Polyhedral Cone

Definition

A polyhedral cone is a geometric object that consists of a finite set of vectors in a vector space, where all non-negative linear combinations of these vectors form the cone. This structure is essential in understanding toric varieties and the combinatorial data associated with fans, as it allows for the description of how various geometric objects are formed from these linear combinations.

5 Must Know Facts For Your Next Test

1. Polyhedral cones can be defined in any vector space, but they are most commonly studied in $ ext{R}^n$, where they can represent various geometric and algebraic structures.
2. The generators of a polyhedral cone are the vectors that form its boundary, and these vectors play a crucial role in defining the properties and relations of the cone.
3. Every polyhedral cone can be associated with a fan, which helps in constructing toric varieties by encoding the necessary combinatorial data.
4. In the context of algebraic geometry, polyhedral cones help to study singularities by providing resolutions through toric varieties.
5. The intersection of polyhedral cones leads to new cones, which can be useful for understanding how different geometric objects interact within a given space.

Review Questions

• How do polyhedral cones relate to the construction of toric varieties?
  • Polyhedral cones are fundamental in constructing toric varieties as they provide the combinatorial framework needed to define these varieties. Each toric variety corresponds to a fan made up of polyhedral cones, where each cone represents a piece of the geometric structure. The relationships between these cones dictate how the overall variety is formed, making them essential for understanding both the geometry and algebraic properties of toric varieties.
• Discuss how polyhedral cones can be used to resolve singularities in algebraic geometry.
  • Polyhedral cones play a significant role in resolving singularities through their connection with toric resolutions. By using the combinatorial data from polyhedral cones, mathematicians can construct toric varieties that serve as smooth approximations of singular spaces. This process allows for studying the local behavior of varieties at singular points and finding resolutions that enable better understanding and manipulation of these complex structures.
• Evaluate the significance of polyhedral cones in understanding both algebraic geometry and combinatorial geometry.
  • Polyhedral cones bridge algebraic and combinatorial geometry by offering tools to study geometric properties through algebraic methods. They allow for clear visualization and organization of data in both disciplines, providing insights into how algebraic varieties can be constructed from combinatorial information. The interplay between polyhedral cones and fans not only aids in constructing toric varieties but also enriches our understanding of singularities and resolutions, making them vital in modern geometric studies.

Related terms

Toric Varieties:

Algebraic varieties that are constructed from fans, which are collections of polyhedral cones, capturing the combinatorial data necessary for their geometry.

Fan:

A collection of polyhedral cones that fit together in a compatible way, providing a way to organize the geometric data necessary for constructing toric varieties.

Normal Fan: A fan that arises from a polyhedron, where each cone corresponds to a face of the polyhedron, allowing for a systematic study of its geometry and combinatorial properties.