Exceptional Locus

Definition

The exceptional locus is a set of points that arises during the process of a resolution of singularities, typically in algebraic geometry. It represents the points in the variety where the resolution introduces new divisors, often related to the complexities of singular points being resolved. Understanding this concept is essential for analyzing how singularities behave under various transformations and how resolutions can simplify the structure of a given space.

5 Must Know Facts For Your Next Test

1. The exceptional locus is often a divisor that can help to control and understand the singularities in the original variety being resolved.
2. In many cases, the exceptional locus can contain multiple components, reflecting the different ways singular points can be addressed through resolution.
3. The exceptional locus is crucial in establishing whether a resolution is 'minimal' or 'terminal', influencing the properties of the resulting space.
4. The size and complexity of the exceptional locus can give insight into how severe the singularities are in the original variety.
5. Studying the exceptional locus is key to understanding how toric varieties relate to more general types of varieties, especially in resolving singularities.

Review Questions

• How does the exceptional locus relate to the resolution of singularities and why is it significant?
  • The exceptional locus plays a vital role in the resolution of singularities as it marks where new divisors are introduced during the resolution process. Its significance lies in helping mathematicians understand how singular points transform into smooth ones, thus providing insights into the overall structure of algebraic varieties. By analyzing the exceptional locus, one can assess both the complexity of singularities and the effectiveness of various resolution techniques.
• In what ways can the structure of the exceptional locus influence the properties of a resolved variety?
  • The structure of the exceptional locus can significantly influence properties such as smoothness, dimension, and intersection theory in a resolved variety. If the exceptional locus is large or complicated, it may indicate that the original variety had complex singularities that need careful handling. Additionally, understanding its composition allows mathematicians to classify different types of resolutions and determine if they are minimal or possess desirable traits like being terminal.
• Evaluate the implications of different forms of exceptional loci when analyzing toric varieties compared to general varieties.
  • Different forms of exceptional loci reveal important distinctions between toric varieties and general varieties during singularity resolutions. For toric varieties, the combinatorial nature allows for well-understood structures that can simplify identifying exceptional loci. In contrast, general varieties may have exceptional loci that are more complex and varied, making them harder to analyze. This evaluation helps mathematicians leverage toric techniques to develop broader approaches for understanding singularities in more complex varieties.

Related terms

Resolution of Singularities: A process in algebraic geometry that replaces a singular variety with a nonsingular one by introducing new variables or spaces.

Divisor: A formal sum of codimension one subvarieties on an algebraic variety, which helps in understanding the structure and properties of the variety.

Toric Variety: A class of algebraic varieties that can be described by combinatorial data, typically involving fans and convex polytopes, which are useful in constructing resolutions.