Singular Locus

Definition

The singular locus of an algebraic variety is the set of points where the variety fails to be smooth, meaning that the local ring of functions at these points does not have a regular sequence. In other words, these points are where the variety exhibits 'singularities' or irregular behavior, which can affect the structure and properties of the variety significantly. Understanding the singular locus is crucial for studying the geometry and topology of algebraic varieties.

5 Must Know Facts For Your Next Test

1. The singular locus can be determined by examining the vanishing of certain partial derivatives of defining equations of the variety.
2. For curves, singular points typically correspond to intersections or cusps, while for surfaces, they can manifest as nodes or self-intersections.
3. The codimension of the singular locus gives insights into how 'badly' the variety behaves at those points relative to its overall dimension.
4. A variety may have a non-empty singular locus and still be irreducible, indicating that irregularities exist but do not decompose the variety into simpler components.
5. The singular locus can provide critical information for resolution processes in algebraic geometry, where one aims to replace a singular variety with a smooth one.

Review Questions

• How do you identify points in the singular locus of an algebraic variety, and what methods are commonly used?
  • To identify points in the singular locus, one typically examines the partial derivatives of the defining equations of the algebraic variety. If these derivatives vanish simultaneously at a point, that point is classified as singular. Common methods include using tools from differential geometry or algebraic geometry, such as Jacobians or Gröbner bases, to analyze when regular sequences fail to exist.
• Discuss the implications of having a non-empty singular locus on the geometric properties of an algebraic variety.
  • Having a non-empty singular locus can significantly affect the geometric properties of an algebraic variety. Singular points may lead to complications in understanding the topology and curvature of the variety. For example, they can hinder smooth parametrizations and affect intersections with other varieties. Moreover, various geometrical constructions and results may need modification or may not hold entirely in the presence of singularities.
• Evaluate how resolving singularities impacts the study of algebraic varieties and their classifications within algebraic geometry.
  • Resolving singularities is a key process in algebraic geometry that transforms a singular variety into a smooth one, allowing for better analysis and classification. This process often involves blowing up points in the singular locus to create new varieties that retain much of the original structure while eliminating irregularities. By doing so, mathematicians gain clearer insights into various properties such as dimension, morphisms, and intersection theory, thus facilitating more precise categorization within algebraic geometry.

Related terms

Smooth Point: A point on an algebraic variety where the local ring is regular, meaning that there exists a regular sequence of parameters corresponding to the dimension of the variety.

Regular Variety: An algebraic variety is regular if it has no singular points, indicating that it is locally similar to Euclidean space and has well-defined geometric properties.

Codimension: The difference between the dimension of an ambient space and the dimension of a subvariety, often used in the context of describing the singular locus within a larger variety.