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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*.

A divisor on a Riemann surface *C* is a formal sum 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 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 is finite, and denoted . The linear system of divisors attached to *D* is the corresponding projective space of dimension .

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:

- ,

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 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

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 β_{i, 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.

- 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.

- Iskovskikh, V.A. (2001) [1994], "Clifford theorem",
*Encyclopedia of Mathematics*, EMS Press