Toric VarietyEdit

Toric varieties sit at the crossroads of algebraic geometry and convex geometry, offering a concrete and highly computable class of varieties built from combinatorial data. They provide a bridge between the world of polyhedral fans in lattices and the more intricate world of algebraic varieties, making many geometric questions accessible through hands-on polyhedral reasoning. As such, toric varieties are a staple tool for researchers who value explicit constructions, transparent singularity analysis, and a clear link between geometry and combinatorics.

Toric varieties can be described in a way that is intrinsic to their torus symmetry: they contain an algebraic torus T ≅ (C^*)^n as a dense open subset, and the action of T on itself extends to the whole variety. This torus-invariant structure provides a dictionary between geometric features and combinatorial objects. For readers familiar with algebraic geometry and convex geometry, the toric dictionary is a natural way to translate geometric problems into questions about lattices, cones, and fans.

Foundations

Definitions and basic construction

A toric variety X is a normal algebraic variety containing T ≅ (C^*)^n as a dense open subset, with an action of T on X that extends its action on itself. The construction is encoded in a fan Σ, a collection of strongly convex rational polyhedral cones in the real vector space N_R = N ⊗ R, where N ≅ Z^n is a lattice and M = Hom(N,Z) is the dual lattice. Each cone σ in Σ gives an affine toric variety U_σ = Spec C[S_σ], where S_σ = σ^∨ ∩ M is the semigroup of integral points in the dual cone. The toric variety X_Σ is then obtained by gluing the U_σ along overlaps determined by face relations among cones in Σ.

Affine pieces and gluing

Affine pieces U_σ correspond to cones σ in the fan, and the global geometry of X_Σ is controlled by how these cones fit together. This construction makes many properties computable from Σ: singularities correspond to the combinatorics of the cones, and smoothness occurs precisely when each σ is generated by part of a Z-basis of N (i.e., Σ is regular).

Invariant divisors and cohomology

Torus-invariant divisors on X_Σ correspond to the one-dimensional cones (rays) ρ in Σ(1). The combinatorics of these rays and their relations determine the divisor class group and, in favorable cases, the Picard group. The Cox construction provides a global coordinate-wise perspective, tying line bundles to degree data via the homogeneous coordinate ring associated to the fan.

Polytopes and duality

There is a tight link between toric varieties and lattice polytopes. If P ⊂ M_R is a full-dimensional lattice polytope, its normal fan Σ_P encodes a projective toric variety X_P together with an ample line bundle L_P. Global sections H^0(X_P, L_P) correspond to lattice points in P, which gives a combinatorial handle on the algebraic geometry of the embedding defined by L_P. This polyhedral viewpoint is central to many computations and conceptual understandings of toric geometry.

Smoothness and singularities

The smoothness of a toric variety is determined combinatorially. X_Σ is smooth if and only if every cone σ ∈ Σ is simplicial and its primitive generators extend to a Z-basis of N (the regularity condition). When this fails, singularities arise, but they are typically torus-invariant and thus amenable to explicit description in terms of the cones.

Constructions and examples

Classical examples

Projective spaces: the projective space projective space is toric, arising from the fan whose cones correspond to the coordinate strata in projective n-space. This provides a baseline example where the torus action is the standard (C^*)^n action on homogeneous coordinates.
Affine spaces and products: the affine space A^n and products like A^1 × P^1 provide simple toric surfaces and higher-dimensional cases.
Hirzebruch surfaces: these are toric surfaces that generalize projective bundles over projective lines and illustrate how toric methods classify and distinguish smooth projective surfaces.
Weighted projective spaces: these are toric orbifolds obtained by a weighted assignment to coordinates; they show how singularities interact with the toric data.

Beyond the basics

More general polarized toric varieties: combinations of a fan with a compatible polytope yield pairs (X_Σ, L) offering explicit projective embeddings.
Toric complete intersections and hypersurfaces: many Calabi–Yau and Fano varieties arise as toric hypersurfaces or complete intersections inside toric ambient spaces.

Combinatorics, geometry, and calculations

The fan–geometry dictionary

The central computational tool is the fan Σ: its cones, rays, and their relations encode the global geometry of X_Σ. The dual cones, semigroups S_σ, and lattice data translate geometric questions into semigroup algebras and combinatorial constraints.

Polytopes and sections

For a polarized toric variety (X_Σ, L), sections of L are described by lattice points in an associated polytope, which makes problems such as counting sections or understanding embedding properties concrete.

Cohomology and intersection theory

In the toric setting, many cohomological and intersection-theoretic questions admit explicit answers in terms of the fan. The Chow ring of a smooth toric variety has a presentation that reflects the combinatorics of Σ, enabling direct calculations of intersection products and characteristic data.

Applications and influence

Mirror symmetry and beyond

Toric geometry plays a pivotal role in mirror symmetry. Batyrev’s construction uses dual reflexive polytopes to produce pairs of Calabi–Yau hypersurfaces in dual toric varieties, providing a concrete playground where predictions about Hodge numbers and other invariants can be tested.

Computational algebraic geometry

Toric methods underpin algorithms for solving systems of polynomial equations, computing Gröbner bases in specific coordinates, and performing effective geometry in low dimensions. The explicit nature of toric data makes many computations tractable.

Connections to broader geometry

Toric varieties shed light on more general classes, such as spherical varieties, and influence the study of degenerations, moduli, and compactifications. They provide a framework in which geometry can be studied via combinatorics, making them a foundational topic for both teaching and research in algebraic geometry.