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