Singular toric variety

Definition

A singular toric variety is a type of algebraic variety that arises from a fan in a lattice, where some of the cones in the fan correspond to points in the variety that are not smooth, meaning they have singularities. These varieties can be understood through combinatorial data and are important for studying their geometric properties, especially in relation to resolutions of singularities and their impact on the structure of the variety.

5 Must Know Facts For Your Next Test

1. Singular toric varieties can arise from fans where certain cones meet at non-smooth points, leading to singularities that need resolution.
2. Understanding singular toric varieties is crucial for applications in mirror symmetry, as they provide insights into dual geometric structures.
3. The resolution of singularities for a singular toric variety often involves constructing a new toric variety that is smooth and retains some properties of the original.
4. These varieties can be classified by their fans, with specific combinatorial features indicating the nature of their singularities.
5. Singular toric varieties serve as important examples in the study of degenerations and their effects on the geometry of families of varieties.

Review Questions

• How do singular toric varieties relate to the concept of smoothness in algebraic geometry?
  • Singular toric varieties highlight the distinction between smoothness and singularity in algebraic geometry. A smooth variety has no singular points, whereas a singular toric variety has specific points where certain cones intersect, leading to non-smooth behavior. These singular points indicate areas where traditional methods may fail, necessitating techniques like resolutions to study these varieties' geometry more deeply.
• Discuss the role of fans in defining singular toric varieties and their impact on understanding these varieties' geometrical properties.
  • Fans play a central role in defining singular toric varieties as they encode the combinatorial data necessary for constructing these varieties. Each cone within a fan corresponds to an affine chart of the toric variety, and singularities arise from how these cones interact. By analyzing the structure of fans, mathematicians can derive essential information about the singular points and their resolutions, allowing for better comprehension of the geometric features of these varieties.
• Evaluate the significance of resolutions of singularities in relation to singular toric varieties and their applications in modern algebraic geometry.
  • Resolutions of singularities are vital for transforming singular toric varieties into smooth ones, enabling mathematicians to apply powerful geometric tools that are typically unavailable for singular cases. This process enhances our understanding of various properties, such as intersection theory and cohomological aspects, while also influencing broader topics like mirror symmetry and deformation theory. The ability to resolve singularities allows researchers to maintain connections between complex geometrical structures and their algebraic counterparts, ultimately advancing both fields.

Related terms

Toric Variety: An algebraic variety that is defined by combinatorial data associated with a fan in a lattice, allowing for a connection between algebraic geometry and combinatorics.

Fan: A collection of cones in a rational polyhedral cone space that encodes the combinatorial data necessary to construct a toric variety.

Resolution of Singularities: A process or technique used to replace a singular variety with a smooth variety, helping to understand and study the properties of the original variety.