde{F}$. The shift $S(\beta )$ localises on $U_{\sigma }$ to the rank-one free module generated in degree $-\beta$, which is the chart description of the divisorial sheaf ${\mathcal{O}}_{X_{\Sigma }}(\beta )$ from [04.11.08]; hence $\widetilde{S(\beta )}={\mathcal{O}}_{X_{\Sigma }}(\beta )$. Two modules give the same sheaf iff they agree after inverting each $x^{\hat{\sigma }}$, that is iff they differ by a submodule supported on $Z(\Sigma )$; this is the stated equivalence, the toric analogue of Serre's $\mathrm{Proj}$ correspondence. $\square$

Bridge. The Cox quotient theorem builds toward the GIT-symplectic dictionary and appears again in the Delzant construction of [04.10.04]; the central insight is that the global gluing of affine charts from [04.11.04] is repackaged as a single quotient of one affine space by one diagonalisable group, and this is exactly the structure ${\mathbb{P}}^n=({\mathbb{A}}^{n+1}\setminus 0)/{\mathbb{C}}^{\ast }$ written for an arbitrary fan. The foundational reason the construction works is the divisor-class exact sequence of [04.11.08]: it simultaneously fixes the grading of $S$, the group $G$ by duality, and the irrelevant ideal $B(\Sigma )$ that carves out the forbidden set. Putting these together, the fan combinatorics of [04.11.04] and the divisor theory of [04.11.08] fuse into one GIT problem in the sense of [04.10.02], and the bridge is dual to the symplectic-reduction shadow recorded by Kempf-Ness in [04.10.04]. This generalises the homogeneous-coordinate description of projective space to all of toric geometry, and is the structure that recurs whenever sheaves on $X_{\Sigma }$ are computed through graded $S$-modules.

Exercises Intermediate+

Advanced results Master

Theorem (GIT chambers and the secondary fan; Cox 1995, Cox-Little-Schenck §14). Fix the affine space ${\mathbb{A}}^{\Sigma (1)}$ with its $G$-action. As the linearising character $\beta \in \mathrm{Cl}(X_{\Sigma }{)}_{\mathbb{R}}$ varies, the semistable locus $({\mathbb{A}}^{\Sigma (1)}{)}^{ss}(\beta )$, hence the GIT quotient, changes only when $\beta$ crosses a wall of the secondary fan (the fan of GIT chambers in $\mathrm{Cl}(X_{\Sigma }{)}_{\mathbb{R}}$). Each top-dimensional chamber yields a simplicial toric variety, all sharing the same total coordinate ring $S$ and group $G$; the distinct quotients are related by toric flips and are the variation of GIT (VGIT) of the $G$-action.

The secondary-fan theorem turns the choice of fan refinement into the choice of a GIT chamber. The varieties $X_{\Sigma }$ obtained for different chambers all have the same Cox data; they differ only in the irrelevant ideal $B(\Sigma )$, equivalently in which forbidden set $Z(\Sigma )$ is removed before quotienting. This is the algebraic incarnation of the toric minimal model program: crossing a wall performs a flip, and the wall-and-chamber structure of $\mathrm{Cl}(X_{\Sigma }{)}_{\mathbb{R}}$ records the birational geometry of the family. The construction is functorial in the lattice data and realises the Mori-theoretic picture combinatorially.

Theorem (sheaf cohomology via graded $S$-modules; Cox 1995 §3, Mustață 2002). For $X_{\Sigma }$ simplicial and complete, the cohomology of a sheaf ${\mathcal{O}}_{X_{\Sigma }}(\beta )$ is computed by the local cohomology of $S$ with support in the irrelevant ideal: there is an isomorphism $$ H^i(X_\Sigma, \mathcal{O}{X\Sigma}(\beta)) ;\cong; H^{i+1}{B(\Sigma)}(S)\beta $$ for $i\ge 1$, the graded local cohomology in degree $\beta$, the toric analogue of the local-cohomology computation of sheaf cohomology on ${\mathbb{P}}^n$.

This theorem makes cohomology on a toric variety into commutative algebra over the single ring $S$. The local cohomology $H_{B(\Sigma )}^{\bullet }(S)$ is computed from the Koszul or Čech complex on the generators of the irrelevant ideal, exactly as the cohomology of line bundles on ${\mathbb{P}}^n$ is computed from $H_{\mathfrak{m}}^{\bullet }(\mathbb{C}[x_0,\dots ,x_n])$. The grading slices out the contribution of each class $\beta$, and the result recovers the lattice-point formulas of [04.11.08] for global sections while extending them to all cohomological degrees.

Theorem (Kempf-Ness shadow; Audin Ch. VI, Cox-Little-Schenck §14.3). Let $X_{\Sigma }$ be a smooth projective toric variety presented as $({\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma ))/G$. Let $K\subseteq G$ be the maximal compact subgroup, a real torus. Then the GIT quotient is canonically homeomorphic to the symplectic (Marsden-Weinstein) quotient ${\mu }_K^{-1}(c)/K$ of ${\mathbb{C}}^{\Sigma (1)}$ by $K$ at a regular value $c$, where ${\mu }_K$ is the moment map of the linear $K$-action. This is the algebraic form of the Delzant construction of [04.10.04].

The Kempf-Ness shadow identifies the Cox quotient with a symplectic reduction. The same affine space ${\mathbb{C}}^{\Sigma (1)}$ carries a linear Hamiltonian action of the compact torus $K$, and reducing at a level $c$ produces a symplectic toric manifold whose moment polytope is the polytope of the polarisation. The GIT-symplectic dictionary of [04.10.04] says the algebraic quotient and the symplectic quotient coincide, so the Cox ring is the algebraic engine sitting beneath Delzant's symplectic construction. The level $c$ corresponds to the ample class $\beta$, and varying $c$ across walls reproduces the secondary-fan chambers on the symplectic side.

Theorem (functoriality and toric morphisms in homogeneous coordinates; Cox 1995 §4). A toric morphism $X_{\Sigma }\to X_{{\Sigma }^{\prime }}$ induced by a fan map $\varphi :N\to N^{\prime }$ is given in homogeneous coordinates by monomial maps $x_{{\rho }^{\prime }}^{\prime }\mapsto {\prod }_{\rho }{x}_{\rho }^{b_{{\rho }^{\prime }\rho }}$, the exponents determined by how the rays of $\Sigma$ map under $\varphi$. The total coordinate rings assemble into a functor, and the morphism is encoded by a graded ring homomorphism compatible with the class-group gradings.

Homogeneous coordinates make toric morphisms into explicit monomial maps, exactly as a morphism ${\mathbb{P}}^n\to {\mathbb{P}}^m$ given by degree-$d$ forms is written in homogeneous coordinates. This is the computational payoff of the Cox construction: maps, embeddings, and rational maps between toric varieties become polynomial bookkeeping in the variables $x_{\rho }$, and the class-group grading tracks which bundles are pulled back. The functoriality refines the fan-morphism functoriality of [04.11.04] to the level of coordinate rings.

Synthesis. The Cox homogeneous coordinate ring is the central insight that unifies the affine-chart picture of [04.11.04], the divisor theory of [04.11.08], and the GIT machinery of [04.10.02] into a single statement: every toric variety is one affine space, minus a forbidden set, modulo one diagonalisable group. The foundational reason this works is the divisor-class exact sequence, which is dual to the torus sequence $1\to G\to ({\mathbb{C}}^{\ast }{)}^{\Sigma (1)}\to T\to 1$; the grading of $S$ by $\mathrm{Cl}(X_{\Sigma })$ and the group $G=\mathrm{Hom}(\mathrm{Cl}(X_{\Sigma }),{\mathbb{C}}^{\ast })$ are the two faces of this duality, and the irrelevant ideal supplies the semistable locus on which the quotient is well-behaved. Putting these together, the secondary fan organises every chamber-quotient of the fixed $G$-action, sheaf cohomology becomes local cohomology of $S$ supported on $B(\Sigma )$, and the whole apparatus is the algebraic shadow of the symplectic Delzant reduction of [04.10.04]. This is exactly the structure ${\mathbb{P}}^n=({\mathbb{A}}^{n+1}\setminus 0)/{\mathbb{C}}^{\ast }$ generalised to an arbitrary fan, and it generalises further to Cox rings of non-toric varieties (Mori dream spaces), where the same total-coordinate-ring-plus-irrelevant-ideal pattern recurs and the bridge is again a GIT quotient by the Picard torus.

Full proof set Master

Proposition (the group $G$ is the kernel of $({\mathbb{C}}^{\ast }{)}^{\Sigma (1)}\to T$), proof. Apply the exact functor ${\mathrm{Hom}}_{\mathbb{Z}}(-,{\mathbb{C}}^{\ast })$ (exact because ${\mathbb{C}}^{\ast }$ is divisible, hence injective as an abelian group) to the divisor-class sequence $0\to M\to {\mathbb{Z}}^{\Sigma (1)}\to \mathrm{Cl}(X_{\Sigma })\to 0$. This yields the exact sequence $$ 1 \to \mathrm{Hom}(\mathrm{Cl}(X_\Sigma), \mathbb{C}^) \to \mathrm{Hom}(\mathbb{Z}^{\Sigma(1)}, \mathbb{C}^) \to \mathrm{Hom}(M, \mathbb{C}^) \to 1. $$ Now $\mathrm{Hom}(\mathbb{Z}^{\Sigma(1)}, \mathbb{C}^) = (\mathbb{C}^)^{\Sigma(1)}$and$\mathrm{Hom}(M, \mathbb{C}^) = T$,so$G = \mathrm{Hom}(\mathrm{Cl}(X_\Sigma), \mathbb{C}^)$isexactlythekernelofthesurjection$(\mathbb{C}^)^{\Sigma(1)} \to T$.Thedefiningequationsof$G$inside$(\mathbb{C}^*)^{\Sigma(1)}$are$\prod_\rho g_\rho^{\langle m, u_\rho\rangle} = 1$for$m$rangingoverabasisof$M$,theimageofthedualmap.$\square$

Proposition (the charts $W_{\sigma }$ cover ${\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma )$), proof. By definition $W_{\sigma }=\{x^{\hat{\sigma }}\ne 0\}=\{x_{\rho }\ne 0\text{ for all }\rho \notin \sigma (1)\}$. A point $p$ lies in some $W_{\sigma }$ iff the set $R(p)=\{\rho :p_{\rho }=0\}$ is contained in $\sigma (1)$ for some cone $\sigma$, that is iff the rays where $p$ vanishes are among the rays of a single cone. The exceptional set is $Z(\Sigma )=V(B(\Sigma ))$ with $B(\Sigma )=⟨x^{\hat{\sigma }}⟩$; a point $p$ lies in $Z(\Sigma )$ iff $x^{\hat{\sigma }}(p)=0$ for every $\sigma$, iff for every cone $\sigma$ some ray $\rho \notin \sigma (1)$ has $p_{\rho }=0$, iff $R(p)$ is contained in no cone. Negating, $p\notin Z(\Sigma )$ iff $R(p)\subseteq \sigma (1)$ for some $\sigma$, iff $p\in W_{\sigma }$ for some $\sigma$. Hence ${\bigcup }_{\sigma }{W}_{\sigma }={\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma )$. $\square$

Proposition (gluing of the chart quotients), proof. For maximal cones $\sigma ,\tau$, the overlap $W_{\sigma }\cap W_{\tau }=W_{\sigma \cap \tau }$ inverts the variables of rays outside $\sigma \cap \tau$, and by the localisation identity (Exercise 5) the degree-zero ring is $(S_{x^{\widehat{\sigma \cap \tau }}}{)}_0=\mathbb{C}[(\sigma \cap \tau {)}^{\vee }\cap M]=\mathcal{O}(U_{\sigma \cap \tau })$. The face $\sigma \cap \tau$ is a common face of $\sigma$ and $\tau$ by the fan axioms of [04.11.04], and the localisation maps $\mathbb{C}[{\sigma }^{\vee }\cap M]\to \mathbb{C}[(\sigma \cap \tau {)}^{\vee }\cap M]$ are exactly the open-immersion comorphisms used in [04.11.04] to glue $U_{\sigma \cap \tau }\hookrightarrow U_{\sigma }$. Hence the chart quotients $W_{\sigma }//G\cong U_{\sigma }$ glue along the same data that glue the $U_{\sigma }$ into $X_{\Sigma }$, producing a global isomorphism $({\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma ))//G\cong X_{\Sigma }$. $\square$

Proposition (the sheaf $\widetilde{S(\beta )}$ is ${\mathcal{O}}_{X_{\Sigma }}(\beta )$), proof. On the chart $U_{\sigma }$ the module $S(\beta )$ localises to $(S_{x^{\hat{\sigma }}})$ shifted by $\beta$, and its degree-zero part $(S_{x^{\hat{\sigma }}}(\beta ){)}_0=(S_{x^{\hat{\sigma }}}{)}_{\beta }$. A degree-$\beta$ Laurent monomial regular on $W_{\sigma }$ corresponds, by the argument of Exercise 5, to an element $m\in M$ with $⟨m,u_{\rho }⟩\ge -a_{\rho }$ for $\rho \in \sigma (1)$, where $\beta =[\sum a_{\rho }{D}_{\rho }]$. These are precisely the sections of ${\mathcal{O}}_{X_{\Sigma }}(D)$ over $U_{\sigma }$ for any divisor $D$ with $[D]=\beta$, by the local description of divisorial sheaves in [04.11.08]. The identification is compatible with restriction to overlaps, so $\widetilde{S(\beta )}={\mathcal{O}}_{X_{\Sigma }}(\beta )$ globally. $\square$

Proposition (the secondary fan governs VGIT), proof sketch. As $\beta$ varies in $\mathrm{Cl}(X_{\Sigma }{)}_{\mathbb{R}}$, the semistable locus is locally constant on chambers and jumps across walls.

Proof. By the Hilbert-Mumford criterion of [04.10.02], semistability of a point $p$ for the character $\beta$ depends only on the sign pattern of the pairings $⟨\beta ,\lambda ⟩$ as $\lambda$ ranges over one-parameter subgroups of $G$ with ${\lim }_{t\to 0}\lambda (t)\cdot p$ existing. For the toric linear action these one-parameter subgroups are indexed by lattice vectors, and the relevant pairings are linear functionals of $\beta$. The set of $\beta$ giving a fixed sign pattern is the relative interior of a rational polyhedral cone — a chamber of the secondary fan, the normal fan of the secondary polytope of the vector configuration $\{u_{\rho }\}$. Within one chamber the semistable locus, and so the GIT quotient, does not change. Crossing a wall flips at least one sign, changing which $x^{\hat{\sigma }}$ generators lie in the irrelevant ideal, hence changing $Z(\Sigma )$ and performing a toric flip on the quotient. The chambers thus enumerate the simplicial fans supported on ${\bigcup }_{\rho }{u}_{\rho }$, and the wall-crossings realise the toric minimal model program combinatorially. $\square$

Connections Master

• Fan and the toric variety $X_{\Sigma }$ 04.11.04. The Cox quotient is the global gluing of [04.11.04] rewritten as a single quotient. Each affine chart $U_{\sigma }=\mathrm{Spec}\mathbb{C}[{\sigma }^{\vee }\cap M]$ reappears as the degree-zero ring $(S_{x^{\hat{\sigma }}}{)}_0$ of a localisation of the Cox ring, and the chart overlaps glue along exactly the face-inclusion comorphisms of the fan unit. This resolves the forward reference to [04.11.15] made in [04.11.04]: the homogeneous-coordinate presentation promised there is constructed here.

• Toric divisor and support function 04.11.08. The divisor-class exact sequence $0\to M\to {\mathbb{Z}}^{\Sigma (1)}\to \mathrm{Cl}(X_{\Sigma })\to 0$ of [04.11.08] is the structural input that fixes the grading of $S$, the group $G$ by duality, and the divisorial sheaves ${\mathcal{O}}_{X_{\Sigma }}(\beta )=\widetilde{S(\beta )}$. The variable $x_{\rho }$ is the homogeneous equation of the toric divisor $D_{\rho }$, so the monomials of $S$ are the effective $T$-invariant divisors, and the lattice-point global-sections formula of [04.11.08] becomes the degree-$\beta$ piece $S_{\beta }$. This resolves the forward reference to [04.11.15] in [04.11.08].

• Geometric invariant theory (GIT) quotients 04.10.02. The Cox quotient is a GIT quotient of ${\mathbb{A}}^{\Sigma (1)}$ by the diagonalisable group $G$, linearised by an ample class $\beta$. The semistable locus is ${\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma )$, computed by the Hilbert-Mumford criterion of [04.10.02], and the chamber structure of the linearisation is the secondary fan governing variation of GIT. The toric case is the most explicit testing ground for GIT, since every ingredient — group, semistable locus, quotient — is read off from fan combinatorics.

• Kempf-Ness theorem 04.10.04. The GIT quotient $({\mathbb{A}}^{\Sigma (1)}\setminus Z(\Sigma ))/G$ is homeomorphic to the symplectic quotient ${\mu }_K^{-1}(c)/K$ of ${\mathbb{C}}^{\Sigma (1)}$ by the maximal compact $K\subseteq G$, by the Kempf-Ness identification of [04.10.04]. This realises the Cox ring as the algebraic engine beneath the symplectic Delzant construction of toric manifolds, with the ample class $\beta$ corresponding to the moment-map level $c$ and the secondary-fan walls corresponding to the walls of the symplectic chamber structure.

• Projective space 04.07.01. The prototype: ${\mathbb{P}}^n=({\mathbb{A}}^{n+1}\setminus 0)/{\mathbb{C}}^{\ast }$ is the Cox quotient of the fan with $n+1$ rays, total coordinate ring $\mathbb{C}[x_0,\dots ,x_n]$, irrelevant ideal $⟨x_0,\dots ,x_n⟩$, and group ${\mathbb{C}}^{\ast }$. The Cox construction is exactly the statement that every toric variety admits homogeneous coordinates in this projective-space sense, and Serre's $\mathrm{Proj}$ correspondence between graded modules and sheaves on ${\mathbb{P}}^n$ generalises to the toric module-sheaf equivalence proved here.

• Toric Picard group 04.11.09. The class group $\mathrm{Cl}(X_{\Sigma })$ grading the Cox ring is computed in [04.11.09], where the Picard subgroup of Cartier classes sits inside $\mathrm{Cl}(X_{\Sigma })$. The ample cone in $\mathrm{Cl}(X_{\Sigma }{)}_{\mathbb{R}}$ that selects the linearisation $\beta$ is the same nef/ample cone studied there, and the secondary-fan chambers refine the Picard-group geometry into GIT chambers.

Historical & philosophical context Master

The homogeneous coordinate ring of a toric variety was introduced by David Cox in The homogeneous coordinate ring of a toric variety (Journal of Algebraic Geometry 4, 1995, pp. 17-50) ^{[Cox 1995]}. Cox's stated aim was to give toric varieties the same computational footing as projective space, where a single graded polynomial ring controls coordinates, line bundles, sheaves, and cohomology through Serre's $\mathrm{Proj}$ correspondence. The construction supplied exactly this: one polynomial ring $S$ graded by the class group, one irrelevant ideal $B(\Sigma )$, and a presentation of $X_{\Sigma }$ as a quotient of affine space by a diagonalisable group. The quotient form generalises the classical ${\mathbb{P}}^n=({\mathbb{A}}^{n+1}\setminus 0)/{\mathbb{C}}^{\ast }$ and made toric geometry tractable for symbolic computation, underpinning the toric packages of Macaulay2, Polymake, and SageMath.

The quotient was independently anticipated from the symplectic side. The construction of a toric manifold as a symplectic reduction ${\mathbb{C}}^d//K$ of affine space by a real torus is the Delzant construction (Thomas Delzant, Hamiltoniens périodiques et image convexe de l'application moment, Bulletin de la Société Mathématique de France 116, 1988, pp. 315-339) ^{[Delzant 1988]}, and Michèle Audin's monograph The Topology of Torus Actions on Symplectic Manifolds (Birkhäuser 1991) ^{[Audin 1991]} developed this reduction in detail. The bridge between Cox's algebraic quotient and Delzant's symplectic quotient is the Kempf-Ness theorem (George Kempf and Linda Ness, The length of vectors in representation spaces, Springer LNM 732, 1979, pp. 233-243), so the Cox ring is the algebraic shadow of a symplectic construction discovered along a parallel route.

The underlying global toric variety had been constructed two decades earlier by Michel Demazure in his 1970 study of the Cremona group ^{[Demazure 1970]}, and the GIT framework that makes the quotient rigorous is Mumford's geometric invariant theory (Mumford-Fogarty-Kirwan, Geometric Invariant Theory, 3rd ed., Springer 1994) ^{[MumfordFogartyKirwan 1994]}. Cox's contribution was to recognise that these strands — Demazure's gluing, Mumford's quotients, and the class-group bookkeeping of toric divisors — combine into a single homogeneous-coordinate description. The philosophical lesson is one of unification: the same variety has a combinatorial description (the fan), an algebraic description (the graded Cox ring with its quotient), and a symplectic description (the Delzant reduction), and the equivalence of these three is the structural backbone of modern toric geometry, extending through Mori dream spaces to the Cox rings of non-toric varieties.

Bibliography Master

@article{Cox1995,
  author  = {Cox, David A.},
  title   = {The homogeneous coordinate ring of a toric variety},
  journal = {Journal of Algebraic Geometry},
  volume  = {4},
  year    = {1995},
  pages   = {17--50}
}

@book{CoxLittleSchenck,
  author    = {Cox, David A. and Little, John B. and Schenck, Henry K.},
  title     = {Toric Varieties},
  publisher = {American Mathematical Society},
  series    = {Graduate Studies in Mathematics},
  volume    = {124},
  year      = {2011}
}

@book{Audin1991,
  author    = {Audin, Mich{\`e}le},
  title     = {The Topology of Torus Actions on Symplectic Manifolds},
  publisher = {Birkh{\"a}user},
  series    = {Progress in Mathematics},
  volume    = {93},
  year      = {1991}
}

@book{MumfordFogartyKirwan1994,
  author    = {Mumford, David and Fogarty, John and Kirwan, Frances},
  title     = {Geometric Invariant Theory},
  edition   = {3rd},
  publisher = {Springer-Verlag},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {34},
  year      = {1994}
}

@incollection{KempfNess1979,
  author    = {Kempf, George and Ness, Linda},
  title     = {The length of vectors in representation spaces},
  booktitle = {Algebraic Geometry (Copenhagen 1978)},
  publisher = {Springer-Verlag},
  series    = {Lecture Notes in Mathematics},
  volume    = {732},
  year      = {1979},
  pages     = {233--243}
}

@article{Delzant1988,
  author  = {Delzant, Thomas},
  title   = {Hamiltoniens p{\'e}riodiques et image convexe de l'application moment},
  journal = {Bulletin de la Soci{\'e}t{\'e} Math{\'e}matique de France},
  volume  = {116},
  year    = {1988},
  pages   = {315--339}
}

@article{Demazure1970,
  author  = {Demazure, Michel},
  title   = {Sous-groupes alg{\'e}briques de rang maximum du groupe de Cremona},
  journal = {Annales scientifiques de l'{\'E}cole normale sup{\'e}rieure (4)},
  volume  = {3},
  year    = {1970},
  pages   = {507--588}
}

@article{Mustata2002,
  author  = {Musta{\c{t}}{\u{a}}, Mircea},
  title   = {Vanishing theorems on toric varieties},
  journal = {Tohoku Mathematical Journal},
  volume  = {54},
  year    = {2002},
  pages   = {451--470}
}

@book{FultonToric,
  author    = {Fulton, William},
  title     = {Introduction to Toric Varieties},
  publisher = {Princeton University Press},
  series    = {Annals of Mathematics Studies},
  volume    = {131},
  year      = {1993}
}