Smoothness

Definition

Smoothness refers to a property of a space where it behaves nicely in terms of differentiability, meaning that it has no abrupt changes, singularities, or 'sharp points.' In algebraic geometry, smoothness implies that the variety is well-behaved at every point, allowing for the application of calculus and differential geometry concepts. This property is essential for understanding how varieties can be manipulated and transformed without encountering issues that arise from singular points.

5 Must Know Facts For Your Next Test

1. A variety is considered smooth if all of its local rings are regular, meaning they have dimension equal to the number of generators of their maximal ideal.
2. In terms of geometry, smooth varieties can be thought of as having a well-defined tangent space at every point, which is not true for singular varieties.
3. Blowing up is a technique used to resolve singularities and achieve smoothness by replacing a singular point with an entire projective space.
4. Toric varieties can exhibit different types of smoothness based on the combinatorial data of their fans; a toric variety is smooth if its associated fan is non-singular.
5. Understanding smoothness is crucial for many areas in algebraic geometry, including intersection theory, deformation theory, and birational geometry.

Review Questions

• What does it mean for a variety to be smooth, and how does this relate to the concept of regular and singular points?
  • For a variety to be smooth means that it has no singular points and behaves nicely under differentiation. A regular point is where the variety maintains this smooth property, while a singular point indicates failure in this regard. The presence of singular points can complicate the study and manipulation of the variety since these points disrupt the expected structure and properties typical of smooth varieties.
• How does blowing up serve as a method for achieving smoothness in algebraic geometry?
  • Blowing up transforms a singular point into a smooth structure by replacing the problematic point with an entire projective space. This method allows mathematicians to 'resolve' the singularity by creating a new variety where the former singularity has been replaced by additional structure. The resulting blow-up retains many properties of the original variety but eliminates issues caused by singular points, leading to a better understanding of the overall geometry.
• Evaluate the implications of smoothness on the study of toric varieties and their associated fans.
  • Smoothness in toric varieties relies heavily on the characteristics of their associated fans. A toric variety is smooth when its fan does not contain any singularities. This relationship highlights how combinatorial data influences geometric properties, showing that understanding fans can provide insights into more complex algebraic structures. Consequently, the study of smoothness in toric varieties opens avenues for exploring connections between algebraic geometry and combinatorics.

Related terms

Regular Point: A point on a variety where the local ring is regular, indicating that the variety is smooth at that point.

Singular Point: A point on a variety where the local ring is not regular, often associated with some form of failure in smoothness.

Resolution of Singularities: A process in algebraic geometry aimed at transforming a singular variety into a smooth one by creating a new variety that has no singular points.