# 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 ≥ –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)}.