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 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) called the Brill–Noether number. The formula can be memorized via the mnemonic (using our desired and Riemann-Roch) 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. ρ ≥ 0 then G^r_d is not empty, and every component has dimension at least ρ. ρ ≥ 1 then G^r_d is connected. ρ < 0 ). ρ > 0 . Other more recent results not necessarily in terms of space G^r_d of linear systems are: ρ ≥ 0 , r ≥ 3 , and n ≥ 1 , the restriction maps are of maximal rank, also known as the maximal rank conjecture. ρ ≥ 0 then there is a curve C interpolating through n general points in \mathbb{P}^r if and only if except in 4 exceptional cases: (d, g, r) ∈ {(5,2,3),(6,4,3),(7,2,5),(10,6,5)}.