Brill–Noether theory

In algebraic geometry, Brill–Noether theory, introduced by, 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 $≥ –D$ 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 $G^r_d$ of linear systems on $C$ of degree $d$ and dimension $r$ are as follows. Other more recent results not necessarily in terms of space $G^r_d$ of linear systems are:
 * George Kempf proved that if $ρ ≥ 0$ then $ρ$ is not empty, and every component has dimension at least $G^r_d$.
 * William Fulton and Robert Lazarsfeld proved that if $ρ ≥ 1$ then $C$ is connected.
 * showed that if $G^r_d$ is generic then $ρ$ is reduced and all components have dimension exactly $G^r_d$ (so in particular $C$ is empty if $ρ < 0$).
 * David Gieseker proved that if $G^r_d$ is generic then $G^r_d$ is smooth. By the connectedness result this implies it is irreducible if $ρ > 0$.


 * 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: $(d, g, r) ∈ {(5,2,3),(6,4,3),(7,2,5),(10,6,5)}.$