Smoothness

Definition

Smoothness refers to a property of varieties where they do not have any singular points, meaning that locally around each point, the variety resembles an affine space. In simpler terms, a smooth variety has no 'sharp edges' or 'corners,' and it behaves nicely with respect to calculus concepts like derivatives. This concept is crucial when studying dimensions and degrees of varieties as well as analyzing singular points and tangent cones, since it helps in understanding the local structure and behavior of the variety.

5 Must Know Facts For Your Next Test

1. A variety is smooth if, at every point, the dimension of the tangent space equals the dimension of the variety itself.
2. If a variety has singular points, it is considered singular; this can lead to complications in determining its properties and behaviors.
3. The smoothness of a variety is essential for applying methods from differential geometry, including studying curves and surfaces.
4. In algebraic geometry, smooth varieties often exhibit better properties than singular ones, such as having well-defined intersection theories.
5. Determining whether a variety is smooth involves checking whether its defining equations have a full rank at each point.

Review Questions

• How does smoothness relate to the concept of singular points in algebraic geometry?
  • Smoothness is directly related to singular points in that a smooth variety must be free of singularities. A singular point is where the local structure fails to resemble an affine space, which means that at these points, derivatives do not behave regularly. Identifying singular points helps determine if a variety is smooth; if any singular points exist, the variety cannot be classified as smooth.
• Discuss how the concept of tangent cones is affected by the smoothness of a variety.
  • Tangent cones serve as approximations of a variety at specific points, particularly at singular points. For smooth varieties, the tangent cone at any point will be well-defined and will correspond neatly to the local behavior of the variety. However, at singular points, the tangent cone may provide incomplete information or be poorly defined, indicating that the local structure is complex and not well-behaved like that of smooth varieties.
• Evaluate the implications of smoothness on intersection theory within algebraic geometry.
  • Smoothness significantly impacts intersection theory because smooth varieties tend to have better-behaved intersections than singular varieties. When working with smooth varieties, one can apply techniques from differential geometry to compute intersection numbers accurately and study their properties. Conversely, if a variety has singular points, intersections might not yield clear results due to complications arising from these irregularities. Thus, understanding smoothness enhances our ability to study and predict behaviors in intersection theory.

Related terms

Singular Point:

A point on a variety where the local behavior does not resemble that of an affine space; often characterized by the failure of derivatives to provide a full rank.

Tangent Cone:

The geometric object that approximates the variety at a singular point, representing the limit of the secant lines approaching that point.

Differentiable Structure: The way in which a variety can be given a smooth structure, allowing for the application of calculus and differential geometry.