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Clifford's theorem on special divisors

Summary

In mathematics, Clifford's theorem on special divisors is a result of William K. Clifford (1878) 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 ${\displaystyle \textstyle D=\sum _{P}m_{P}P}$  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 ${\displaystyle L(D)}$  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 ${\displaystyle L(D)}$  is finite, and denoted ${\displaystyle \ell (D)}$ . The linear system of divisors attached to D is the corresponding projective space of dimension ${\displaystyle \ell (D)-1}$ .

The other significant invariant of D is its degree d, which is the sum of all its coefficients.

A divisor is called special if (K − D) > 0, where K is the canonical divisor.[1]

Clifford's theorem states that for an effective special divisor D, one has:

${\displaystyle 2(\ell (D)-1)\leq d}$ ,

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 ${\displaystyle d-2(\ell (D)-1)}$  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 ${\displaystyle \lfloor {\tfrac {g-1}{2}}\rfloor .}$

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, The Royal Society, 169: 663–681, 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. Graduate Texts in Mathematics. Vol. 229. 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. Graduate Texts in Mathematics. Vol. 52. ISBN 0-387-90244-9.