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 "H"¹ 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 "H"⁰ 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 harvtxt|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.

References

*citation|id=MR|0563378
last=Griffiths|first= Phillip|last2= Harris|first2= Joseph
title=On the variety of special linear systems on a general algebraic curve.
journal=Duke Math. J.|volume= 47 |year=1980|issue= 1|pages= 233–272
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