Quotient Singularities

Definition

Quotient singularities are specific types of singular points on algebraic varieties that arise from the action of a finite group on a smooth variety. These singularities are characterized by their local structure, which resembles the quotient of a smooth space by a group action, resulting in an isolated singularity that often has desirable properties in terms of resolutions and minimal models.

5 Must Know Facts For Your Next Test

1. Quotient singularities can be classified according to their associated finite group actions, leading to different types of singularities based on the group structure.
2. These singularities are particularly significant because they often admit simple resolutions, which means they can be 'smoothed out' effectively without introducing new complications.
3. They arise frequently in the study of algebraic varieties and can provide insights into the geometry and topology of these spaces.
4. A common example of a quotient singularity is the cone over a projective variety, which can be studied through its local rings and their properties under quotienting by a group action.
5. Quotient singularities play a crucial role in understanding the behavior of algebraic varieties under deformation and contribute to the development of minimal model theory.

Review Questions

• How do quotient singularities relate to resolutions of singularities in algebraic geometry?
  • Quotient singularities often allow for straightforward resolutions because their local structure is relatively simple, resembling the quotient of a smooth variety. When applying the resolution process, one can replace the singularity with a smooth variety while preserving important geometric properties. Understanding how to resolve these singularities is crucial for studying the overall structure of algebraic varieties.
• Discuss the role of quotient singularities in classifying singular points on algebraic varieties.
  • Quotient singularities are integral to the classification of singular points because they fall into specific categories based on group actions. Their classification helps mathematicians identify which types of resolutions or modifications are needed for a given variety. By studying these singularities, one can gain insight into more complex cases and understand how various types fit into the broader picture of algebraic geometry.
• Evaluate the impact of quotient singularities on minimal model theory and their implications for algebraic varieties.
  • Quotient singularities significantly influence minimal model theory as they typically exhibit favorable properties, such as being canonical or terminal. These characteristics are essential for constructing minimal models, where one seeks to simplify the variety while retaining its essential features. By understanding how quotient singularities behave under different transformations, mathematicians can make strides in resolving complex questions about algebraic varieties' structure and classification.

Related terms

Resolution of Singularities: A process in algebraic geometry that replaces a singular variety with a new variety that has no singularities, allowing for a better understanding of its geometric structure.

Canonical Singularity: A type of quotient singularity that appears in the classification of singularities, which satisfies certain numerical criteria making them particularly well-behaved in the context of minimal models.

Terminal Singularities: Singular points on varieties that are more severe than canonical singularities, meaning they have certain positivity properties that are useful in the context of minimal model theory.