Brill–Noether theory

Summary

Brill–Noether theory studies the linear systems on an algebraic curve. 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.