Brill–Noether theory
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. In classical language, special divisors move on the curve in a "larger than expected" linear system of divisors.
Throughout, we consider a projective smooth curve over the complex numbers (or over some other algebraically closed field).
The condition to be a special divisor D can be formulated in sheaf cohomology terms, as the non-vanishing of the H1 cohomology of the sheaf of sections of the invertible sheaf or line bundle associated to D. This means that, by the Riemann–Roch theorem, the H0 cohomology or space of holomorphic sections is larger than expected.
Alternatively, by Serre duality, the condition is that there exist holomorphic differentials with divisor ≥ –D on the curve.