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 Template:Mvar 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 Template:Mvar 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 Template:Mvar. 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 Template:Mvar, the moduli space for curves Template:Mvar of genus Template:Mvar 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 Template:Mvar, as a function of Template:Mvar, 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 Template:Mvar, and the subset of Pic(C) corresponding to divisor classes of divisors Template:Mvar, with given values Template:Mvar of deg(D) and Template:Mvar of l(D) – 1 in the notation of the Riemann–Roch theorem. There is a lower bound Template:Mvar for the dimension dim(d, r, g) of this subscheme in Pic(C):

$\dim (d,r,g)\ge \rho =g-(r+1)(g-d+r)$

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 Template:Mvar and for d ≥ 1, r ≥ 0 the basic results about the space Template:Tmath of linear systems on Template:Mvar of degree Template:Mvar and dimension Template:Mvar are as follows.

• George Kempf proved that if ρ ≥ 0 then Template:Tmath is not empty, and every component has dimension at least Template:Mvar.
• William Fulton and Robert Lazarsfeld proved that if ρ ≥ 1 then Template:Tmath is connected.
• showed that if Template:Mvar is generic then Template:Tmath is reduced and all components have dimension exactly Template:Mvar (so in particular Template:Tmath is empty if ρ < 0).
• David Gieseker proved that if Template:Mvar is generic then Template:Tmath is smooth. By the connectedness result this implies it is irreducible if ρ > 0.

Other more recent results not necessarily in terms of space Template:Tmath of linear systems are:

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

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

Maximal rank conjecture

Let $C\subset$ ${\mathbb{\mathbb{P}}}^r$ 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 $H^0({{𝒪}}_{{\mathbb{\mathbb{P}}}^r}(m))\to H^0({{𝒪}}_{\mathbb{ℂ}}(m))$ 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

• Template:Cite thesis