Divisor

Definition

A divisor is a formal mathematical object associated with algebraic varieties, representing a formal sum of codimension one subvarieties. It helps in understanding the structure of varieties by encoding information about their points, particularly in terms of their multiplicities and intersections. Divisors are crucial in the study of algebraic curves, surfaces, and their functions, especially when analyzing line bundles and the behavior of rational functions on these spaces.

5 Must Know Facts For Your Next Test

1. Divisors can be classified into effective divisors, which represent actual subvarieties, and Cartier divisors, which relate to line bundles and their sections.
2. On an algebraic curve, a divisor can be represented as a finite sum of points, where each point has an associated integer that indicates its multiplicity.
3. In the context of surfaces, divisors help in understanding intersections and can be used to compute intersection numbers, which are essential in various geometric and topological studies.
4. The Riemann-Roch theorem relates the dimensions of spaces of sections of line bundles associated with divisors, providing powerful tools for analyzing curves and surfaces.
5. Divisors play a key role in defining the notion of degree, which quantifies how many times a divisor intersects with an effective divisor or how many times a rational function vanishes.

Review Questions

• How do divisors help in understanding the geometry of algebraic curves?
  • Divisors help in understanding the geometry of algebraic curves by providing a way to represent points on the curve along with their multiplicities. They allow for the analysis of rational functions on the curve, indicating where these functions vanish or have poles. This representation aids in applying tools like the Riemann-Roch theorem to derive important information about function spaces on curves.
• Discuss the significance of effective divisors in the classification of algebraic surfaces.
  • Effective divisors are significant in classifying algebraic surfaces because they provide concrete information about the surface's structure through its points and their multiplicities. They allow mathematicians to study intersections between surfaces and understand how these intersections contribute to topological invariants. The use of effective divisors also leads to applications in intersection theory, enhancing our understanding of surface classifications.
• Evaluate how the Riemann-Roch theorem utilizes divisors to connect geometry with algebra in both curves and surfaces.
  • The Riemann-Roch theorem utilizes divisors to bridge geometry and algebra by relating the algebraic properties of line bundles to geometric features of curves and surfaces. By establishing relationships between divisors and spaces of sections, this theorem enables us to compute dimensions and understand how divisors affect function behaviors on varieties. This connection is vital for deriving results about genus, dimension counting, and other invariants that arise from studying curves and surfaces.

Related terms

Rational Function: A function defined by the ratio of two polynomials, which can be associated with divisors by determining where the function has zeros and poles.

Line Bundle: A geometric object that allows for the study of divisors by providing a way to understand sections of the space over which a divisor is defined.

Codimension: The difference between the dimension of a space and the dimension of a subspace, which is relevant in defining the properties of divisors on varieties.