Quotient Variety

Definition

A quotient variety is a type of algebraic variety formed by taking an algebraic variety and factoring it by the action of an algebraic group, resulting in a new space that captures the structure of the original variety up to the symmetries defined by the group action. This process essentially groups points in the original variety that are related through the group action, creating a new variety that reflects the intrinsic properties of the original while allowing for a simplification of its structure. Quotient varieties provide a powerful way to study geometric objects under symmetry and facilitate connections between geometry and group theory.

5 Must Know Facts For Your Next Test

1. Quotient varieties are constructed by taking an algebraic variety and dividing it by an equivalence relation induced by the action of an algebraic group.
2. The points in a quotient variety correspond to orbits of points in the original variety, capturing how these points relate to each other under the group action.
3. Quotient varieties can be used to study geometric properties that are invariant under group actions, providing insight into both the original variety and the nature of the group.
4. When the action of the group is free, meaning no two points in an orbit are fixed, the quotient variety behaves particularly well and retains many properties of the original variety.
5. Quotient varieties often arise in various contexts such as moduli problems, where they can parameterize families of geometric objects up to certain equivalences.

Review Questions

• How does a quotient variety simplify the study of an algebraic variety under group actions?
  • A quotient variety simplifies the study of an algebraic variety by collapsing points that are related through a group action into single equivalence classes or orbits. This allows mathematicians to focus on the intrinsic properties shared by these related points rather than examining each point individually. By analyzing this new structure, one can gain insights into both the original variety's properties and how these properties behave under symmetry.
• Discuss how the concept of orbit space relates to the construction of quotient varieties.
  • The concept of orbit space is closely tied to quotient varieties since both involve understanding how points in a space relate through a group's action. When forming a quotient variety, we consider each orbit as a point in the new space, effectively identifying all points in an orbit as equivalent. This relationship highlights how quotient varieties represent not just individual geometric objects but collections of them based on symmetry, making it easier to study properties that hold across these orbits.
• Evaluate how understanding quotient varieties can influence modern research in algebraic geometry and related fields.
  • Understanding quotient varieties can significantly influence modern research by providing tools for classifying geometric objects in relation to their symmetries. This has implications for various fields, including string theory, algebraic topology, and complex geometry. By enabling researchers to tackle problems regarding moduli spaces and deformation theory through the lens of symmetry, quotient varieties help bridge gaps between different areas of mathematics and foster deeper insights into their interconnected structures.

Related terms

Algebraic Group: An algebraic group is a group that is also an algebraic variety, meaning its elements can be represented as solutions to polynomial equations.

Group Action: A group action is a formal way of describing how elements of a group correspond to transformations of a set or space, preserving the structure of the set.

Orbit Space: The orbit space is the set of orbits formed by a group acting on a space, where each orbit consists of points that are equivalent under the group action.