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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.

Statement

A divisor on a Riemann surface C is a formal sum $\textstyle D=\sum _{P}m_{P}P$ of points P on C with integer coefficients. One considers a divisor as a set of constraints on meromorphic functions in the function field of C, defining $L(D)$ as the vector space of functions having poles only at points of D with positive coefficient, at most as bad as the coefficient indicates, and having zeros at points of D with negative coefficient, with at least that multiplicity. The dimension of $L(D)$ is finite, and denoted $\ell (D)$ . The linear system of divisors attached to D is the corresponding projective space of dimension $\ell (D)-1$ .

The other significant invariant of D is its degree d, which is the sum of all its coefficients.

A divisor is called special if ℓ(K − D) > 0, where K is the canonical divisor.^[1]

Clifford's theorem states that for an effective special divisor D, one has:

2(\ell (D)-1)\leq d

,

and that equality holds only if D is zero or a canonical divisor, or if C is a hyperelliptic curve and D linearly equivalent to an integral multiple of a hyperelliptic divisor.

The Clifford index of C is then defined as the minimum of $d-2(\ell (D)-1)$ 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 $\lfloor {\tfrac {g-1}{2}}\rfloor .$

The Clifford index measures how far the curve is from being hyperelliptic. It may be thought of as a refinement of the gonality: in many cases the Clifford index is equal to the gonality minus 2.^[2]

Green's conjecture

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 β_{i, 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. There are numerous partial results.^[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.^[6] The conjecture for arbitrary curves remains open.