Projective Toric Variety

Definition

A projective toric variety is a specific type of algebraic variety that is defined by combinatorial data from a convex polytope and can be embedded into projective space. These varieties provide a bridge between algebraic geometry and combinatorial geometry, allowing for the study of geometric properties using the underlying combinatorial structure of polytopes.

5 Must Know Facts For Your Next Test

1. Projective toric varieties can be associated with the vertices of a polytope, where each vertex corresponds to a specific coordinate point in projective space.
2. These varieties can be used to study properties such as smoothness and singularities through their combinatorial structure, making them useful tools in algebraic geometry.
3. A key feature of projective toric varieties is their dense torus action, which allows for the understanding of their structure using invariant theory.
4. The embedding of projective toric varieties into projective space can be described using homogeneous coordinate rings, which encode the relationships between the coordinates of points in the variety.
5. The connection between polytopes and projective toric varieties provides insight into how geometric properties influence algebraic ones and vice versa.

Review Questions

• How do polytopes relate to the construction of projective toric varieties, and what role do they play in defining these varieties?
  • Polytopes serve as the foundational geometric objects from which projective toric varieties are constructed. Each vertex of a polytope corresponds to a point in projective space, allowing the algebraic structure of the variety to be derived from its combinatorial properties. The vertices and edges of the polytope determine the fan that glues together affine pieces to form the overall structure of the projective toric variety.
• Discuss how the properties of projective toric varieties are influenced by their combinatorial data derived from polytopes.
  • The combinatorial data obtained from polytopes directly influences several properties of projective toric varieties, including their smoothness and singularities. By examining the arrangement of vertices and edges in a polytope, one can infer important information about the geometric configuration of the corresponding variety. This interplay allows for powerful techniques in understanding the geometry of projective toric varieties through their underlying combinatorial structures.
• Evaluate the significance of dense torus actions in projective toric varieties and how they facilitate the study of these varieties within algebraic geometry.
  • Dense torus actions play a crucial role in projective toric varieties by providing a framework for understanding their geometric structures through invariant theory. These actions allow for the identification of orbits corresponding to different points in projective space, facilitating analysis of symmetries and regularities within the variety. The study of these actions not only deepens our understanding of projective toric varieties but also establishes connections between algebraic geometry and other mathematical fields like combinatorics and representation theory.

Related terms

Polytope:

A polytope is a geometric object with flat sides, which exists in any number of dimensions, and serves as the fundamental building block for defining toric varieties.

Fan:

A fan is a collection of cones that describe how to glue together affine toric varieties to form a projective toric variety, providing the necessary combinatorial data.

Toric Variety: A toric variety is an algebraic variety that is constructed from combinatorial data associated with a fan or a polytope, often exhibiting nice geometric and algebraic properties.