In algebraic geometry, Brill–Noether theory, introduced by Alexander von Brill and Max Noether (1874), is the study of special divisors, certain divisors on a curve C that determine more compatible functions than would be predicted.
Throughout, we consider a projective smooth curve over the complex numbers (or over some other algebraically closed field).
Alternatively, by Serre duality, the condition is that there exist holomorphic differentials with divisor ≥ –D on the curve.
The basic statement can be formulated in terms of the Picard variety Pic(C) of a smooth curve C, and the subset of Pic(C) corresponding to divisor classes of divisors D, with given values d of deg(D) and r of l(D) – 1 in the notation of the Riemann–Roch theorem.
There is a lower bound ρ for the dimension dim(d, r, g) of this subscheme in Pic(C): called the Brill–Noether number.