Editing Brill–Noether theory (section)

{{short description|Field of algebraic geometry}}

In [[algebraic geometry]], '''Brill–Noether theory''', introduced by {{harvs|txt|author1-link=Alexander von Brill|author2-link=Max Noether|last1=von Brill|first1=Alexander| last2=Noether|first2=Max|year=1874}}, is the study of '''special divisors''', certain [[divisor on an algebraic curve|divisors]] on a [[Algebraic curve|curve]] {{mvar|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 number]]s (or over some other [[algebraically closed field]]).

The condition to be a special divisor {{mvar|D}} can be formulated in [[sheaf cohomology]] terms, as the non-vanishing of the {{math|''H''<sup>1</sup>}} [[cohomology]] of the sheaf of sections of the [[invertible sheaf]] or [[line bundle]] associated to {{mvar|D}}. This means that, by the [[Riemann–Roch theorem]], the {{math|''H''<sup>0</sup>}} cohomology or space of holomorphic sections is larger than expected.

Alternatively, by [[Serre duality]], the condition is that there exist [[holomorphic differential]]s with divisor {{math|≥ –''D''}} on the curve.