Ample Line Bundle

Definition

An ample line bundle is a line bundle that has the property that some positive tensor power of it can embed the variety into projective space. This means that an ample line bundle can be thought of as providing enough global sections to allow for a geometric realization of the variety in a higher-dimensional projective space, which connects to key ideas such as divisors, sheaves, and the geometry of varieties.

5 Must Know Facts For Your Next Test

1. An ample line bundle can be used to give an embedding of a variety into projective space, making it a fundamental concept in algebraic geometry.
2. If a line bundle is ample, then it has enough sections so that for some positive integer $m$, the sections of the bundle $L^m$ generate a map into projective space.
3. The property of being ample is related to positivity; specifically, it implies that the corresponding divisor associated with the line bundle is effective.
4. Ample line bundles play a crucial role in defining and understanding the notion of birational geometry and morphisms between varieties.
5. The Nakai-Moishezon criterion provides a practical way to check if a line bundle is ample by examining its intersection numbers with curves on the variety.

Review Questions

• How does an ample line bundle facilitate the embedding of a variety into projective space?
  • An ample line bundle allows for an embedding into projective space by ensuring that there exists some positive integer $m$ such that the global sections of the tensor power $L^m$ generate enough maps to projective coordinates. This is essential because it means we can visualize and study the geometric properties of the variety in a more manageable context, leveraging tools from projective geometry.
• Discuss how the property of ampleness relates to the positivity of divisors on algebraic varieties.
  • Ampleness directly relates to positivity through the associated divisor of the line bundle. If a line bundle is ample, its corresponding divisor is effective, meaning it can be represented as a sum of subvarieties with non-negative coefficients. This connection indicates that ample line bundles not only influence embeddings but also provide insights into the structure and properties of divisors within algebraic geometry.
• Evaluate the importance of the Nakai-Moishezon criterion in determining whether a line bundle is ample and how this affects the study of algebraic varieties.
  • The Nakai-Moishezon criterion is significant because it offers a clear method for verifying whether a line bundle is ample by looking at its intersection numbers with curves on the variety. This tool allows mathematicians to make conclusions about the geometric properties of varieties without needing explicit embeddings. The ability to determine ampleness helps in classifying varieties and understanding their behavior under various morphisms, playing a critical role in advanced studies of algebraic geometry.

Related terms

Line Bundle:

A line bundle is a vector bundle of rank one over a topological space, typically used to study divisors and sheaves on algebraic varieties.

Projective Space:

Projective space is a mathematical space that extends the concept of usual Euclidean space by adding 'points at infinity,' allowing for the study of properties invariant under projection.

Divisor:

A divisor is a formal sum of codimension one subvarieties, which serves as a way to generalize the notion of functions on algebraic varieties.