Special divisor

In mathematics, in the theory of algebraic curves, certain divisors on a curve C are particular, in the sense of determining more compatible functions than would be predicted. These are the special divisors. In classical language, they move on the curve in a larger linear system of divisors.

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 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 larger than expected.

Alternatively, by Serre duality, the condition is that there exist holomorphic differentials with divisor ≥ −D on the curve.

Brill-Noether theory

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 was stated by the German geometers Ludwig Brill and Max Noether in 1879. A rigorous proof was first given by Griffiths & Harris (1980).

These formulations can be carried over into higher dimensions, and there is now a corresponding Brill-Noether theory for some classes of algebraic surfaces.


* Griffiths, Phillip & Harris, Joseph (1980), "On the variety of special linear systems on a general algebraic curve.", Duke Math. J. 47 (1): 233--272, MR0563378

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