Toric Variety

Definition

A toric variety is a special type of algebraic variety that is constructed from combinatorial data associated with convex polyhedra, particularly fans. They are particularly useful in algebraic geometry because they provide a way to study geometric objects through combinatorial methods, revealing connections between algebra, geometry, and topology.

5 Must Know Facts For Your Next Test

1. Toric varieties can be realized as projective varieties when constructed from fans that correspond to polyhedral cones.
2. They allow the study of singularities through combinatorial techniques, making it easier to analyze their geometric properties.
3. Any toric variety can be represented in terms of its associated fan, providing a direct link between geometry and combinatorics.
4. Toric varieties can often be defined over arbitrary fields, which broadens their applicability in various mathematical contexts.
5. They play a crucial role in mirror symmetry and string theory, revealing deep connections between different areas of mathematics.

Review Questions

• How do fans relate to the construction of toric varieties and what role do they play in understanding their properties?
  • Fans are critical for constructing toric varieties as they encode the combinatorial data necessary for defining the variety's structure. Each cone in a fan corresponds to a chart in the toric variety, allowing for the organization of geometric information. This relationship helps mathematicians understand how the variety behaves under various transformations and how singularities can be resolved within this framework.
• Discuss the importance of studying singularities in toric varieties and how combinatorial techniques facilitate this analysis.
  • Studying singularities in toric varieties is essential because it allows mathematicians to understand the local behavior of these varieties at problematic points. Combinatorial techniques simplify this process by providing tools to analyze the geometry through the fan's structure. For instance, one can identify how singular points correspond to specific configurations in the fan, allowing for resolutions that can be more easily computed using combinatorial methods.
• Evaluate the impact of toric varieties on modern mathematics, specifically regarding their applications in mirror symmetry and string theory.
  • Toric varieties have significantly influenced modern mathematics by providing frameworks for understanding complex relationships between algebraic geometry and other fields like mirror symmetry and string theory. Their ability to bridge combinatorial data with geometric structures leads to profound insights into duality and symmetries in these theories. As researchers explore these connections further, toric varieties continue to reveal new avenues for exploration in both mathematics and theoretical physics, showcasing their vital role in contemporary studies.

Related terms

Fan: A fan is a collection of cones that satisfy certain intersection properties, used to define toric varieties and organize the combinatorial data involved.

Polytope: A polytope is a geometric object with flat sides in any number of dimensions, which can be represented as a convex hull of a finite set of points, and serves as a basis for constructing toric varieties.

Toric Morphism: A toric morphism is a special type of morphism between toric varieties that respects their combinatorial structures and can be described using the corresponding fans.