Brill–Noether theory
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 Alexander von Brill and Max Noether in Brill & Noether (1874). A rigorous proof was first given by Griffiths & Harris (1980). The basic statement can be formulated in terms of the Picard variety Pic(C), and the subset of Pic(C) corresponding to divisor classes of divisors D, with given values n of deg(D) and r of l(D) in the notation of the Riemann–Roch theorem. There is a lower bound for the dimension dim(n, r, g) of this subset in Pic(C) (which is a subscheme):
- dim(n, r, g) ≥ r(n − r + 1) − (r − 1)g
called the Riemann–Brill–Noether theorem. Modern work shows that the bound is tight, and indeed equality holds for curves with generic moduli.[1]
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.
Notes
- ^ "Algebraic curve", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
References
- E. Arbarello (1985). Geometry of Algebraic Curves Volume I. Grundlehren de mathematischen Wisenschaften 267. ISBN 0-387-90997-4.
{{cite book}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - 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.
{{cite journal}}: Invalid|ref=harv(help); More than one of|author2=and|last2=specified (help) - Griffiths, Phillip; Harris, Joseph (1980). "On the variety of special linear systems on a general algebraic curve". Duke Math. J. 47 (1): 233–272. MR 0563378.
{{cite journal}}: Invalid|ref=harv(help) - P. Griffiths (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 245. ISBN 0-471-05059-8.
{{cite book}}: Unknown parameter|coauthors=ignored (|author=suggested) (help) - Robin Hartshorne (1977). Algebraic Geometry. Graduate Texts in Mathematics. Vol. 52. New York: Springer-Verlag. p. 345. ISBN 0-387-90244-9.