In mathematics, Clifford's theorem on special divisors is a result of William K. Clifford (1878) on algebraic curves, showing the constraints on special linear systems on a curve C. A divisor on a Riemann surface C is a formal sum
The linear system of divisors attached to D is the corresponding projective space of dimension
taken over all special divisors (except canonical and trivial), and Clifford's theorem states this is non-negative.
It can be shown that the Clifford index for a generic curve of genus g is equal to the floor function
[2] A conjecture of Mark Green states that the Clifford index for a curve over the complex numbers that is not hyperelliptic should be determined by the extent to which C as canonical curve has linear syzygies.
In detail, one defines the invariant a(C) in terms of the minimal free resolution of the homogeneous coordinate ring of C in its canonical embedding, as the largest index i for which the graded Betti number βi, i + 2 is zero.
Green and Robert Lazarsfeld showed that a(C) + 1 is a lower bound for the Clifford index, and Green's conjecture states that equality always holds.
[3] Claire Voisin was awarded the Ruth Lyttle Satter Prize in Mathematics for her solution of the generic case of Green's conjecture in two papers.
[4][5] The case of Green's conjecture for generic curves had attracted a huge amount of effort by algebraic geometers over twenty years before finally being laid to rest by Voisin.