Fan

Definition

In algebraic geometry, a fan is a collection of cones that provides a way to describe a toric variety. It allows for the systematic study of these varieties by organizing their geometric properties into combinatorial data, which can be analyzed using techniques from polyhedral geometry. Fans serve as the building blocks for toric varieties, making them fundamental in understanding the relationships between algebraic varieties and their associated combinatorial structures.

5 Must Know Facts For Your Next Test

1. A fan is typically defined in a rational vector space and consists of finitely many cones that are closed under linear combinations.
2. Each cone in a fan corresponds to a torus-invariant affine open subset of the associated toric variety.
3. Fans can be used to describe smooth or singular toric varieties, allowing mathematicians to analyze their properties through combinatorial methods.
4. The construction of a toric variety from a fan involves gluing together affine pieces associated with each cone in the fan.
5. Fans can also encode information about the intersection theory of toric varieties, which is crucial for studying their geometric behavior.

Review Questions

• How does the structure of a fan influence the properties of the corresponding toric variety?
  • The structure of a fan directly influences the properties of the corresponding toric variety by determining its combinatorial layout and how its affine pieces are glued together. Each cone in the fan represents an open subset that contributes to the overall topology and geometry of the variety. This connection allows for insights into aspects such as smoothness and singularities based on the arrangement and relationships of cones within the fan.
• Discuss the importance of cones within a fan and how they relate to torus-invariant open subsets in toric varieties.
  • Cones within a fan are crucial because they define local geometric structures that correspond to specific regions in the associated toric variety. Each cone's apex represents a torus-invariant point, while its base outlines an open subset where the variety behaves nicely. This relationship facilitates the understanding of how different cones can interact and overlap, ultimately influencing the global geometry and topological features of the entire toric variety.
• Evaluate how polyhedral geometry contributes to the understanding and classification of fans in algebraic geometry.
  • Polyhedral geometry plays a significant role in understanding and classifying fans by providing tools and concepts that allow mathematicians to analyze geometric objects with flat faces. This discipline helps characterize fans' structure through their cones and intersections, leading to insights about how these elements interact with algebraic varieties. By leveraging polyhedral methods, researchers can explore deeper relationships between combinatorial data encoded by fans and algebraic properties, fostering advancements in both theoretical and applied aspects of algebraic geometry.

Related terms

Toric Variety: A type of algebraic variety that is constructed from a fan, characterized by its combinatorial and geometric properties.

Cone: A geometric object that is formed by taking a point (the apex) and all line segments connecting it to a flat subset (the base). In the context of fans, cones represent the local structure of toric varieties.

Polyhedral Geometry: The study of geometric shapes with flat faces, which plays an essential role in understanding the properties and applications of fans in algebraic geometry.