Divisor

Definition

A divisor is a formal mathematical object that represents a way to encode the idea of a 'point' or 'subvariety' on an algebraic variety, capturing how functions behave near those points. It provides a way to study the properties of algebraic varieties through their intersections and the associated function field. Understanding divisors is crucial for exploring rational maps, examining genus, and transitioning into modern algebraic geometry concepts like schemes.

5 Must Know Facts For Your Next Test

1. Divisors are expressed as formal sums of points with integer coefficients, allowing the treatment of multiplicities and intersections.
2. They can be categorized into effective divisors, which represent actual points on the variety, and principal divisors, arising from meromorphic functions.
3. The Riemann-Roch theorem links the divisor of a curve to its linear system, providing essential insights into the dimensions of spaces of sections of line bundles.
4. In birational geometry, divisors help establish relationships between different varieties, indicating when they can be considered equivalent in terms of function fields.
5. Divisors also play a key role in defining divisor class groups, which classify divisors up to linear equivalence and are important in understanding the geometry of varieties.

Review Questions

• How does the concept of divisors relate to rational maps and their behaviors on algebraic varieties?
  • Divisors provide a structured way to analyze rational maps by allowing us to study how these maps behave at different points on an algebraic variety. They encode information about where functions might have poles or zeros, revealing how rational maps interact with the structure of the variety. Understanding divisors helps identify points where rational maps are not defined and allows us to determine birational equivalences based on shared divisor properties.
• Discuss how the Riemann-Roch theorem incorporates divisors to inform about the genus of algebraic curves.
  • The Riemann-Roch theorem connects divisors directly to the genus by providing a formula that relates the dimension of spaces of meromorphic functions to the divisor class. The theorem shows how the presence of zeros and poles (captured through divisors) affects the overall structure of an algebraic curve. By analyzing these relationships, one can derive critical insights into the genus, which indicates how complex or simple a curve is.
• Evaluate how the introduction of schemes changes the understanding of divisors in modern algebraic geometry.
  • The introduction of schemes expands the traditional notion of divisors by allowing for more general types of spaces that incorporate both algebraic and topological ideas. In this context, divisors become more flexible and can represent not just points but also more complex substructures within schemes. This broader perspective enhances our ability to study varieties under various morphisms and reveals deeper connections between algebraic properties and geometric intuition, fundamentally changing how we approach divisor theory.

Related terms

Rational Map: A rational map is a function between varieties that is defined by rational functions, which can sometimes fail to be well-defined everywhere on the domain.

Genus: The genus is a topological invariant that classifies algebraic curves based on their number of holes, reflecting their complexity.

Scheme: A scheme is a fundamental concept in modern algebraic geometry, generalizing varieties to include more general spaces that can accommodate a wider range of geometric phenomena.