Clifford's theorem on special divisors

Clifford's theorem on special divisors 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.

Contents 1 Statement 2 Green's conjecture 3 Notes 4 References 5 External links 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 ℓ(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 ^ 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 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. External links Iskovskikh, V.A. (2001) [1994], "Clifford theorem", Encyclopedia of Mathematics, EMS Press hide vte Topics in algebraic curves Rational curves Five points determine a conicProjective lineRational normal curveRiemann sphereTwisted cubic Elliptic curves Analytic theory Elliptic functionElliptic integralFundamental pair of periodsModular form Arithmetic theory Counting points on elliptic curvesDivision polynomialsHasse's theorem on elliptic curvesMazur's torsion theoremModular elliptic curveModularity theoremMordell–Weil theoremNagell–Lutz theoremSupersingular elliptic curveSchoof's algorithmSchoof–Elkies–Atkin algorithm Applications Elliptic curve cryptographyElliptic curve primality Higher genus De Franchis theoremFaltings's theoremHurwitz's automorphisms theoremHurwitz surfaceHyperelliptic curve Plane curves AF+BG theoremBézout's theoremBitangentCayley–Bacharach theoremConic sectionCramer's paradoxCubic plane curveFermat curveGenus–degree formulaHilbert's sixteenth problemNagata's conjecture on curvesPlücker formulaQuartic plane curveReal plane curve Riemann surfaces Belyi's theoremBring's curveBolza surfaceCompact Riemann surfaceDessin d'enfantDifferential of the first kindKlein quarticRiemann's existence theoremRiemann–Roch theoremTeichmüller spaceTorelli theorem Constructions Dual curvePolar curveSmooth completion Structure of curves Divisors on curves Abel–Jacobi mapBrill–Noether theoryClifford's theorem on special divisorsGonality of an algebraic curveJacobian varietyRiemann–Roch theoremWeierstrass pointWeil reciprocity law Moduli ELSV formulaGromov–Witten invariantHodge bundleModuli of algebraic curvesStable curve Morphisms Hasse–Witt matrixRiemann–Hurwitz formulaPrym varietyWeber's theorem Singularities AcnodeCrunodeCuspDelta invariantTacnode Vector bundles Birkhoff–Grothendieck theoremStable vector bundleVector bundles on algebraic curves Categories: Algebraic curvesTheorems in algebraic geometryUnsolved problems in geometry