Affine toric variety

Definition

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.

5 Must Know Facts For Your Next Test

1. Affine toric varieties can be thought of as spaces that encode both geometric and algebraic information through their associated fans.
2. The coordinate ring of an affine toric variety is generated by monomials corresponding to the rays of the fan.
3. These varieties can often be visualized using combinatorial objects like polyhedra, with the geometric features reflecting the underlying algebraic properties.
4. Each affine toric variety has a natural action by the algebraic torus, which reflects its structure and symmetries.
5. The study of affine toric varieties has significant applications in areas like mirror symmetry and combinatorial geometry, linking seemingly disparate fields.

Review Questions

• How does the structure of an affine toric variety relate to its associated fan?
  • The structure of an affine toric variety is fundamentally tied to its associated fan, which provides the combinatorial data necessary to define it. Each cone in the fan corresponds to specific algebraic properties and geometric features in the affine space. This relationship allows us to translate geometric intuition into algebraic terms, showcasing how the arrangement of cones can influence the shape and behavior of the variety.
• Discuss how monomials in the coordinate ring of an affine toric variety are related to the rays of its fan.
  • Monomials in the coordinate ring of an affine toric variety correspond directly to the rays of its fan. Each ray represents a direction in which a monomial can be formed, and these rays form the basis for generating the coordinate ring. This relationship emphasizes how combinatorial aspects of the fan translate into algebraic structures, showcasing a deep connection between geometry and algebra in this context.
• Evaluate the impact of affine toric varieties on broader mathematical concepts such as mirror symmetry and combinatorial geometry.
  • Affine toric varieties play a crucial role in connecting various mathematical domains, particularly in mirror symmetry and combinatorial geometry. Their rich structure enables mathematicians to explore dualities between different geometrical objects, leading to insights about symmetries and transformations. Moreover, the combinatorial aspects inherent in these varieties allow for innovative approaches to solving problems within both algebra and geometry, demonstrating their versatility and importance across multiple fields.

Related terms

Toric variety: A variety that is constructed from combinatorial data associated with cones in a fan, encompassing both affine and projective cases.

Fan: A collection of cones that represents the combinatorial data needed to define a toric variety, where each cone corresponds to a different aspect of the geometry.

Polyhedral cone: A set of vectors that can be expressed as non-negative linear combinations of a finite set of generators, serving as the building blocks for fans in toric geometry.