Cartier Divisor

Definition

A Cartier divisor on a scheme is a formal sum of codimension one subvarieties that can be locally represented by a global section of the sheaf of rational functions. This concept plays a crucial role in the study of algebraic varieties, as it allows us to translate geometric information into algebraic terms. Cartier divisors can be thought of as generalizations of classical divisors, and they help us understand the behavior of rational functions and their poles on varieties.

5 Must Know Facts For Your Next Test

1. Cartier divisors provide a way to define line bundles on algebraic varieties, linking algebraic geometry with topology.
2. They are locally defined by the vanishing of sections in the sheaf of regular functions, which relates them closely to the geometry of the underlying space.
3. Every Weil divisor can be represented as a Cartier divisor on a smooth variety, but this is not true in general for singular varieties.
4. Cartier divisors can be classified into effective and non-effective types based on the positivity of their coefficients.
5. The concept of Cartier divisors is essential for defining intersections and calculating intersection numbers in algebraic geometry.

Review Questions

• Compare and contrast Cartier divisors and Weil divisors in terms of their definitions and applications in algebraic geometry.
  • Cartier divisors and Weil divisors both represent formal sums of codimension one subvarieties, but they differ in their approach. Cartier divisors are defined using local sections of sheaves, allowing for a more algebraic perspective, while Weil divisors focus on linear equivalence classes from a geometric standpoint. In practice, Cartier divisors are particularly useful for defining line bundles on smooth varieties, whereas Weil divisors provide insights into the overall structure of the variety.
• Discuss how Cartier divisors facilitate the understanding of rational functions and their poles within algebraic varieties.
  • Cartier divisors serve as a bridge between geometry and algebra by allowing us to study rational functions through their associated divisors. Since Cartier divisors can be represented by local sections of sheaves, we can express the poles and zeros of rational functions in terms of these divisors. This connection enables mathematicians to apply tools from algebra to analyze geometric properties such as intersections and singularities, making Cartier divisors invaluable for understanding how rational functions behave on varieties.
• Evaluate the importance of effective Cartier divisors in relation to line bundles and intersection theory within algebraic geometry.
  • Effective Cartier divisors play a significant role in algebraic geometry because they represent actual subvarieties rather than just formal sums. Their non-negative coefficients correspond to geometric objects that can be used to define line bundles on varieties. This connection is crucial for intersection theory, where effective Cartier divisors help calculate intersection numbers and understand how different subvarieties interact. By studying effective Cartier divisors, mathematicians gain insights into both the geometric structure and algebraic properties of varieties.

Related terms

Weil Divisor: A Weil divisor is a formal sum of codimension one subvarieties that represents a linear equivalence class on a variety, focusing on the structure of divisors in a more geometric sense.

Rational Function: A rational function is a function defined by the ratio of two polynomial functions, which plays a key role in the study of divisors and their associated sheaves.

Effective Cartier Divisor: An effective Cartier divisor is a Cartier divisor whose coefficients are non-negative integers, representing an actual subvariety rather than just a formal sum.