Brill–Noether theory

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 H¹ 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 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):

${\displaystyle \dim(d,r,g)\geq \rho =g-(r+1)(g-d+r)}$

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

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

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

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

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

• Eric Larson (2017) proved that if ρ ≥ 0, r ≥ 3, and n ≥ 1, the restriction maps ${\displaystyle H^{0}({\mathcal {O}}_{\mathbb {P} ^{r}}(n))\rightarrow H^{0}({\mathcal {O}}_{C}(n))}$ are of maximal rank, also known as the maximal rank conjecture.^[1][2]

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

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

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