---
tag: Z9DN23DAE2
kind: concept
name: Weil divisor
msc: [14C20]
sources:
  - url: https://en.wikipedia.org/wiki/Divisor_(algebraic_geometry)
---

## Summary

A Weil divisor on a suitable scheme $X$ is a formal sum with integer coefficients of finitely many closed subvarieties of codimension one.
The zeros and poles of a rational function, counted with multiplicity, form a principal Weil divisor.
The quotient of all Weil divisors by the principal ones is the divisor class group, which equals the Picard group on smooth varieties.

## Explanations

### L1 human

On a curve, a divisor is a finite list of points with integer weights, such as $2P - 3Q$.
On a surface, the points are replaced by curves, and in general by subvarieties of codimension one.
A rational function gives a divisor: its zeros with positive weight and its poles with negative weight.
For example, on the projective line, the function $x^2/(x - 1)$ has divisor $2[0] - [1] - [\infty]$.