Brill–Noether theory
In the theory of algebraic curves, 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. The condition to be a special divisor "D" can be formulated in sheaf cohomology terms, as the non-vanishing of the "H" cohomology of the sheaf of the sections of the invertible sheaf or line bundle associated to "D". This means that, by the Riemann–Roch theorem , the "H" 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. For smooth curves "C" and for "d"≥1, "r"≥0 the basic results about the space  G of linear systems on "C" of degree "d" and dimension "r" are as follows.