Algebraic Cycle

Definition

An algebraic cycle is a formal finite sum of subvarieties of a given algebraic variety, where each subvariety is multiplied by an integer that represents its 'multiplicity'. These cycles provide a framework for studying the geometry and topology of varieties through intersection theory and help in defining important algebraic invariants, such as Chow rings.

5 Must Know Facts For Your Next Test

1. Algebraic cycles can be classified into various types, including 0-cycles (points), 1-cycles (curves), and higher-dimensional cycles corresponding to subvarieties of increasing dimension.
2. The Chow group of a variety is the group of algebraic cycles modulo rational equivalence, which allows us to group cycles that can be continuously deformed into one another.
3. Algebraic cycles are crucial in defining intersection numbers, which represent how two subvarieties intersect within a given variety, leading to important results in geometry.
4. One can perform operations on algebraic cycles such as addition and multiplication, leading to the structure of the Chow ring, which encodes the interaction of these cycles.
5. The study of algebraic cycles connects to both algebraic geometry and topology, as they bridge the gap between discrete algebraic structures and continuous geometric phenomena.

Review Questions

• How do algebraic cycles contribute to the understanding of intersection theory in algebraic geometry?
  • Algebraic cycles serve as fundamental objects in intersection theory, as they represent the geometric aspects of subvarieties within a variety. By analyzing how these cycles intersect, one can derive intersection numbers that quantify the nature and multiplicity of these intersections. This understanding helps in exploring deeper geometric properties and relationships among different varieties.
• Discuss the relationship between Chow rings and algebraic cycles. How do they enhance our understanding of algebraic varieties?
  • Chow rings are constructed from the algebraic cycles of a variety and provide an algebraic framework for performing operations on these cycles. They allow us to define concepts like addition and multiplication of cycles, which leads to a richer understanding of the geometric relationships among them. By studying Chow rings, mathematicians can extract invariants that characterize varieties and reveal underlying structures not immediately apparent from just looking at the varieties themselves.
• Evaluate the significance of rational equivalence in the context of algebraic cycles and Chow groups. How does this concept impact their classification?
  • Rational equivalence plays a critical role in the classification of algebraic cycles by identifying when two cycles can be considered equivalent due to continuous deformation. This concept allows us to form Chow groups, which encapsulate the structure of algebraic cycles modulo rational equivalence. By analyzing these groups, mathematicians can uncover deep connections between geometry, topology, and algebra, thereby enhancing our comprehension of complex varieties and their interrelationships.

Related terms

Chow Ring: The Chow ring is an algebraic structure that encapsulates information about algebraic cycles on a variety, allowing for operations like addition and multiplication of cycles.

Intersection Theory: Intersection theory studies how subvarieties intersect within a given variety, providing essential tools to understand the properties of cycles and their interactions.

Homology: Homology is a mathematical concept that associates a sequence of abelian groups or modules with a topological space, providing information about its shape and structure, which relates to algebraic cycles through the study of singular homology.