Brill–Noether theory - Misplaced Pages

(Redirected from Brill–Noether theorem) Field of algebraic geometry

In algebraic geometry, 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.

Throughout, we consider a projective smooth curve over the complex numbers (or over some other algebraically closed field).

The condition to be a special divisor D can be formulated in sheaf cohomology terms, as the non-vanishing of the H cohomology of the sheaf of sections of the invertible sheaf or line bundle associated to D. This means that, by the Riemann–Roch theorem, the H 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)\geq \rho =g-(r+1)(g-d+r)

called the Brill–Noether number. The formula can be memorized via the mnemonic (using our desired $h^{0}(D)=r+1$ and Riemann-Roch)

g-(r+1)(g-d+r)=g-h^{0}(D)h^{1}(D)

For smooth curves C and for d ≥ 1, r ≥ 0 the basic results about the space ⁠ $G_{d}^{r}$ ⁠ of linear systems on C of degree d and dimension r are as follows.

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

Other more recent results not necessarily in terms of space ⁠ $G_{d}^{r}$ ⁠ of linear systems are:

Eric Larson (2017) proved that if ρ ≥ 0, r ≥ 3, and n ≥ 1, the restriction maps $H^{0}({\mathcal {O}}_{\mathbb {P} ^{r}}(n))\rightarrow H^{0}({\mathcal {O}}_{C}(n))$ are of maximal rank, also known as the maximal rank conjecture.

Eric Larson and Isabel Vogt (2022) proved that if ρ ≥ 0 then there is a curve C interpolating through n general points in ⁠ $\mathbb {P} ^{r}$ ⁠ if and only if $(r-1)n\leq (r+1)d-(r-3)(g-1),$ except in 4 exceptional cases: (d, g, r) ∈ {(5,2,3),(6,4,3),(7,2,5),(10,6,5)}.

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. Vol. 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. S2CID 120777748. 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.
Eduardo Casas-Alvero (2019). Algebraic Curves, the Brill and Noether way. Universitext. Springer. ISBN 9783030290153.
Philip A. Griffiths; Joe Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 245. ISBN 978-0-471-05059-9.

Notes

Larson, Eric (2018-09-18). "The Maximal Rank Conjecture". arXiv:1711.04906 .
Hartnett, Kevin (2018-09-05). "Tinkertoy Models Produce New Geometric Insights". Quanta Magazine. Retrieved 2022-08-28.
Larson, Eric; Vogt, Isabel (2022-05-05). "Interpolation for Brill--Noether curves". arXiv:2201.09445 .
"Old Problem About Algebraic Curves Falls to Young Mathematicians". Quanta Magazine. 2022-08-25. Retrieved 2022-08-28.

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Divisors on curves	Abel–Jacobi map Brill–Noether theory Clifford's theorem on special divisors Gonality of an algebraic curve Jacobian variety Riemann–Roch theorem Weierstrass point Weil reciprocity law
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