Self-intersection numbers

Definition

Self-intersection numbers are integers that represent the intersection of a geometric object with itself, particularly in the context of algebraic varieties and their embeddings. They provide valuable information about the geometry and singularities of these varieties, indicating how a divisor or a curve interacts with itself. Understanding self-intersection numbers is crucial when dealing with toric resolutions, as they help classify singularities and understand how resolutions modify the geometry.

5 Must Know Facts For Your Next Test

1. Self-intersection numbers can be computed using intersection theory, where they arise as coefficients in the intersection product.
2. In the case of a smooth curve on a surface, the self-intersection number can be interpreted as the degree of the curve when embedded in the surface.
3. The self-intersection number of a divisor is essential for determining whether it is effective, as a negative self-intersection indicates certain properties about the variety.
4. Self-intersection numbers play a key role in the classification of singularities in toric geometry, helping to distinguish between different types of singular points.
5. In toric resolutions, understanding self-intersection numbers aids in analyzing how singularities are resolved and how new geometry is formed through blow-ups.

Review Questions

• How do self-intersection numbers inform our understanding of singularities in algebraic varieties?
  • Self-intersection numbers provide crucial insights into the nature of singularities by indicating how divisors interact with themselves within an algebraic variety. A positive self-intersection number often suggests a more regular or 'nice' behavior, while negative values may signal more complicated geometric structures or potential singular points. By analyzing these numbers, mathematicians can classify singularities and determine their resolutions.
• Discuss the relationship between self-intersection numbers and toric resolutions in geometry.
  • Self-intersection numbers are pivotal in the context of toric resolutions as they help characterize singularities encountered in algebraic varieties. When performing a toric resolution, these numbers guide the process by revealing how certain singularities can be transformed into simpler structures. This understanding assists in identifying appropriate blow-up operations to achieve a smoother resolution and provides insight into how the geometry evolves during this process.
• Evaluate the significance of self-intersection numbers in classifying different types of toric varieties.
  • Self-intersection numbers play a significant role in classifying toric varieties by offering information about their geometric properties and singularities. The values of these numbers can determine whether a variety is smooth or has particular types of singularities, influencing how we categorize them into distinct classes. By assessing these intersection numbers, mathematicians can draw connections between the combinatorial data represented by toric fans and the actual geometry of the corresponding varieties, enriching our understanding of their structure and behavior.

Related terms

Divisor: A formal sum of subvarieties in an algebraic variety, used to study properties like intersection theory and linear systems.

Toric Varieties: A class of algebraic varieties that can be described combinatorially using convex polytopes and fans, often simplifying the study of their properties.

Resolution of Singularities: A process that replaces a singular variety with a smoother one, allowing for better understanding and manipulation of its geometric properties.