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 $H 1$ 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 $H 0$ cohomology or space of holomorphic sections is larger than expected.

Alternatively, by Serre duality, the condition is that there exist holomorphic differentials with divisor $\geq - D$ 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 ρ = 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 $d \geq 1$ , $r \geq 0$ 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 $ρ \geq 0$ then ⁠ $G_{d}^{r}$ ⁠ is not empty, and every component has dimension at least $ρ$ .
William Fulton and Robert Lazarsfeld proved that if $ρ \geq 1$ 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 $ρ \geq 0$ , $r \geq 3$ , and $n \geq 1$ , the restriction maps $H^{0} (𝒪_{ℙ^{r}} (n)) \to H^{0} (𝒪_{C} (n))$ are of maximal rank, also known as the maximal rank conjecture.

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

Maximal rank conjecture

Let $C \subset$ $ℙ^{r}$ be a general curve of genus g embedded via a general linear series of degree d. The Maximal Rank Conjecture asserts that the restriction maps $H^{0} (𝒪_{ℙ^{r}} (m)) \to H^{0} (𝒪_{ℂ} (m))$ are of maximal rank; this determines the Hilbert function of $C$ .

Its first proof was published by Eric Larson on 14 Nov 2017 whilst a graduate student at MIT.

References

Barbon, Andrea (2014). Algebraic Brill–Noether Theory (PDF) (Master's thesis). Radboud University Nijmegen.
Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip 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.
Griffiths, Philip A.; Harris, Joe (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 [math.AG].
↑ 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 [math.AG].
↑ "Old Problem About Algebraic Curves Falls to Young Mathematicians". Quanta Magazine. 2022-08-25. Retrieved 2022-08-28.
↑ Larson, Eric (2020-08-01). "The Maximal Rank Conjecture for sections of curves". Journal of Algebra. 555: 223–245. arXiv:1208.2730. doi:10.1016/j.jalgebra.2020.03.006. ISSN 0021-8693.
↑ Larson, Eric (2018-09-18), The Maximal Rank Conjecture, arXiv:1711.04906
↑ "Old Problem About Algebraic Curves Falls to Young Mathematicians". Quanta Magazine. 2022-08-25. Retrieved 2025-03-07.

[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.

[5] Larson, Eric (2020-08-01). "The Maximal Rank Conjecture for sections of curves". Journal of Algebra. 555: 223–245. arXiv:1208.2730. doi:10.1016/j.jalgebra.2020.03.006. ISSN 0021-8693.

[6] Larson, Eric (2018-09-18), The Maximal Rank Conjecture, arXiv:1711.04906

[7] "Old Problem About Algebraic Curves Falls to Young Mathematicians". Quanta Magazine. 2022-08-25. Retrieved 2025-03-07.