Nef Canonical Divisor

Definition

A nef canonical divisor is a divisor that is numerically effective, meaning it does not intersect negatively with any curves in the variety. It plays a crucial role in the classification of varieties and in understanding their geometric properties. In particular, a nef canonical divisor can help determine whether a variety has certain desirable features, such as being Fano or Kähler, and it is closely connected to the study of singularities, particularly canonical and terminal ones.

5 Must Know Facts For Your Next Test

1. A nef canonical divisor is important for understanding the geometry of a variety and its birational properties.
2. If a canonical divisor is nef, it suggests that the variety may have nice properties like being Kähler or having good minimal models.
3. The existence of a nef canonical divisor can prevent certain types of singularities from occurring in a variety.
4. Nefness can be tested using intersection theory, specifically by checking its behavior against curves in the variety.
5. In the context of minimal model theory, nef canonical divisors play a critical role in determining whether one can construct a minimal model from a given variety.

Review Questions

• How does the concept of a nef canonical divisor relate to the classification of varieties?
  • The nef canonical divisor is integral to the classification of varieties because it provides insight into their geometric structure and properties. A variety with a nef canonical divisor may exhibit favorable characteristics such as being Kähler or having an ample divisor. This classification helps researchers understand the possible morphisms between varieties and their respective behaviors under deformation.
• Discuss the significance of nefness in determining the presence of singularities in algebraic varieties.
  • Nefness is crucial in analyzing singularities because if a canonical divisor is nef, it can imply that certain undesirable singularities, such as non-terminal ones, are absent. This is significant since terminal singularities lead to favorable outcomes in minimal model theory. By studying the intersection properties of the canonical divisor with curves, mathematicians can derive conclusions about the singularities present on the variety and their implications for its overall structure.
• Evaluate the impact of nef canonical divisors on minimal model theory and its applications in algebraic geometry.
  • Nef canonical divisors have a profound impact on minimal model theory by guiding researchers in constructing minimal models from given varieties. When a canonical divisor is identified as nef, it often allows for the successful resolution of singularities and leads to better understanding and classification of varieties. The applications extend to various fields within algebraic geometry, including studying Fano varieties and understanding how different structures relate through birational transformations, ultimately enhancing our grasp of geometric properties.

Related terms

Numerically Effective (Nef): A divisor is numerically effective if it intersects non-negatively with every curve on the variety.

Fano Variety: A Fano variety is a type of algebraic variety with a positive first Chern class, indicating the existence of ample divisors.

Terminal Singularities:

Terminal singularities are a class of singularities that satisfy specific positivity conditions and are important for minimal model theory.