Divisor Sheaf

Definition

A divisor sheaf is a specific type of sheaf on a scheme that encodes information about effective divisors and their associated properties, such as local functions that have poles or zeros along a divisor. It allows for the study of algebraic properties in a geometric setting by associating to each open set of the scheme a ring of functions that exhibit behavior related to the divisor. This concept is crucial in understanding how divisors can be interpreted within the language of sheaves and schemes.

5 Must Know Facts For Your Next Test

1. Divisor sheaves can be constructed from effective divisors by associating sections of functions that have prescribed poles and zeros.
2. The stalks of a divisor sheaf at points correspond to local rings of functions that vanish or are allowed to have poles along the divisor.
3. Divisor sheaves are closely related to the concept of Cartier divisors, which are defined locally in terms of the sheaf of regular functions.
4. The global sections of a divisor sheaf can give insight into the intersection theory and cohomological properties of the scheme.
5. Divisor sheaves play a significant role in the formulation of Riemann-Roch theorem in the context of schemes, linking geometry and algebra.

Review Questions

• How do divisor sheaves relate to effective divisors and what role do they play in the study of schemes?
  • Divisor sheaves serve as an algebraic tool that encodes information about effective divisors by associating sections of functions that reflect the behavior around poles and zeros specified by the divisor. They allow mathematicians to study these divisors within the framework of schemes, providing insights into their local and global properties. This relationship helps bridge the gap between algebraic geometry and the more general setting of schemes.
• Discuss the importance of stalks in understanding divisor sheaves and how they contribute to local properties.
  • The stalks of a divisor sheaf provide vital information about its local structure at points in the scheme. By examining these stalks, one can see how functions behave around specific points, particularly regarding their poles and zeros determined by the associated effective divisor. This local analysis is crucial for understanding how divisor sheaves interact with other geometric structures on schemes, contributing to broader concepts like cohomology and intersection theory.
• Evaluate how divisor sheaves facilitate the application of the Riemann-Roch theorem within the context of schemes.
  • Divisor sheaves are instrumental in applying the Riemann-Roch theorem to schemes by establishing a link between geometry and algebraic properties encoded in effective divisors. This connection enables one to calculate dimensions of spaces of sections of line bundles associated with divisors, allowing for conclusions about global properties from local data. The insights gained from this evaluation reveal deeper relationships between curves, surfaces, and their respective geometric characteristics.

Related terms

Effective Divisor: An effective divisor is a formal sum of irreducible closed subvarieties with non-negative integer coefficients, representing a divisor that can be thought of as having 'positive' contributions.

Locally Free Sheaf:

A locally free sheaf is a sheaf of modules that locally looks like a free module of finite rank, providing an algebraic structure that can be used to study vector bundles on schemes.

Rational Function: A rational function is a quotient of two polynomials, which may exhibit poles or zeros at certain points, making it essential for defining local properties related to divisors.