Affine toric variety

Definition

An affine toric variety is a type of algebraic variety that is associated with a fan in a lattice, often represented as an open subset of a vector space. These varieties arise from combinatorial data related to convex polyhedra and are characterized by their coordinate rings, which can be expressed in terms of monomials corresponding to the rays of the fan. They serve as an important example in the study of toric varieties, bridging algebraic geometry and combinatorial geometry.

5 Must Know Facts For Your Next Test

1. Affine toric varieties can be constructed from fans by associating each cone with an affine open set in the variety.
2. The coordinate ring of an affine toric variety is generated by monomials corresponding to the rays of the fan, reflecting its combinatorial nature.
3. These varieties are closely related to convex geometry, as their construction is often based on the properties of convex polytopes.
4. Affine toric varieties allow for a geometric interpretation of algebraic concepts, enabling connections between algebraic geometry and combinatorial methods.
5. They serve as examples of more general toric varieties and are essential in understanding the structure and classification of these varieties.

Review Questions

• How do the properties of fans influence the structure of affine toric varieties?
  • Fans are essential in defining affine toric varieties because they provide the combinatorial data needed to construct these varieties. Each cone in a fan corresponds to an open set in the affine toric variety, influencing how the variety is shaped and what geometric properties it possesses. This relationship highlights the deep connection between combinatorial structures and algebraic geometry.
• Discuss the role of monomial coordinates in the coordinate ring of an affine toric variety and how they relate to the underlying combinatorial data.
  • Monomial coordinates in the coordinate ring of an affine toric variety are derived from the rays of the fan associated with the variety. Each ray corresponds to a generator of the coordinate ring, leading to a structure that reflects both geometric and combinatorial aspects. This means that studying monomial coordinates allows us to analyze not just algebraic properties but also the combinatorial nature that defines the affine toric variety's construction.
• Evaluate how understanding affine toric varieties can advance our knowledge in both algebraic geometry and combinatorial geometry, particularly in applications like mirror symmetry or toric degenerations.
  • Understanding affine toric varieties enriches both algebraic and combinatorial geometry by offering insights into how geometric structures can be encoded using combinatorial data. This knowledge facilitates significant applications, such as mirror symmetry, where dual relationships between different geometries are explored. Additionally, studying toric degenerations provides tools for understanding complex algebraic varieties through simpler combinatorial frameworks, making these concepts pivotal in modern mathematical research.

Related terms

Fan: A fan is a collection of cones that fit together in a coherent way, used to define toric varieties through their combinatorial structure.

Lattice: A lattice is a discrete subgroup of Euclidean space that spans the entire space, providing a framework for constructing toric varieties.

Toric Variety: A toric variety is a variety that contains an algebraic torus as a dense subset and is defined by combinatorial data from fans and cones.