Rational Singularity

Definition

A rational singularity is a type of singular point on a variety where the local ring at that point behaves like a rational number in terms of its cohomological properties. This means that the singularity can be resolved in such a way that the resulting resolution retains certain desirable features, making it manageable in the study of algebraic varieties and their geometry. Rational singularities are particularly interesting because they allow for specific techniques like toric resolutions to be employed, which can simplify the analysis of singularities.

5 Must Know Facts For Your Next Test

1. Rational singularities arise in various contexts, particularly when dealing with projective varieties or schemes with certain cohomological properties.
2. They can often be characterized by the vanishing of specific cohomological groups, indicating that certain forms of 'complexity' are absent at the singularity.
3. One important property of rational singularities is that they are preserved under certain operations, such as taking products with smooth varieties.
4. The existence of rational singularities allows for the use of techniques from toric geometry, which can provide combinatorial approaches to resolving singularities.
5. Rational singularities are closely linked to notions of duality in algebraic geometry, particularly when considering how they interact with the canonical class of a variety.

Review Questions

• How do rational singularities relate to cohomological properties in algebraic geometry?
  • Rational singularities are characterized by the vanishing of certain cohomological groups, indicating that the local ring at the singularity behaves well in terms of its algebraic structure. This implies that despite being singular points, these varieties have desirable properties that allow them to be handled more easily within cohomological frameworks. Understanding this relationship is crucial for applying techniques such as toric resolutions effectively.
• Discuss how toric resolutions can be used to resolve rational singularities and what advantages they provide.
  • Toric resolutions leverage the combinatorial nature of toric varieties to provide clear pathways for resolving rational singularities. By constructing a resolution through a fan associated with the singularity, one can create a smoother variety that retains key properties from the original. This approach simplifies complex calculations and helps clarify the geometric structure surrounding the singular point, making it easier to analyze and understand.
• Evaluate the impact of rational singularities on broader algebraic geometric theories and practices.
  • Rational singularities play a significant role in advancing algebraic geometry by providing frameworks through which complicated structures can be simplified and studied. Their properties facilitate various techniques in both theoretical and practical applications, influencing areas such as intersection theory and moduli problems. The ability to resolve these singularities while retaining essential characteristics allows mathematicians to build upon existing theories, pushing forward our understanding of both classical and modern algebraic varieties.

Related terms

Toric Variety: A type of algebraic variety that is constructed from combinatorial data associated with a fan, allowing for geometric interpretations of toric ideals and their resolutions.

Cohomology: A mathematical tool used to study topological spaces through algebraic invariants, providing insight into their structure and properties.

Resolution of Singularities: A process that replaces a singular variety with a smoother one, allowing for better understanding and manipulation of the original variety's geometry.