Arithmetic intersection theory

Definition

Arithmetic intersection theory is a framework that studies how algebraic cycles intersect on algebraic varieties, specifically in the context of arithmetic geometry. It connects geometry with number theory, allowing mathematicians to analyze the behavior of cycles in relation to various arithmetic properties, such as divisors and rational points. This theory is crucial for understanding deep relationships between algebraic geometry and number theory.

5 Must Know Facts For Your Next Test

1. Arithmetic intersection theory extends classical intersection theory by incorporating notions from number theory, making it applicable to both complex and arithmetic varieties.
2. One of the key goals of arithmetic intersection theory is to define and compute intersection numbers, which represent how cycles interact within an algebraic variety.
3. This theory plays a vital role in understanding the distribution of rational points on varieties, influencing results in the area of Diophantine geometry.
4. The Grothendieck-Riemann-Roch theorem provides a significant connection between topology and intersection theory, showing how the Chern classes relate to intersections of cycles.
5. Applications of arithmetic intersection theory include proofs of important conjectures like the Bogomolov conjecture and advancements in the study of motives.

Review Questions

• How does arithmetic intersection theory enhance our understanding of the relationship between algebraic cycles and their intersections?
  • Arithmetic intersection theory provides a framework to study the interactions between algebraic cycles through the lens of both geometry and number theory. It helps mathematicians calculate intersection numbers, revealing how different cycles behave when they intersect. By connecting these intersections with arithmetic properties, the theory enriches our comprehension of the structure and relationships present in algebraic varieties.
• Discuss the role of Chow groups in arithmetic intersection theory and how they facilitate the analysis of algebraic cycles.
  • Chow groups are fundamental to arithmetic intersection theory as they categorize algebraic cycles based on their rational equivalence. This classification allows mathematicians to focus on specific features of cycles, such as their intersections. By analyzing Chow groups, one can derive significant results regarding the properties of cycles, ultimately leading to deeper insights into how these cycles intersect within various varieties.
• Evaluate how advancements in arithmetic intersection theory have impacted our understanding of rational points on algebraic varieties and their applications in number theory.
  • Advancements in arithmetic intersection theory have significantly deepened our understanding of rational points on algebraic varieties. By linking geometric properties with number-theoretic concepts, researchers have been able to explore new areas in Diophantine geometry. The insights gained from this interplay have led to important applications, including resolving conjectures about rational solutions and informing broader mathematical theories surrounding motives and cycles.

Related terms

Algebraic cycle: An algebraic cycle is a formal sum of subvarieties of an algebraic variety, which can be used to study properties like intersection and cohomology.

Chow groups: Chow groups are mathematical structures that classify algebraic cycles on a variety up to rational equivalence, serving as a foundation for intersection theory.

Numerical equivalence: Numerical equivalence is a relation between cycles that considers them equivalent if they have the same intersection numbers with all other cycles.