In mathematics, the Brill-Noether theory in algebraic geometry is the theory of special divisors on generic algebraic curves. It is of interest mainly in the case of genus g ≥ 3. In conceptual terms, for g given, the moduli space for curves of genus g should contain an open, dense subset parametrizing those curves with the minimum in the way of special divisors. The point of the theory is to 'count constants', for those curves: to predict the dimension of the space of special divisors (up to linear equivalence) of a given degree d, as a function of g, that must be present on a curve of that genus.

The theory is named for the German geometers Ludwig Brill and Max Noether . The results were given in nineteenth century style; the whole theory was updated and modern proofs given by Phillip Griffiths and others.

The condition to be a special divisor D can be formulated in sheaf theory terms, as the vanishing of the H¹ cohomology of the sheaf of the 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 as large as possible (there is the minimum obstruction to taking a section). Alternatively, by Serre duality, the condition is that no holomorphic differential with divisor ≥ −D exists on the curve. These formulations can be carried over into higher dimensions, and there is now a corresponding Brill-Noether theory for some cases of algebraic surfaces.