12.2 Toric varieties
3 min read•july 30, 2024
Toric varieties are fascinating geometric objects that blend algebra, geometry, and combinatorics. They're built from fans, which are collections of cones in a lattice, and contain an algebraic torus as a dense open subset.
These varieties have deep connections to convex polytopes and offer a wealth of examples in algebraic geometry. Their combinatorial nature makes them ideal for studying singularities, cohomology, and intersection theory, with applications ranging from to moduli spaces.
Toric varieties from combinatorial data
Construction of toric varieties
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- Toric varieties contain an algebraic torus as a dense open subset
- The action of the torus on itself extends to an action on the entire variety
- Toric varieties are constructed from fans
- Fans are collections of strongly convex rational polyhedral cones in a lattice
- Torus-invariant prime divisors of a toric variety correspond bijectively to the rays (1-dimensional cones) of the
Affine and projective toric varieties
- Affine toric varieties are in one-to-one correspondence with strongly convex rational polyhedral cones in a lattice
- Projective toric varieties are in one-to-one correspondence with complete fans
- Complete fans have cones that cover the entire space
- The gluing of affine toric varieties to form a toric variety is determined by the combinatorial data of the fan
Toric varieties and convex polytopes
Relationship between toric varieties and convex polytopes
- Toric varieties are closely related to convex polytopes
- Convex polytopes are the convex hulls of finitely many points in a lattice
- The normal fan of a convex is a fan that encodes the combinatorial structure of the polytope and gives rise to a toric variety
- The toric variety associated with a convex polytope is projective if and only if the polytope is a lattice polytope
- Lattice polytopes have vertices with integer coordinates
Geometric and combinatorial properties
- The moment map of a is a map from the variety to the polytope
- Provides a geometric link between the two objects
- The faces of a polytope correspond to the torus-invariant subvarieties of the associated toric variety
- Combinatorial properties of a polytope can be studied using the associated toric variety
- Properties include the face lattice and Ehrhart polynomial
Geometric and algebraic properties of toric varieties
Singularities and homological properties
- Toric varieties are normal algebraic varieties
- Locally isomorphic to affine spaces
- The singularities of a toric variety can be characterized combinatorially in terms of the cones in the fan
- Toric varieties are Cohen-Macaulay
- Their local rings have a certain homological property
Cohomology and intersection theory
- The cohomology ring of a smooth projective toric variety can be described combinatorially in terms of the fan
- Toric varieties admit a natural action of the algebraic torus
- Leads to a decomposition into torus orbits
- Intersection theory on a toric variety can be studied using the combinatorics of the fan
- Includes the intersection numbers of torus-invariant divisors
Applications of toric varieties
Examples and counterexamples in algebraic geometry
- Toric varieties provide a rich source of examples and counterexamples in algebraic geometry
- Examples include Fano varieties, Calabi-Yau varieties, and varieties with prescribed singularities
- Toric geometry can be used to study the geometry of linear systems and the structure of the cone of effective divisors on a variety
Applications in computational algebraic geometry and beyond
- Toric varieties appear naturally in the study of Newton polytopes and sparse polynomial systems
- Have applications in computational algebraic geometry
- Toric varieties have been used to construct explicit examples of mirror symmetry
- Mirror symmetry is a duality between certain pairs of Calabi-Yau varieties
- Toric methods have been applied to the study of moduli spaces
- Examples include the moduli space of curves and the moduli space of stable vector bundles
- Toric varieties have connections to other areas of mathematics
- Areas include combinatorics, symplectic geometry, and mathematical physics
Key Terms to Review (17)
Affine toric variety: An affine toric variety is a type of algebraic variety that is associated with a fan, which is a collection of cones in a rational vector space. These varieties are defined as subsets of affine space, and they are built from the combinatorial data of the fan, allowing for a connection between geometry and algebra. Affine toric varieties can be realized as the zero sets of certain polynomial functions, providing a bridge between geometric intuition and algebraic structures.
Associated Polyhedron: An associated polyhedron is a geometric representation that captures the combinatorial structure of a toric variety, which is formed by taking the convex hull of a set of lattice points corresponding to the generators of the torus. This polyhedron plays a critical role in understanding the properties and features of toric varieties, linking algebraic geometry with convex geometry.
Batyrev's Theorem: Batyrev's Theorem is a significant result in algebraic geometry that relates to the mirror symmetry of certain types of varieties, particularly in the context of toric varieties. It establishes a correspondence between the counting of rational curves on a Fano variety and the geometry of its dual variety, suggesting deep connections between algebraic geometry and theoretical physics, especially string theory. This theorem is vital for understanding the structure of toric varieties, which are built from combinatorial data and have applications in both algebraic and arithmetic geometry.
Complete intersection: A complete intersection refers to a type of algebraic variety that can be defined as the common zero set of a specific number of homogeneous polynomials whose degrees are such that their total number matches the dimension of the variety. This concept connects various properties like depth, regular sequences, and Cohen-Macaulay rings, highlighting how varieties can be constructed in a structured manner. Complete intersections are also crucial in the study of toric varieties, where they represent certain combinatorial and geometric configurations within projective spaces.
Cox Ring: A Cox ring is a mathematical tool used in algebraic geometry, particularly in the study of toric varieties. It is defined as a graded ring associated with a toric variety, which encodes information about its geometric properties and allows for the construction of new varieties through various operations. Cox rings are instrumental in understanding the relationships between the geometry of varieties and their combinatorial data, such as fans or polytopes.
Coxeter complexes: Coxeter complexes are combinatorial structures that arise from the study of reflection groups and their associated geometry. They are constructed from the vertices of a simplicial complex, where the edges represent reflections across hyperplanes corresponding to the group elements. This relationship links Coxeter complexes to toric varieties, as they can provide a way to understand the geometry of these varieties through the lens of combinatorial data.
Fan: In algebraic geometry, a fan is a collection of cones that provides a way to describe a toric variety. It allows for the systematic study of these varieties by organizing their geometric properties into combinatorial data, which can be analyzed using techniques from polyhedral geometry. Fans serve as the building blocks for toric varieties, making them fundamental in understanding the relationships between algebraic varieties and their associated combinatorial structures.
Giorgio p. s. di gregorio: Giorgio P. S. Di Gregorio is an influential mathematician known for his contributions to the study of toric varieties and their applications in algebraic geometry. His work has focused on understanding the geometric and combinatorial aspects of these varieties, particularly through the lens of fan theory and polyhedral geometry, bridging the gap between algebraic geometry and combinatorial geometry.
Goresky-MacPherson Formula: The Goresky-MacPherson formula is a powerful result in algebraic geometry that relates the intersection cohomology of a singular variety to its topological properties. It provides a way to compute the intersection cohomology of complex algebraic varieties, particularly those that are not smooth, using their stratifications. This formula is especially relevant when dealing with toric varieties, as it allows for the calculation of their cohomology using combinatorial data from the associated fan.
Mirror Symmetry: Mirror symmetry is a phenomenon in algebraic geometry that suggests a deep connection between certain types of geometric spaces, specifically between a pair of Calabi-Yau manifolds. This concept implies that the complex geometry of one manifold can be mirrored by the symplectic geometry of another, revealing intricate relationships between their mathematical properties and physical interpretations, especially in string theory.
Mori Dream Spaces: Mori dream spaces are a special class of varieties in algebraic geometry that arise in the study of the minimal model program. They are characterized by their ability to compactify certain types of toric varieties and are closely linked to the concepts of Kähler-Einstein metrics and the interplay between geometry and algebraic structures. These spaces provide insights into the behavior of the canonical bundle and allow mathematicians to explore how birational transformations can affect geometric properties.
Polytope: A polytope is a geometric object with flat sides, existing in any number of dimensions, that generalizes the concept of polygons and polyhedra. Polytopes are crucial in various mathematical fields, including algebraic geometry, where they help in understanding toric varieties by providing a combinatorial structure that relates to algebraic properties.
Projective toric variety: A projective toric variety is a type of algebraic variety that is constructed from combinatorial data related to a fan in a torus, specifically realized as a projective space. These varieties arise from the geometry of convex polytopes and serve as a bridge between algebraic geometry and combinatorics. They provide an effective way to study both the geometric properties of varieties and the associated toric ideals through their combinatorial structures.
Smoothness: Smoothness refers to a property of a space where it behaves nicely in terms of differentiability, meaning that it has no abrupt changes, singularities, or 'sharp points.' In algebraic geometry, smoothness implies that the variety is well-behaved at every point, allowing for the application of calculus and differential geometry concepts. This property is essential for understanding how varieties can be manipulated and transformed without encountering issues that arise from singular points.
Support function: The support function is a mathematical concept used to describe a convex set by providing a way to evaluate how far the set extends in a given direction. This function plays a crucial role in the study of toric varieties, as it relates to the geometry of the associated polyhedral cones and their properties. In this context, the support function allows for a deeper understanding of how points and edges interact within toric varieties and connects to concepts like fans and divisors.
Torus action: A torus action is a type of group action where a torus, which is typically the product of circles, acts on a variety in a way that preserves its structure. This concept is crucial in understanding the geometry of toric varieties, as the actions can simplify complex geometrical structures and relate algebraic properties with combinatorial aspects through fans and polytopes.
Victor Guillemin: Victor Guillemin is a mathematician renowned for his contributions to toric geometry, particularly in understanding the connections between combinatorial data and geometric structures. His work emphasizes the importance of toric varieties, which are algebraic varieties defined by combinatorial data associated with convex polytopes, linking algebraic geometry with polyhedral combinatorics.