What is the relationship between divisors and line bundles?

On a smooth variety, divisors up to linear equivalence correspond exactly to isomorphism classes of line bundles; a divisor's class in the Picard group is the line bundle whose sections vanish along that divisor.

What does the Riemann-Roch theorem tell you?

For a divisor on a smooth projective curve, it gives the dimension of the space of rational functions with poles bounded by the divisor in terms of the divisor's degree and the curve's genus, a fundamental counting result.

Divisors and Riemann-Roch

Divisors record the zeros and poles of functions on a variety, line bundles package them geometrically, and the Riemann-Roch theorem counts the functions with prescribed pole behavior in terms of geometric invariants.

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Definition

A divisor on a variety is a formal combination of codimension-one subvarieties encoding zeros and poles; line bundles are their geometric counterparts, and the Riemann-Roch theorem relates the dimension of the space of sections of a divisor to its degree, the genus, and the canonical divisor.

Scope

This topic develops Weil and Cartier divisors, linear equivalence, the divisor class group and the Picard group, and the correspondence between divisors and line bundles (invertible sheaves). It treats linear systems and the maps to projective space they define, the canonical divisor, and the genus of a curve, culminating in the Riemann-Roch theorem for curves and the role of Serre duality. Higher-dimensional and Grothendieck-Hirzebruch generalizations are indicated as the natural extension.

Core questions

How do Weil and Cartier divisors encode the zero and pole behavior of rational functions?
Why are divisors up to linear equivalence the same data as line bundles?
How do linear systems determine maps from a variety to projective space?
What does the Riemann-Roch theorem compute, and how does Serre duality enter?

Key concepts

Weil and Cartier divisors; linear equivalence
Divisor class group and the Picard group
Line bundles (invertible sheaves) and linear systems
Canonical divisor and genus of a curve
Riemann-Roch theorem and Serre duality

Clinical relevance

Divisors and Riemann-Roch are the computational heart of the theory of curves and underlie the construction of error-correcting Goppa codes, the arithmetic of elliptic curves, and the classification of algebraic surfaces and higher-dimensional varieties.

History

Riemann's inequality on the dimension of function spaces (1857) was completed by his student Roch into the Riemann-Roch theorem; Hirzebruch's mid-20th-century generalization and Grothendieck's relative version embedded it in modern cohomological algebraic geometry.

Key figures

Bernhard Riemann
Gustav Roch
Friedrich Hirzebruch

Seminal works

hartshorne1977
eisenbud1995

Frequently asked questions

What is the relationship between divisors and line bundles?: On a smooth variety, divisors up to linear equivalence correspond exactly to isomorphism classes of line bundles; a divisor's class in the Picard group is the line bundle whose sections vanish along that divisor.
What does the Riemann-Roch theorem tell you?: For a divisor on a smooth projective curve, it gives the dimension of the space of rational functions with poles bounded by the divisor in terms of the divisor's degree and the curve's genus, a fundamental counting result.