Singular Point

Definition

A singular point on a variety is a point where the variety fails to be 'smooth' or 'regular,' meaning that it does not have a well-defined tangent space. This concept is crucial in understanding the behavior of varieties at these points, as they can exhibit unusual geometrical properties and affect the overall structure of the variety. Singular points are often identified using tangent cones and play a significant role in classifying varieties as normal or Cohen-Macaulay.

5 Must Know Facts For Your Next Test

1. A singular point can be characterized by the failure of the Jacobian matrix to have full rank at that point.
2. Not all varieties have singular points; smooth varieties are entirely free of singularities.
3. The study of singular points is vital in resolving singularities, which helps to produce smoother models of the variety.
4. The presence of singular points can affect various algebraic properties of the variety, such as its dimension and intersection theory.
5. In the context of Cohen-Macaulay varieties, singular points can indicate a lack of depth in the local rings, complicating certain algebraic operations.

Review Questions

• How does a singular point relate to the concept of tangent cones, and what role do they play in understanding varieties?
  • A singular point on a variety can be analyzed using the tangent cone at that point. The tangent cone provides a way to visualize how the variety behaves near the singular point. If the tangent cone does not adequately represent the local structure of the variety due to a lack of smoothness, it highlights the presence of a singularity. This relationship is essential for classifying varieties and studying their geometric properties.
• Discuss how singular points can affect whether a variety is classified as normal or Cohen-Macaulay.
  • Singular points directly influence whether a variety is considered normal or Cohen-Macaulay. In normal varieties, local rings should be integrally closed, which means they cannot have singular points that disrupt this condition. On the other hand, Cohen-Macaulay varieties may allow for certain types of singularities but must still satisfy specific depth criteria. The nature and behavior of these singular points provide insight into the overall algebraic structure of the varieties involved.
• Evaluate the implications of having singular points on a variety's geometric properties and algebraic structure.
  • Having singular points on a variety has significant implications for both its geometric properties and algebraic structure. Geometrically, these points may lead to unexpected behavior such as non-trivial topology or failure to maintain local homogeneity. Algebraically, they can complicate issues like dimension, intersection theory, and depth in local rings, affecting how we perform computations or study intersections with other varieties. Understanding these implications is crucial for developing a comprehensive grasp of how varieties function in Algebraic Geometry.

Related terms

Tangent Cone:

The tangent cone at a point on a variety provides a geometric approximation of the variety near that point, helping to analyze its smoothness or singularity.

Normal Variety: A normal variety is one where every local ring is integrally closed, meaning there are no singular points that disrupt the overall regularity of the variety.

Cohen-Macaulay Variety: A Cohen-Macaulay variety has desirable depth properties and behaves well under intersection with other varieties, which often influences the nature of its singular points.