Smooth variety

Definition

A smooth variety is a type of algebraic variety that has no singular points, meaning it behaves nicely in terms of its geometric and algebraic properties. Smoothness ensures that at every point in the variety, the local structure resembles that of an affine space, which is crucial for various mathematical concepts like intersection theory and cohomology. This concept plays a vital role in understanding line bundles and their classifications, as well as in advanced results like the Grothendieck-Riemann-Roch theorem.

5 Must Know Facts For Your Next Test

1. Smooth varieties can be defined over arbitrary fields, including both algebraically closed fields and more general fields, extending their applicability.
2. The notion of smoothness is closely linked to the concept of regular functions; a variety is smooth if its defining equations have derivatives that do not vanish simultaneously at any point.
3. In terms of dimension, a smooth variety has well-defined dimensions at every point, making it easier to apply dimensionality arguments in algebraic geometry.
4. Smooth varieties play a crucial role in defining line bundles; every line bundle on a smooth projective variety can be understood in terms of divisor class groups.
5. The Grothendieck-Riemann-Roch theorem fundamentally relies on the smoothness of varieties, allowing one to compute important characteristics like Chern classes and pushforward maps.

Review Questions

• How does the definition of smooth varieties impact their role in intersection theory?
  • Smooth varieties provide a clear structure that allows for precise calculations in intersection theory. Since smooth varieties do not have singular points, intersections between such varieties can be analyzed using tools like dimension counting and transversality. This simplicity enables mathematicians to draw conclusions about how different varieties intersect without encountering complications arising from singularities.
• Discuss how the classification of line bundles on smooth varieties differs from those on singular varieties.
  • The classification of line bundles on smooth varieties is significantly more straightforward due to their well-behaved geometry. In smooth varieties, each line bundle corresponds to a divisor class, allowing for clear identification of line bundles via cohomological methods. In contrast, singular varieties often require more complex techniques because singularities can lead to complications in understanding divisors and their associated bundles, thus complicating classification efforts.
• Evaluate the significance of smoothness in the application of the Grothendieck-Riemann-Roch theorem and its implications for cohomological properties.
  • Smoothness is essential for applying the Grothendieck-Riemann-Roch theorem because it ensures that the necessary geometric conditions are met for computing Chern classes and pushforward operations. The theorem yields deep insights into the relationship between the geometry of a smooth variety and its cohomological properties. As a result, understanding how these properties manifest in smooth varieties provides valuable information for broader algebraic geometric contexts, facilitating advancements in both theoretical and practical aspects of the field.

Related terms

Singular Variety: A singular variety is an algebraic variety that contains points where the local structure fails to resemble an affine space, often leading to complications in studying its properties.

Cohomology:

Cohomology is a mathematical tool used to study the global properties of spaces by assigning algebraic invariants, helping understand features like the number of holes or connected components.

Tangent Space: The tangent space at a point on a variety consists of all possible directions in which one can move away from that point, providing insight into the local geometry and smoothness of the variety.