Self-intersection

Definition

Self-intersection refers to a point or set of points where a geometric object intersects itself. In the context of arithmetic surfaces, self-intersection can describe how divisors on these surfaces interact with one another, providing important information about the geometry and arithmetic properties of the surface. This concept is crucial for understanding intersection theory and the behavior of curves on surfaces.

5 Must Know Facts For Your Next Test

1. In arithmetic geometry, self-intersection numbers are often computed using the intersection theory on projective varieties.
2. Self-intersection can indicate how a divisor intersects itself in a more algebraic sense, giving rise to important invariants like the degree of the divisor.
3. When dealing with smooth curves on an arithmetic surface, self-intersection can be understood through intersection numbers, often represented using notation like $D^2$ for a divisor $D$.
4. Self-intersection helps in understanding singularities on surfaces; for instance, if the self-intersection number is negative, it can signal certain types of singular behavior.
5. The computation of self-intersection plays a vital role in defining various characteristic classes, which are crucial in linking algebraic geometry with topology.

Review Questions

• How does self-intersection relate to the properties of divisors on arithmetic surfaces?
  • Self-intersection provides crucial information about how divisors interact with themselves on arithmetic surfaces. When analyzing a divisor $D$, its self-intersection number $D^2$ can reveal insights into its geometry, such as whether it is effective or how it contributes to singularities on the surface. Understanding this relationship allows mathematicians to utilize intersection theory effectively in the study of arithmetic surfaces.
• Discuss the implications of positive versus negative self-intersection numbers in the context of arithmetic geometry.
  • In arithmetic geometry, a positive self-intersection number often suggests that a divisor is effective and can represent an actual curve within the surface. Conversely, a negative self-intersection indicates potential singularities or complex interactions between divisors. These implications are crucial for understanding not only the geometric properties but also how they relate to broader concepts like stability and duality within the framework of algebraic geometry.
• Evaluate the role of self-intersection in connecting arithmetic geometry with topology through characteristic classes.
  • Self-intersection plays a significant role in bridging arithmetic geometry and topology by influencing characteristic classes, which encapsulate geometric data in algebraic settings. The computation of self-intersection numbers aids in defining Chern classes and other invariants that reveal topological information about varieties. This connection illustrates how algebraic properties can lead to topological insights, enriching both fields and demonstrating their interdependence.

Related terms

Divisor: A formal sum of subvarieties of codimension one on a variety, which can be used to study properties of the variety.

Intersection Theory:

A branch of mathematics that studies the intersection of geometric objects, such as varieties or divisors, and the properties that arise from these intersections.

Chern Class: An invariant associated with a vector bundle or a complex manifold that captures information about the geometry and topology of the manifold.