Smooth point

Definition

A smooth point on a variety is a point where the local structure of the variety behaves nicely, specifically where the tangent space has the expected dimension. At a smooth point, the variety does not exhibit any singular behavior, which means that it locally resembles an affine space and allows for well-defined tangent vectors. Understanding smooth points is crucial as they are essential in defining properties like normality and Cohen-Macaulayness of varieties.

5 Must Know Facts For Your Next Test

1. A variety is considered smooth at a point if the Jacobian matrix of defining equations has full rank at that point.
2. The concept of smooth points is closely linked to differentiability in calculus; just like differentiable functions have well-defined tangents, smooth varieties have well-defined tangent spaces.
3. In the context of normal varieties, having smooth points is important because singularities can lead to complications in defining geometric and algebraic properties.
4. Smooth points are typically dense in irreducible components of varieties, meaning they are often found in abundance relative to singular points.
5. Cohen-Macaulay varieties often exhibit smooth points, as their geometric properties can help ensure favorable algebraic conditions.

Review Questions

• How does the definition of a smooth point relate to the concepts of tangent spaces and singular points?
  • A smooth point is characterized by having a well-defined tangent space, which indicates that the local geometry around that point resembles Euclidean space. In contrast, a singular point lacks this property, meaning that its tangent space may not have the expected dimension or may not exist at all. This relationship highlights how understanding smooth points helps differentiate between regular and irregular behavior in algebraic varieties.
• Discuss the implications of having smooth points on the normality of a variety.
  • Having smooth points contributes significantly to the normality of a variety. A normal variety has no singular points in its structure, allowing for better algebraic manipulation and analysis. Smooth points imply that locally around those points, the structure behaves predictably, which helps ensure that all local rings of the variety meet the required conditions for normality. Thus, the presence of smooth points serves as an indicator of a variety's overall regularity.
• Evaluate how the presence of smooth points affects the Cohen-Macaulay property of a variety and what this means for its applications in algebraic geometry.
  • The presence of smooth points often enhances the Cohen-Macaulay property of a variety by ensuring that its local rings satisfy certain depth conditions. Varieties with abundant smooth points tend to maintain favorable relationships between their dimensions and co-dimensions, which is central to achieving Cohen-Macaulay status. This property is essential in algebraic geometry as it ensures good behavior under various transformations and provides a robust framework for understanding intersections, resolutions, and other geometric properties.

Related terms

Singular point: A point on a variety where the local structure fails to be smooth, often characterized by a lack of well-defined tangent space.

Tangent space: The vector space that approximates the variety at a point, representing directions in which one can move away from that point.

Cohen-Macaulay variety: A type of algebraic variety that has desirable depth properties, indicating that the local rings have dimension equal to their Krull dimension.