---
tag: 82ZR6BCJXV
kind: concept
name: Brill–Noether theory
msc: [14H51, 14C20]
sources:
  - url: https://en.wikipedia.org/wiki/Brill%E2%80%93Noether_theory
same_as:
  wikidata: Q3032324
  wikipedia: Brill–Noether theory
---

## Summary

Brill–Noether theory studies the linear systems on an [[algebraic curve|XJM2MT4S3Y]].
For a given degree $d$ and dimension $r$, it asks which line bundles of degree $d$ have at least $r + 1$ independent sections.
The answer for a general curve of genus $g$ is governed by the Brill–Noether number $\rho = g - (r + 1)(g - d + r)$.
Such line bundles exist on every curve when $\rho \ge 0$, and they do not exist on a general curve when $\rho < 0$.
The theory organizes the maps of curves to projective spaces and is a central tool in the study of moduli of curves.

## Explanations

### L1 human

A curve can be mapped to a line, a plane or a higher projective space in many ways, and each map comes from a line bundle with enough sections.
Brill–Noether theory asks which maps of a given degree into a given space exist.
A simple count of conditions predicts the answer, and for a typical curve the prediction is right.
For instance, the count says that a typical curve of genus $3$ has no map of degree $2$ to the line, so it is not hyperelliptic.