In mathematics, Clifford's theorem on special divisors is a result of txt on algebraic curves, showing the constraints on special linear systems on a curve C.

Statement

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.[1]

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.[2]

Green's conjecture

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.

Notes

  1. ^ Hartshorne p.296
  2. ^ Eisenbud (2005) p.178
  3. ^ Eisenbud (2005) pp. 183-4.
  4. ^ Green's canonical syzygy conjecture for generic curves of odd genus - Claire Voisin
  5. ^ Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface - Claire Voisin
  6. ^ Satter Prize

References

  • Arbarello, Enrico; Cornalba, Maurizio; Griffiths, Phillip A.; Harris, Joe (1985). Geometry of Algebraic Curves Volume I. Grundlehren de mathematischen Wisenschaften 267. ISBN 0-387-90997-4.
  • Clifford, William K. (1878), "On the Classification of Loci", Philosophical Transactions of the Royal Society of London. 169: 663–681, The Royal Society, doi:10.1098/rstl.1878.0020. ISSN 0080-4614. JSTOR 109316
  • Eisenbud, David (2005). The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry. Vol. 229. Graduate Texts in Mathematics. New York, NY: Springer-Verlag. ISBN 0-387-22215-4. Zbl 1066.14001.
  • Fulton, William (1974). Algebraic Curves. Mathematics Lecture Note Series. W.A. Benjamin. p. 212. ISBN 0-8053-3080-1.
  • Griffiths, Phillip A. & Harris, Joe (1994). Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. p. 251. ISBN 0-471-05059-8.
  • Hartshorne, Robin (1977). Algebraic Geometry. Vol. 52. Graduate Texts in Mathematics. ISBN 0-387-90244-9.