Clifford's theorem
From Wikipedia, the free encyclopedia
In mathematics, Clifford's theorem on special divisors is a result of W. K. Clifford on algebraic curves, showing the constraints on special linear systems on a curve C.
If D is a divisor on C, then D is (abstractly) a formal sum of points P on C (with integer coefficients), and in this application a set of constraints to be applied to functions on C (if C is a Riemann surface, these are meromorphic functions, and in general lie in the function field of C). Functions in this sense have a divisor of zeros and poles, counted with multiplicity; a divisor D is here of interest as a set of constraints on functions, insisting that poles at given points are only as bad as the negative coefficients in D indicate, and that zeros at points in D with a positive coefficient have at least that multiplicity. The dimension of the vector space
- L(D)
of such functions is finite, and denoted l(D). Conventionally the linear system of divisors attached to D is then attributed dimension r(D) = l(D) −1, which is the dimension of the projective space parametrizing it.
The other significant invariant of D is its degree, d, which is the sum of all its coefficients.
In this notation, Clifford's theorem is the statement
- l(D) − 1 ≤ d/2,
for a special divisor D ≠ 0, together with the information that the case of equality here is only for C a hyperelliptic curve, and D an integral multiple of its canonical divisor K.
The Clifford index of C then defined as the minimum value of the d − 2r(D), taken over all special divisors. Clifford's theorem is then the statement that this is non-negative. The Clifford index for a generic curve of genus g is known to be the floor function of
- (g − 1)/2.
