Brill–Noether theory

In the theory of algebraic curves, Brill–Noether theory, introduced by Alexander von Brill and Max 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 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.

Main theorems of Brill–Noether theory

For a 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) − 1 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 C 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.

• George Kempf proved that if ρ≥0 then G^r
  _d is not empty, and every component has dimension at least ρ.
• William Fulton and Robert 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).
• David 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.

References

• Barbon, Andrea (2014). Algebraic Brill–Noether Theory (PDF) (Master's thesis). Radboud University Nijmegen.
• Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Philip A.; Harris, Joe (1985). "The Basic Results of the Brill–Noether Theory". Geometry of Algebraic Curves. Grundlehren der Mathematischen Wissenschaften 267. Volume I. pp. 203–224. doi:10.1007/978-1-4757-5323-3_5. ISBN 0-387-90997-4.
• von Brill, Alexander; 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 Mathematical Journal. 47 (1): 233–272. doi:10.1215/s0012-7094-80-04717-1. MR 0563378.
• Philip A. Griffiths; Joe Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 245. ISBN 978-0-471-05059-9.