Toric varieties

Definition

Toric varieties are a special class of algebraic varieties that are defined by combinatorial data related to fans, which consist of cones in a lattice. These varieties provide a bridge between algebraic geometry and combinatorial geometry, allowing for the study of geometric objects through their associated combinatorial structures. They have applications in various areas including intersection theory, mirror symmetry, and computational methods in algebraic geometry.

5 Must Know Facts For Your Next Test

1. Toric varieties can be constructed from a fan by associating each cone in the fan with an affine toric variety, leading to a globally defined variety.
2. They allow for effective computation in algebraic geometry, particularly in determining properties like cohomology through the use of combinatorial techniques.
3. In mirror symmetry, toric varieties serve as a framework for understanding duality between pairs of geometrical structures, enhancing our understanding of their underlying algebraic properties.
4. The defining polytopes of toric varieties can encode significant geometric information, such as intersection numbers and curvature properties.
5. Toric varieties are often used as test cases for algorithms in computational algebraic geometry due to their explicit combinatorial structure.

Review Questions

• How do toric varieties relate to fans and what role do they play in the construction of these varieties?
  • Toric varieties are constructed from fans, which are collections of cones in a lattice that satisfy specific intersection conditions. Each cone in the fan corresponds to an affine toric variety, allowing for a piecewise definition that can be glued together to form a globally defined variety. This relationship highlights how combinatorial data encapsulated by fans directly translates into geometric structures, making toric varieties a central theme where algebraic and combinatorial geometries intersect.
• Discuss the significance of toric varieties in computational methods within algebraic geometry.
  • Toric varieties are significant in computational methods because they provide an explicit way to represent complex geometric objects through simple combinatorial data. By leveraging their structure, one can efficiently compute important invariants such as sheaf cohomology using algorithms that exploit their polyhedral nature. This makes them particularly useful for researchers and practitioners looking to apply computational techniques to problems in algebraic geometry without getting bogged down by more complicated structures.
• Evaluate the implications of toric varieties on mirror symmetry and their role in understanding duality between geometrical structures.
  • Toric varieties play a crucial role in mirror symmetry by providing a clear framework for examining duality between pairs of geometrical structures. The combinatorial data that defines these varieties allows mathematicians to establish correspondences between different algebraic objects, leading to insights about their properties. Understanding this duality helps bridge concepts between algebraic geometry and theoretical physics, especially in string theory, where such symmetries are essential for deriving physical insights from mathematical formulations.

Related terms

Fans: A fan is a collection of cones in a vector space that satisfy certain intersection properties and is used to construct toric varieties.

Polyhedral Geometry: A branch of mathematics dealing with the study of polyhedra and their properties, closely linked to the combinatorial aspects of toric varieties.

Homogeneous Coordinates: A system of coordinates used in projective geometry that allows for the representation of points at infinity and is essential in the study of projective toric varieties.