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Cox RingEdit

The Cox ring, also known as the total coordinate ring, is a unifying algebraic object that encodes the global sections of all line bundles on a normal variety X over an algebraically closed field k. Named after David A. Cox, who introduced the construction in the context of toric varieties, the Cox ring packages the information about linear systems into a single graded algebra. It is built from the divisor class group of X and serves as a bridge between classical projective geometry and modern birational geometry.

Concretely, if X is a normal variety with divisor class group Cl(X) that is finitely generated, the Cox ring is defined as R(X) = ⊕_{[D] in Cl(X)} H^0(X, O_X(D)). The grading is by Cl(X): a section s in H^0(X, O_X(D)) is assigned degree [D], and the ring structure comes from the natural identification O_X(D1) ⊗ O_X(D2) ≅ O_X(D1 + D2). This makes R(X) into a multi-graded k-algebra whose components are the global sections of the various line bundles (or divisors) on X. See Line bundle and Global section for background on the objects appearing in the construction. In practice, one often works under the additional assumption that Cl(X) is finitely generated, which keeps the grading manageable and allows for concrete algebraic manipulation.

The concept is intimately tied to toric geometry. For a toric variety X corresponding to a fan, the Cox ring is a polynomial ring in as many variables as there are rays of the fan, with a grading by the divisor class group that records how each variable contributes to a given Weil divisor. This recovers the familiar homogeneous coordinate ring of projective space when X is projective space P^n. More generally, the toric case motivates the idea that Cox rings assemble all coordinate data of line bundles in a way that naturally supports torus actions and quotients Toric variety.

Definition

  • Let X be a normal variety over k with divisor class group Cl(X) finitely generated. The total coordinate ring (Cox ring) is R(X) = ⊕_{[D] ∈ Cl(X)} H^0(X, O_X(D)). The multiplication is induced by the tensor product of sections, using the isomorphism O_X(D1) ⊗ O_X(D2) ≅ O_X(D1 + D2).

  • The ring is naturally graded by Cl(X): the piece H^0(X, O_X(D)) has degree [D]. If Cl(X) has torsion, one often passes to a suitable representative or uses a refined construction to keep the grading well-defined.

  • One motivation for the construction is to view X as a quotient of an affine, or at least simpler, space by a torus action: Spec(R(X)) carries an action of the quasi-torus Spec(k[Cl(X)]) whose quotient (in a suitable sense, e.g., a Geometric Invariant Theory quotient) recovers X. See Spec and Geometric invariant theory for related notions.

Construction and basic properties

  • The universal torsor viewpoint: Spec(R(X)) with its Cl(X)-grading encodes the total space of all line bundles on X in a single algebraic object. The action of the torus T = Spec(k[Cl(X)]) reflects the different line bundle degrees, and X can be realized as a suitable quotient of this total space.

  • Finite generation and Mori dream spaces: If R(X) is finitely generated as a k-algebra, X has particularly well-behaved birational geometry. In this case, X is often called a Mori dream space, meaning the effective cone of divisors is polyhedral and a finite number of birational models (SQMs) capture the birational geometry via GIT quotients of Spec(R(X)). See Mori dream space.

  • Relationship to linear systems: The summands H^0(X, O_X(D)) are the global sections of line bundles, i.e., they form linear systems |D|. The Cox ring organizes all these systems in a single graded algebra, so questions about base loci, movable divisors, and effective cone can be studied through the algebraic structure of R(X). See Linear system and Divisor for related concepts.

  • Relation to the classical homogeneous coordinate ring: In the case X = Proj of a graded ring, the Cox ring specializes to a familiar homogeneous coordinate ring. For projective spaces and many toric varieties, the Cox ring is particularly tractable and provides a direct computational handle.

Examples

  • Projective space: For X = P^n, the divisor class group is isomorphic to Z, and the Cox ring is R(P^n) ≅ k[x_0, ..., x_n], graded by total degree. The variety X can be recovered as Proj of this Cox ring.

  • Toric varieties: If X is a complete toric variety with rays corresponding to a set Σ(1), then R(X) is a polynomial ring in variables x_ρ for ρ ∈ Σ(1), with degrees determined by the class group. The toric variety arises as the good quotient of Spec(R(X)) by the associated torus action. See Toric variety for background and examples.

  • Other examples and non-examples: Beyond toric and projective cases, Cox rings occur for many classes of varieties, such as certain del Pezzo surfaces and blow-ups. The finite generation of the Cox ring (and hence the Mori dream space property) can fail in interesting cases, illustrating the limits of the construction as a universal packing of linear systems. See Del Pezzo surface and Blow-up for related constructions.

Finite generation and limitations

  • Finite generation is powerful but not universal. While many canonical classes of varieties (notably many toric and Fano-type spaces) have finitely generated Cox rings, there are naturally occurring examples where R(X) is not finitely generated. This non-finite generation signals obstacles to treating X via a single, well-behaved global quotient, and it motivates refined approaches within birational geometry. See Birational geometry and Mori dream space for context.

  • The study of finite generation touches on deep questions about the structure of effective and movable divisors, the chamber structure of the movable cone, and the global behavior of linear systems. The Cox ring thus serves both as a practical computational tool and as a theoretical lens on the birational landscape of X.

Applications in birational geometry

  • Via the Spec of the Cox ring and the torus action, X can be analyzed through GIT-type quotients, enabling explicit descriptions of birational models, flips, and divisorial contractions. This program connects with the minimal model program and offers a way to organize birational models in a coherent, algebraic framework.

  • The Cox ring encodes information about all line bundles simultaneously, so questions about base loci, base components, and effective divisors can be translated into questions about generators and relations in R(X). This translation has proven fruitful in both conceptual understanding and computational practice.

See also