Support function

Definition

The support function is a mathematical concept used to describe a convex set by providing a way to evaluate how far the set extends in a given direction. This function plays a crucial role in the study of toric varieties, as it relates to the geometry of the associated polyhedral cones and their properties. In this context, the support function allows for a deeper understanding of how points and edges interact within toric varieties and connects to concepts like fans and divisors.

5 Must Know Facts For Your Next Test

1. The support function for a convex set can be defined as $h_{C}(u) = ext{sup}\{\langle x, u\rangle : x \in C\}$, where $u$ is a direction vector and $\langle x, u\rangle$ is the dot product.
2. In toric varieties, support functions are directly linked to the properties of associated fans and how these fans describe the geometry of the variety.
3. Support functions help in analyzing the relationship between the convex hull of points and their projections onto different directions.
4. They are essential for determining the dual properties of toric varieties, particularly when dealing with divisors and line bundles.
5. The support function provides insight into the convex geometry underlying toric varieties, allowing for the exploration of their algebraic and geometric properties.

Review Questions

• How does the support function relate to the geometry of toric varieties?
  • The support function is vital in understanding the geometry of toric varieties because it describes how far the variety extends in different directions. This function connects to the associated fan's properties, helping to define the structure and shape of the variety. By analyzing these directions through the support function, one can gain insights into both the convex sets involved and how they relate to algebraic structures within toric varieties.
• Discuss the role of support functions in determining dual properties of toric varieties.
  • Support functions play a significant role in determining the dual properties of toric varieties by providing a way to analyze relationships between various geometrical elements. For instance, they are used to connect convex hulls with dual cones, allowing mathematicians to explore how line bundles and divisors behave under transformation. This understanding enhances one's grasp of both algebraic and geometric aspects within toric varieties.
• Evaluate how support functions can be utilized to explore further connections between convex geometry and algebraic structures in toric varieties.
  • Support functions serve as a bridge between convex geometry and algebraic structures by linking the geometric properties of convex sets with algebraic representations in toric varieties. By using support functions to analyze these connections, one can uncover deeper relationships between polytopes and their corresponding algebraic objects. This exploration reveals how geometric insights can inform algebraic questions and vice versa, making support functions essential for advancing our understanding of both fields.

Related terms

Toric variety: A type of algebraic variety that is defined by combinatorial data from a fan, typically associated with a convex polytope.

Fan: A collection of cones that cover a certain space, used to define toric varieties through their combinatorial properties.

Convex cone: A set that is closed under positive scalar multiplication and addition, often used in defining the structure of toric varieties.