Fan structure

Definition

A fan structure is a combinatorial object that consists of a collection of cones in a vector space, organized in such a way that they fit together to form a fan. This concept is crucial for toric geometry, where it helps in studying varieties by connecting algebraic properties with geometric aspects, particularly when analyzing resolutions of singularities and the interaction between different algebraic objects.

5 Must Know Facts For Your Next Test

1. Fan structures are built from cones, which can be thought of as geometric shapes formed by rays emanating from the origin in a vector space.
2. Each cone in a fan represents a local chart for the corresponding toric variety, allowing for easier analysis of singularities.
3. The fan structure encodes the combinatorial data necessary to construct a toric variety, linking algebraic and geometric properties.
4. Studying fan structures can provide insight into how different singularities behave and how they can be resolved through toric varieties.
5. Fans can also be categorized into non-singular and singular fans, which have distinct implications for the geometry of the associated toric varieties.

Review Questions

• How does the concept of fan structure relate to the construction of toric varieties?
  • The fan structure is foundational for constructing toric varieties because it provides the combinatorial data necessary for their definition. Each cone within the fan corresponds to an affine piece of the toric variety, allowing for a piecewise definition. This connection between combinatorics and geometry enables mathematicians to analyze and understand the local structure of these varieties effectively.
• Discuss the role of cones within fan structures and their importance in resolving singularities.
  • Cones are integral components of fan structures, serving as local models for the associated toric varieties. Each cone can represent different aspects of the variety's geometry. When resolving singularities, these cones help identify local charts that simplify the complex interactions present at those points, allowing mathematicians to construct resolutions that eliminate or mitigate singular behavior.
• Evaluate the implications of having singular versus non-singular fans on the properties of associated toric varieties.
  • The distinction between singular and non-singular fans has significant implications for the properties of associated toric varieties. Non-singular fans lead to well-behaved varieties that possess desirable geometric properties, like smoothness. In contrast, singular fans may correspond to varieties with more complex singularities that can complicate analysis and resolution processes. Understanding these implications helps in developing techniques for managing singularities effectively in algebraic geometry.

Related terms

Toric Variety: An algebraic variety that can be constructed from a fan, where each cone corresponds to an affine piece of the variety.

Cones: Geometric objects formed by taking a point and all rays emanating from it in a vector space, which are used to construct fan structures.

Toric Resolution: A process of resolving singularities of an algebraic variety by constructing a toric variety that serves as a resolution.