11.1 Toric varieties and polytopes

Toric varieties are algebraic varieties containing a torus as a dense open subset. They're built from combinatorial data like fans and polytopes, making them accessible to study using discrete methods. This connection to combinatorics provides a rich class of examples in algebraic geometry.

Polytopes and fans encode the geometry of toric varieties. Lattice polytopes give rise to projective toric varieties, while fans describe affine toric varieties. The interplay between polytope combinatorics and toric variety geometry is central to toric geometry.

Definition of toric varieties

• Toric varieties are algebraic varieties that contain a torus as a dense open subset and the action of the torus on itself extends to an action on the entire variety
• Toric varieties provide a rich class of examples in algebraic geometry and have connections to combinatorics, convex geometry, and representation theory
• Toric varieties can be constructed from combinatorial data such as fans and polytopes, making them accessible to study using discrete methods

Toric varieties from fans

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• A fan is a collection of cones in a real vector space that satisfies certain compatibility conditions
• Each cone in a fan corresponds to an affine toric variety, and the fan encodes how these affine pieces glue together to form the toric variety
• The rays (1-dimensional cones) in a fan correspond to the torus-invariant divisors on the toric variety

Toric varieties from polytopes

• A polytope is a convex hull of a finite set of points in a real vector space
• The normal fan of a polytope is a fan that encodes the combinatorial structure of the polytope
• The toric variety associated to a polytope is the same as the toric variety associated to its normal fan

Affine toric varieties

• An affine toric variety is a toric variety that can be described as the spectrum of a semigroup algebra
• Each cone in a fan corresponds to an affine toric variety, which is the spectrum of the semigroup algebra of the dual cone
• Affine toric varieties are the building blocks of general toric varieties

Projective toric varieties

• A projective toric variety is a toric variety that admits a torus-equivariant embedding into a projective space
• Projective toric varieties can be constructed from polytopes: the toric variety associated to a lattice polytope is projective
• Examples of projective toric varieties include projective spaces, products of projective spaces, and weighted projective spaces

Polytopes and toric geometry

• Polytopes and fans are the combinatorial objects that encode the geometry of toric varieties
• The interplay between the combinatorics of polytopes and the geometry of toric varieties is a central theme in toric geometry

Lattice polytopes

• A lattice polytope is a polytope whose vertices have integer coordinates
• Lattice polytopes are the polytopes that give rise to projective toric varieties
• The lattice points in a lattice polytope correspond to a basis of the space of global sections of an ample line bundle on the associated toric variety

Normal fan of a polytope

• The normal fan of a polytope is a fan that encodes the combinatorial structure of the polytope
• The cones in the normal fan correspond to the faces of the polytope
• The normal fan of a polytope determines the toric variety associated to the polytope

Polytopes vs fans

• Polytopes and fans are dual objects: the normal fan of a polytope encodes the same combinatorial data as the polytope itself
• Some toric varieties (e.g., affine toric varieties) are more naturally described using fans, while others (e.g., projective toric varieties) are more naturally described using polytopes
• The choice of whether to work with polytopes or fans often depends on the specific problem or context

Moment polytopes

• The moment polytope of a projective toric variety is the image of the variety under the moment map associated to the torus action
• The moment polytope of a projective toric variety is a lattice polytope that encodes the same data as the fan of the variety
• Moment polytopes are a useful tool for studying the geometry and topology of toric varieties, such as their cohomology and intersection theory

Orbits and torus action

• The torus action on a toric variety determines its structure and properties
• Understanding the orbits of the torus action and their closures is crucial for studying toric varieties

Torus orbits in toric varieties

• The torus acts on a toric variety with finitely many orbits
• There is a one-to-one correspondence between the orbits of the torus action and the cones in the fan of the toric variety
• The dimension of an orbit is equal to the codimension of the corresponding cone in the fan

Orbit closures and faces

• The closure of a torus orbit in a toric variety is a toric subvariety
• The orbit closures correspond to the faces of the polytope associated to the toric variety
• The inclusion relations between orbit closures are determined by the face relations in the polytope or fan

Torus invariant divisors

• A torus invariant divisor on a toric variety is a divisor that is invariant under the torus action
• Torus invariant divisors correspond to the rays (1-dimensional cones) in the fan of the toric variety
• The torus invariant divisors generate the Picard group (group of line bundles) of the toric variety

Torus equivariant morphisms

• A torus equivariant morphism between toric varieties is a morphism that commutes with the torus actions on the varieties
• Torus equivariant morphisms correspond to morphisms of fans or polytopes that respect the torus action
• Torus equivariant morphisms are a key tool for studying maps between toric varieties and their properties

Toric resolution of singularities

• Toric varieties can have singularities, but these singularities can be resolved using toric methods
• Toric resolution of singularities is a powerful technique that has applications beyond toric varieties themselves

Cones and affine toric varieties

• Affine toric varieties are determined by cones in the fan of the toric variety
• Singularities of affine toric varieties correspond to non-smooth cones in the fan
• Resolving the singularities of an affine toric variety amounts to subdividing the corresponding cone into smooth cones

Refinement of fans

• A refinement of a fan is a fan that subdivides the cones of the original fan into smaller cones
• Refinements of fans correspond to toric birational morphisms that resolve singularities
• A toric variety is smooth if and only if its fan consists of smooth cones

Toric resolution of toric varieties

• Any toric variety can be resolved by a smooth toric variety via a toric birational morphism
• The resolution can be obtained by refining the fan of the original toric variety into a smooth fan
• Toric resolutions are not unique, but there exists a minimal resolution that is unique up to isomorphism

Resolution of singularities

• Toric resolution of singularities is a special case of the general problem of resolution of singularities in algebraic geometry
• Toric methods provide a constructive approach to resolution of singularities for a large class of algebraic varieties
• Toric resolution has applications to the study of singularities, birational geometry, and the minimal model program

Cohomology of toric varieties

• The cohomology of toric varieties can be studied using combinatorial techniques
• The cohomology rings of toric varieties have a rich structure that reflects the combinatorics of the associated polytopes or fans

Cohomology ring of smooth projective toric varieties

• The cohomology ring of a smooth projective toric variety is isomorphic to the Stanley-Reisner ring of the associated fan
• The Stanley-Reisner ring is a quotient of a polynomial ring by an ideal determined by the combinatorial structure of the fan
• The cohomology ring is generated by the classes of torus invariant divisors, with relations coming from the linear dependence of divisors

Intersection theory on toric varieties

• Intersection theory on toric varieties can be studied using the combinatorics of polytopes
• The intersection numbers of torus invariant divisors can be computed using the mixed volume of the corresponding polytopes
• The intersection theory of toric varieties is closely related to the theory of mixed subdivisions and mixed Hodge structures

Chow rings and polytopes

• The Chow ring of a toric variety is a ring that encodes the intersection theory of algebraic cycles on the variety
• For a smooth projective toric variety, the Chow ring is isomorphic to the cohomology ring and can be described using the associated polytope
• The Chow ring of a singular toric variety can be studied using the combinatorics of the associated fan and its subdivisions

Toric varieties over finite fields

• Toric varieties can be defined over any field, including finite fields
• The cohomology and intersection theory of toric varieties over finite fields have arithmetic analogues that involve counting points
• Toric varieties over finite fields have applications to coding theory, cryptography, and the study of zeta functions of varieties

Toric degenerations

• Toric degenerations are a way of approximating a general algebraic variety by a toric variety
• Toric degenerations have applications to the study of moduli spaces, enumerative geometry, and tropical geometry

Toric degenerations of projective varieties

• A toric degeneration of a projective variety is a flat family of varieties that specializes to a toric variety
• Toric degenerations can be constructed using Gröbner bases and initial ideals of the defining equations of the variety
• The special fiber of a toric degeneration encodes information about the original variety, such as its intersection theory and cohomology

Gröbner bases and initial ideals

• A Gröbner basis is a special generating set of an ideal in a polynomial ring that depends on a choice of monomial order
• The initial ideal of an ideal with respect to a monomial order is the ideal generated by the initial terms of the elements of the ideal
• Gröbner bases and initial ideals are key tools for studying toric degenerations and their properties

SAGBI bases and toric ideals

• A SAGBI (Subalgebra Analogue of Gröbner Basis for Ideals) basis is a special generating set of a subalgebra of a polynomial ring that depends on a choice of monomial order
• Toric ideals are the defining ideals of affine toric varieties and can be studied using SAGBI bases
• SAGBI bases and toric ideals have applications to the study of toric degenerations and their relation to tropical geometry

Tropical geometry and toric degenerations

• Tropical geometry is a piecewise-linear analogue of algebraic geometry that arises as a limit of toric degenerations
• The tropicalization of an algebraic variety is a piecewise-linear object that encodes information about the toric degenerations of the variety
• Tropical geometry provides a new perspective on the study of algebraic varieties and their moduli spaces, and has connections to combinatorics, topology, and mathematical physics