Clifford's theorem on special divisors

Statement

A divisor on a Riemann surface C is a formal sum ${\displaystyle D=\sum _P{m}_{P}P}$ 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 ${\displaystyle L(D)}$ 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 ${\displaystyle L(D)}$ is finite, and denoted ${\displaystyle \ell (D)}$. The linear system of divisors attached to D is the corresponding projective space of dimension ${\displaystyle \ell (D)-1}$.

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.

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

${\displaystyle 2(\ell (D)-1)\le d}$,

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 ${\displaystyle d-2(\ell (D)-1)}$ 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 ${\displaystyle \lfloor \frac{g-1}{2}\rfloor .}$

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.