In mathematics, Clifford's theorem on special divisors is a result of on algebraic curves, showing the constraints on special linear systems on a curve C.
styleD=\sumPmPP
L(D)
L(D)
\ell(D)
\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)\led
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)
\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]
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 βi, 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.
. David Eisenbud . The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry . . 229 . 2005 . New York, NY . . 0-387-22215-4 . 1066.14001 .
. William Fulton (mathematician) . Algebraic Curves . Mathematics Lecture Note Series . W.A. Benjamin . 1974 . 0-8053-3080-1 . 212 .
. Phillip A.. Griffiths . Phillip Griffiths . Joe. Harris . Joe Harris (mathematician) . Principles of Algebraic Geometry . Wiley Classics Library . Wiley Interscience . 1994 . 0-471-05059-8 . 251 .
. Robin Hartshorne . Algebraic Geometry . . 52 . 1977 . 0-387-90244-9 .