Projective toric variety

Definition

A projective toric variety is a type of algebraic variety that is constructed from a combinatorial object known as a polytope. These varieties are defined by their relation to fans, which are collections of cones in a vector space that encode the combinatorial and geometric data necessary to study the variety. Projective toric varieties provide a way to understand how geometry interacts with combinatorics through the lens of polytopes and their associated torus actions.

5 Must Know Facts For Your Next Test

1. Projective toric varieties are obtained by taking the projectivization of affine toric varieties, which means they can be viewed as compact objects in projective space.
2. The combinatorial structure of a projective toric variety is determined by the fan associated with the polytope, allowing for a clear relationship between algebraic geometry and combinatorial geometry.
3. Every projective toric variety has an associated group action by the algebraic torus, which provides insights into its geometric properties, such as its dimension and singularities.
4. Projective toric varieties can be used to study various geometric properties, including intersection theory and deformation theory, making them essential tools in algebraic geometry.
5. They serve as examples in mirror symmetry, where the interplay between the geometry of projective toric varieties and their duals reveals deeper relationships between different areas in mathematics.

Review Questions

• How does the structure of a polytope influence the characteristics of a projective toric variety?
  • The structure of a polytope directly influences the characteristics of a projective toric variety through its associated fan. The fan encodes the combinatorial data necessary to construct the variety, determining aspects such as its dimension and singularity structure. By examining the vertices and edges of the polytope, one can gain insights into how these features manifest in the geometric properties of the corresponding projective toric variety.
• Discuss the significance of fans in defining projective toric varieties and how they relate to algebraic geometry.
  • Fans play a crucial role in defining projective toric varieties as they provide a combinatorial framework for constructing these geometries. Each cone in a fan corresponds to an open set in the variety, allowing us to piece together the global structure from local data. This relationship bridges combinatorics and algebraic geometry, showing how abstract combinatorial objects can yield concrete geometric representations and facilitate the study of complex algebraic structures.
• Evaluate the role of projective toric varieties in contemporary mathematical research, particularly in relation to mirror symmetry and intersection theory.
  • Projective toric varieties are increasingly important in contemporary mathematical research due to their applications in mirror symmetry and intersection theory. In mirror symmetry, these varieties allow mathematicians to establish connections between seemingly unrelated geometric structures by examining their dual relationships. Furthermore, they provide a rich framework for studying intersection numbers and their implications in various contexts, such as enumerative geometry and string theory, highlighting their versatility and relevance across different fields in mathematics.

Related terms

Polytope: A polytope is a geometric object with flat sides, which exists in any number of dimensions, and can be represented as the convex hull of a finite set of points.

Fan:

A fan is a collection of cones that are compatible with each other, used to define toric varieties through their combinatorial structure.

Torus Action: A torus action refers to the action of a torus (a product of circles) on a variety, which helps in understanding the symmetries and properties of the variety.