# Clifford's theorem on special divisors

In mathematics, **Clifford's theorem on special divisors** is a result of W. K. Clifford (1878) on algebraic curves, showing the constraints on special linear systems on a curve *C*.

## Statement

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 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* − 2*r*(*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

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, 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.^{[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

- ↑ 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

## References

- 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, 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.**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.

## External links

- Iskovskikh, V.A. (2001), "Clifford theorem", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4