Divisorial Contractions

Definition

Divisorial contractions are morphisms that contract certain divisors on a variety, typically allowing one to simplify the structure of a space by collapsing the image of these divisors. This concept is crucial in the classification of singularities, particularly when identifying canonical and terminal singularities, which involve understanding how these contractions affect the properties of varieties and their singular points.

5 Must Know Facts For Your Next Test

1. Divisorial contractions are often used in birational geometry to simplify the classification of varieties by removing problematic singularities.
2. In the context of canonical and terminal singularities, divisorial contractions help determine whether a variety has certain desirable properties regarding its singularities.
3. When performing a divisorial contraction, it's important to understand how the image of the contracted divisor behaves and what new singularities may arise.
4. The existence of a divisorial contraction can indicate that a variety is not minimal with respect to its canonical class, providing insight into its geometry.
5. These contractions are closely related to other birational transformations, which alter the underlying variety while preserving important geometric features.

Review Questions

• How do divisorial contractions influence the classification of canonical and terminal singularities?
  • Divisorial contractions play a key role in classifying canonical and terminal singularities by allowing mathematicians to simplify complex structures. By contracting specific divisors, one can analyze the resulting variety for its singular points. This process helps in determining if the singularities present are canonical or terminal, thus providing essential information about the variety's geometry and its behavior under deformation.
• Discuss how divisorial contractions relate to the resolution of singularities in algebraic geometry.
  • Divisorial contractions are an integral part of the resolution of singularities process. When attempting to resolve singularities, one often uses these contractions to modify the variety, reducing complexity by collapsing problematic divisors. This transformation allows for a clearer understanding of the geometry involved and facilitates moving toward a nonsingular model, which is crucial for various applications in algebraic geometry.
• Evaluate the implications of utilizing divisorial contractions in birational geometry and their effects on varieties' properties.
  • Using divisorial contractions in birational geometry has significant implications for understanding the properties of varieties. These contractions help establish connections between different varieties by showing how one can be transformed into another through specific morphisms. This not only aids in revealing deeper geometric insights but also influences classification theories by highlighting how certain varieties can be simplified or altered while preserving key features. As such, analyzing these transformations provides a powerful tool for researchers aiming to unravel complex algebraic structures.

Related terms

Divisor:

A divisor is a formal sum of irreducible subvarieties of codimension one, often used to study the properties of varieties and their intersection theory.

Singularity: A singularity refers to a point on a variety where it fails to be well-behaved, often characterized by certain algebraic conditions that cause irregular behavior.

Resolution of Singularities: The process of transforming a singular variety into a nonsingular one, often using divisorial contractions as a step in simplifying the structure.