# Brill–Noether theory

In the theory of algebraic curves, **Brill–Noether theory**, introduced by Brill and 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.

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.

## Main theorems of Brill–Noether theory

For given genus *g*, the moduli space for curves *C* of genus *g* should contain a dense subset parameterizing those curves with the minimum in the way of special divisors. One goal 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 basic statement can be formulated in terms of the Picard variety Pic(*C*) of a smooth curve *C*, and the subset of Pic(*C*) corresponding to divisor classes of divisors *D*, with given values *d* of deg(*D*) and *r* of *l*(*D*) in the notation of the Riemann–Roch theorem. There is a lower bound ρ for the dimension dim(*d*, *r*, *g*) of this subscheme in Pic(*C*):

- dim(
*d*,*r*,*g*) ≥ ρ = g − (r+1)(g − d+r)

called the **Brill–Noether number**.
For smooth curves *G* and for *d*≥1, *r*≥0 the basic results about the space *G**r**d* of linear systems on *C* of degree *d* and dimension *r* are as follows.

- Kempf proved that if ρ≥0 then
*G**r**d*is not empty, and every component has dimension at least ρ. - Fulton and Lazarsfeld proved that if ρ≥1 then
*G**r**d*is connected. - Griffiths & Harris (1980) showed that if
*C*is generic then*G**r**d*is reduced and all components have dimension exactly ρ (so in particular*G**r**d*is empty if ρ<0). - Gieseker proved that if
*C*is generic then*G**r**d*is smooth. By the connectedness result this implies it is irreducible if*ρ*> 0.

The problem formulation can be carried over into higher dimensions, and there is now a corresponding Brill–Noether theory for some classes of algebraic surfaces. Algebraic geometer Montserrat Teixidor i Bigas has written several papers about this topic, including "Brill–Noether Theory for stable vector bundles*; ^{[1]} "A Riemann Singularity Theorem for generalized Brill–Noether loci";^{[2]} "Brill–Noether theory for vector bundles of rank 2" ^{[3]} and "Brill–Noether theory for stable vector bundles".^{[4]}*

## References

- E. Arbarello; M. Cornalba; P.A. Griffiths; J. Harris (1985).
*Geometry of Algebraic Curves Volume I*. Grundlehren de mathematischen Wisenschaften 267. ISBN 0-387-90997-4. - Brill, Alexander von; Noether, Max (1874). "Ueber die algebraischen Functionen und ihre Anwendung in der Geometrie".
*Mathematische Annalen*.**7**(2): 269–316. doi:10.1007/BF02104804. JFM 06.0251.01. Retrieved 2009-08-22. - Griffiths, Phillip; Harris, Joseph (1980). "On the variety of special linear systems on a general algebraic curve.".
*Duke Math. J*.**47**(1): 233–272. doi:10.1215/s0012-7094-80-04717-1. MR 0563378. - P. Griffiths; J. Harris (1994).
*Principles of Algebraic Geometry*. Wiley Classics Library. Wiley Interscience. p. 245. ISBN 0-471-05059-8.