Ample divisor

Definition

An ample divisor is a type of divisor on a projective variety that ensures the positivity necessary for certain geometric properties, particularly in the context of embedding the variety into projective space. Ample divisors are essential for the study of minimal models and birational geometry because they help establish the existence of embeddings and play a crucial role in understanding the properties of varieties under morphisms.

5 Must Know Facts For Your Next Test

1. An ample divisor $D$ on a projective variety $X$ allows for the embedding of $X$ into projective space, meaning that the global sections of its associated line bundle can provide sufficient information about $X$.
2. The positivity condition of an ample divisor ensures that it intersects other divisors positively, helping to create a rich geometric structure within the variety.
3. Ample divisors can be characterized by their ability to yield enough global sections when raised to high powers, leading to significant insights about the geometry of the underlying variety.
4. In birational geometry, ample divisors are used to define minimal models, where one seeks to simplify varieties by contracting or resolving certain features without losing essential properties.
5. The theory surrounding ample divisors also connects to the cone of curves, where they play a critical role in understanding how curves on varieties interact with their associated linear systems.

Review Questions

• How do ample divisors facilitate embeddings of varieties into projective space?
  • Ample divisors are fundamental in ensuring that projective varieties can be embedded into projective space by providing sufficient global sections from their associated line bundles. When a divisor is ample, it guarantees that when you take high enough powers of this divisor, the resulting sections can be used to construct a map from the variety to projective space. This ability to produce embeddings is crucial for studying various geometric properties and classifications within algebraic geometry.
• Discuss the role of ample divisors in minimal models and birational geometry.
  • In minimal models and birational geometry, ample divisors serve as essential tools for understanding how varieties can be simplified while preserving their core geometric features. These divisors help establish which morphisms can be considered 'nice' or favorable by ensuring positivity and enabling contractions or resolutions. The presence of ample divisors indicates directions in which one can modify or simplify a variety without losing important information about its structure, making them integral to the development of minimal models.
• Evaluate how the concept of Kodaira dimension relates to ample divisors and the classification of varieties.
  • Kodaira dimension is intricately linked to ample divisors as it measures how the global sections of powers of these divisors grow, providing insight into the geometric behavior of varieties. A variety's Kodaira dimension helps classify its complexity based on its growth patterns associated with ample line bundles. By understanding these relationships, one can derive significant conclusions about the variety's structure and its place within broader classification schemes, illustrating how ample divisors contribute to our comprehension of algebraic varieties and their geometric properties.

Related terms

divisor: A divisor is a formal sum of subvarieties of codimension one on a variety, which provides a way to describe how functions behave with respect to these subvarieties.

line bundle: A line bundle is a vector bundle of rank one, associated with a divisor, which allows for the interpretation of divisors in terms of sections that can be defined globally or locally.

Kodaira dimension: Kodaira dimension is an invariant that measures the growth of the space of global sections of powers of an ample line bundle, playing a key role in classifying varieties.