Brill-Noether theory
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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.