Smooth Variety

Definition

A smooth variety is a type of algebraic variety where all the points have a well-defined tangent space, meaning there are no singular points. This characteristic ensures that the variety behaves nicely in terms of geometry and algebra, allowing for important operations such as intersections and blowing up to proceed smoothly. Smooth varieties are particularly significant because they can be used to resolve singularities, leading to deeper insights in projective geometry.

5 Must Know Facts For Your Next Test

1. Smooth varieties have a dimension equal to the dimension of their tangent space at every point, ensuring no abrupt changes in structure.
2. In algebraic geometry, smooth varieties are important because they allow for the application of tools like intersection theory without complications from singularities.
3. When blowing up a singular variety, one often replaces singular points with smooth varieties, making them easier to study and analyze.
4. Smooth varieties can often be classified into different types based on their geometric and topological properties, such as being affine or projective.
5. The presence of smooth varieties in a family of varieties indicates that the family behaves well under deformation, meaning small changes lead to small changes in structure.

Review Questions

• How does the concept of a smooth variety relate to the process of blowing up and resolving singularities?
  • The concept of a smooth variety is crucial when discussing blowing up and resolving singularities because smooth varieties provide a well-behaved framework for these processes. When we blow up a singular variety, we typically replace points of singularity with smooth varieties, allowing us to eliminate problematic behavior at those points. This transformation helps to simplify the geometry and allows us to apply various mathematical techniques that rely on smoothness.
• Discuss the properties that distinguish smooth varieties from singular varieties and why these properties are essential in algebraic geometry.
  • Smooth varieties are distinguished from singular varieties primarily by their tangent spaces, which are well-defined at all points in a smooth variety. This smoothness allows for better control over geometric operations like intersection and allows for the application of more sophisticated mathematical tools. In contrast, singular varieties can exhibit irregular behavior that complicates these operations, making it difficult to study their structure and properties. The regularity provided by smooth varieties is essential for developing theories in algebraic geometry that are applicable across various contexts.
• Evaluate the implications of working with smooth varieties in the context of projective geometry and how this affects our understanding of algebraic structures.
  • Working with smooth varieties in projective geometry has profound implications for our understanding of algebraic structures. Smooth varieties allow mathematicians to leverage powerful techniques from differential geometry while studying properties like dimension and cohomology. These insights lead to clearer classifications of algebraic structures and facilitate the resolution of complex problems related to singularities. Furthermore, the ability to connect local properties of smooth varieties to global features through techniques like deformation theory provides a comprehensive framework for exploring relationships among various types of algebraic objects.

Related terms

Singular Variety: An algebraic variety that contains points where the tangent space fails to be well-defined, resulting in irregular or 'singular' behavior.

Blow-Up: A geometric transformation that replaces a point (often a singularity) on a variety with a projective space, effectively 'resolving' the singularity.

Projective Variety:

An algebraic variety that is defined as a subset of projective space, often used to study properties that are invariant under projective transformations.