Clifford's theorem on special divisors

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 positive coefficients in D indicate, and that zeros at points in D with a negative 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.
A divisor is called special if ℓ(K − D) > 0, where K is the canonical divisor.^{[1]}
In this notation, Clifford's theorem is the statement that for a special divisor D ≠ 0,
 ℓ(D) − 1 ≤ d/2,
together with the information that the case of equality here is only for C a hyperelliptic curve, and D an integral multiple of the canonical divisor K.
The Clifford index of C is 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 nonnegative. The Clifford index for a generic curve of genus g is known to be the floor function of
A conjecture of Michael 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, the invariant a(C) is determined by 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 Lazarsfeld showed that a(C) + 1 is a lower bound for the Clifford index, and Green's conjecture is that equality always holds. There are numerous partial results.^{[2]}
References
 E. Arbarello; M. Cornalba, P.A. Griffiths, J. Harris (1985). Geometry of Algebraic Curves Volume I. Grundlehren de mathematischen Wisenschaften 267. ISBN 0387909974.
 William Fulton (1974). Algebraic Curves. Mathematics Lecture Note Series. W.A. Benjamin. p. 212. ISBN 0805330814.
 P.A. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 251. ISBN 0471050598.
 Robin Hartshorne (1977). Algebraic Geometry. Graduate Texts in Mathematics. 52. ISBN 0387902449.
 Iskovskikh, V.A. (2001), "Clifford theorem", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/C/c022490.htm
Notes
 ^ Hartshorne p.296
 ^ David Eisenbud, The Geometry of Syzygies (2005), pp. 1834.
Categories: Algebraic curves
 Theorems in geometry
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