Rational Singularities

Definition

Rational singularities are specific types of singular points in algebraic varieties where the local cohomology vanishes in a certain way, often making the singularity appear 'nice' or 'manageable.' These singularities allow for certain desirable properties, such as being rational over the base field, which helps in the study of depth, regular sequences, and Cohen-Macaulay rings.

5 Must Know Facts For Your Next Test

1. Rational singularities can be characterized by their vanishing local cohomology, meaning that they exhibit certain nice behavior from a topological perspective.
2. These singularities are often found in algebraic varieties that are defined over fields with characteristic zero, like the complex numbers.
3. One important property is that if a variety has rational singularities, it is often Cohen-Macaulay, thus linking these concepts together.
4. The existence of rational singularities implies that resolutions of these singularities can lead to well-behaved geometrical structures.
5. Rational singularities can be useful in applications such as deformation theory and mirror symmetry, as they simplify calculations involving singular spaces.

Review Questions

• How do rational singularities relate to the concept of depth in rings?
  • Rational singularities often imply that the underlying ring is Cohen-Macaulay, where the depth equals the dimension. This connection highlights that understanding depth can provide insights into the nature of singular points. Additionally, since rational singularities have vanishing local cohomology, they contribute to achieving a well-defined notion of depth within algebraic varieties.
• Discuss how regular sequences interact with rational singularities and their implications for algebraic geometry.
  • Regular sequences are critical in determining the structure of rings and their depth. In the context of rational singularities, if a ring associated with a variety has a regular sequence, it leads to a better understanding of its singularity behavior. Specifically, if the variety is Cohen-Macaulay and has rational singularities, regular sequences help define effective resolutions and manage its geometry effectively.
• Evaluate the significance of rational singularities in modern algebraic geometry and their applications beyond theoretical constructs.
  • Rational singularities have considerable importance in modern algebraic geometry due to their role in simplification and resolution techniques. They aid in the study of deformation theory and mirror symmetry, impacting how mathematicians understand complex structures. By linking these singular points to Cohen-Macaulay properties and local cohomology behaviors, researchers can derive concrete results applicable to both theoretical exploration and practical problems within algebraic varieties.

Related terms

Cohen-Macaulay Rings: A class of rings where the depth equals the Krull dimension, which plays a crucial role in understanding the structure of varieties and their singularities.

Regular Sequences: A sequence of elements in a ring that forms a system of parameters, allowing us to explore properties like depth and Cohen-Macaulayness.

Depth: A measure of the 'size' of a ring's ideal and its relation to the number of non-zero divisors, providing insight into the structure and singularities of algebraic varieties.