Clifford's theorem on special divisors
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
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 that for an effective special divisor D,
- ℓ(D) − 1 ≤ d/2,
together with the information that the case of equality here is only for D zero or canonical, or C 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 value of the d − 2r(D), taken over all special divisors (that are not canonical or trivial). Clifford's theorem is then the statement that this is non-negative. The Clifford index for a generic curve of genus g is the floor function of
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, 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 βi, 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.
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. 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. The conjecture for arbitrary curves remains open.
- Hartshorne p.296
- Eisenbud (2005) p.178
- Eisenbud (2005) pp. 183-4.
- Green's canonical syzygy conjecture for generic curves of odd genus - Claire Voisin
- Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface - Claire Voisin
- Satter Prize
- E. Arbarello; M. Cornalba; P.A. Griffiths; J. Harris (1985). Geometry of Algebraic Curves Volume I. Grundlehren de mathematischen Wisenschaften 267. ISBN 0-387-90997-4.
- Clifford, W. K. (1878), "On the Classification of Loci", Philosophical Transactions of the Royal Society of London, The Royal Society, 169: 663–681, ISSN 0080-4614, JSTOR 109316, doi:10.1098/rstl.1878.0020
- Eisenbud, David (2005). The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry. Graduate Texts in Mathematics. 229. New York, NY: Springer-Verlag. ISBN 0-387-22215-4. Zbl 1066.14001.
- William Fulton (1974). Algebraic Curves. Mathematics Lecture Note Series. W.A. Benjamin. p. 212. ISBN 0-8053-3080-1.
- P.A. Griffiths; J. Harris (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 251. ISBN 0-471-05059-8.
- Robin Hartshorne (1977). Algebraic Geometry. Graduate Texts in Mathematics. 52. ISBN 0-387-90244-9.