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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
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curves had attracted a huge amount of effort by algebraic geometers over twenty years before finally being laid to rest by Voisin. The conjecture for
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states that the
Clifford index for a curve over the complex numbers that is not hyperelliptic should be determined by the extent to which
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The
Clifford index measures how far the curve is from being hyperelliptic. It may be thought of as a refinement of the
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for her solution of the generic case of Green's conjecture in two papers. The case of Green's conjecture for
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Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface - Claire Voisin
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The
Geometry of Syzygies. A second course in commutative algebra and algebraic geometry
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Green's canonical syzygy conjecture for generic curves of odd genus - Claire Voisin
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with integer coefficients. One considers a divisor as a set of constraints on
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linearly equivalent to an integral multiple of a hyperelliptic divisor.
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states that equality always holds. There are numerous partial results.
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537:: in many cases the Clifford index is equal to the gonality minus 2.
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as the vector space of functions having poles only at points of
1484:
827:. Mathematics Lecture Note Series. W.A. Benjamin. p. 212.
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as the coefficient indicates, and having zeros at points of
853:. Wiley Classics Library. Wiley Interscience. p. 251.
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has linear syzygies. In detail, one defines the invariant
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726:. Grundlehren de mathematischen Wisenschaften 267.
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524:{\displaystyle \lfloor {\tfrac {g-1}{2}}\rfloor .}
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577:in its canonical embedding, as the largest index
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107:{\displaystyle \textstyle D=\sum _{P}m_{P}P}
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744:(1878), "On the Classification of Loci",
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29:
620:Ruth Lyttle Satter Prize in Mathematics
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1364:Clifford's theorem on special divisors
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398:is zero or a canonical divisor, or if
22:Clifford's theorem on special divisors
922:
724:Geometry of Algebraic Curves Volume I
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540:
179:that multiplicity. The dimension of
421:is then defined as the minimum of
383:{\displaystyle 2(\ell (D)-1)\leq d}
287:The other significant invariant of
13:
1533:Vector bundles on algebraic curves
1456:Weber's theorem (Algebraic curves)
1053:Hasse's theorem on elliptic curves
1043:Counting points on elliptic curves
14:
1579:
894:
851:Principles of Algebraic Geometry
394:and that equality holds only if
175:with negative coefficient, with
1144:Hurwitz's automorphisms theorem
793:. Vol. 229. New York, NY:
464:{\displaystyle d-2(\ell (D)-1)}
1563:Theorems in algebraic geometry
1369:Gonality of an algebraic curve
1280:Differential of the first kind
754:, The Royal Society: 663–681,
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1568:Unsolved problems in geometry
1523:Birkhoff–Grothendieck theorem
1222:Nagata's conjecture on curves
1093:Schoof–Elkies–Atkin algorithm
967:Five points determine a conic
877:Graduate Texts in Mathematics
791:Graduate Texts in Mathematics
704:
330:states that for an effective
36:, showing the constraints on
1083:Supersingular elliptic curve
47:
7:
1290:Riemann's existence theorem
1217:Hilbert's sixteenth problem
1109:Elliptic curve cryptography
1022:Fundamental pair of periods
908:Encyclopedia of Mathematics
571:homogeneous coordinate ring
167:with positive coefficient,
10:
1584:
1420:Moduli of algebraic curves
901:Iskovskikh, V.A. (2001) ,
664:Eisenbud (2005) pp. 183-4.
565:) in terms of the minimal
277:{\displaystyle \ell (D)-1}
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1114:Elliptic curve primality
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230:{\displaystyle \ell (D)}
1446:Riemann–Hurwitz formula
1410:Gromov–Witten invariant
1270:Compact Riemann surface
1058:Mazur's torsion theorem
208:is finite, and denoted
26:William K. Clifford
1063:Modular elliptic curve
760:10.1098/rstl.1878.0020
714:; Cornalba, Maurizio;
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38:special linear systems
977:Rational normal curve
843:Griffiths, Phillip A.
716:Griffiths, Phillip A.
655:Eisenbud (2005) p.178
630:curves remains open.
526:
466:
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245:is the corresponding
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124:meromorphic functions
109:
1528:Stable vector bundle
1389:Weil reciprocity law
1379:Riemann–Roch theorem
1359:Brill–Noether theory
1295:Riemann–Roch theorem
1212:Genus–degree formula
1073:Mordell–Weil theorem
1048:Division polynomials
742:Clifford, William K.
489:
425:
344:
298:A divisor is called
253:
212:
201:{\displaystyle L(D)}
183:
156:{\displaystyle L(D)}
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1340:Structure of curves
1232:Quartic plane curve
1154:Hyperelliptic curve
1134:De Franchis theorem
1078:Nagell–Lutz theorem
596:is zero. Green and
583:graded Betti number
404:hyperelliptic curve
312: −
1347:Divisors on curves
1139:Faltings's theorem
1088:Schoof's algorithm
1068:Modularity theorem
903:"Clifford theorem"
873:Algebraic Geometry
610:Green's conjecture
541:Green's conjecture
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328:Clifford's theorem
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1441:Hasse–Witt matrix
1384:Weierstrass point
1331:Smooth completion
1300:TeichmĂĽller space
1202:Cubic plane curve
1122:
1121:
1036:Arithmetic theory
1017:Elliptic integral
1012:Elliptic function
869:Hartshorne, Robin
712:Arbarello, Enrico
598:Robert Lazarsfeld
512:
322:canonical divisor
80:
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1558:Algebraic curves
1374:Jacobian variety
1344:
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1247:Riemann surfaces
1237:Real plane curve
1197:Cramer's paradox
1177:BĂ©zout's theorem
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951:algebraic curves
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879:. Vol. 52.
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646:Hartshorne p.296
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618:was awarded the
545:A conjecture of
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482:is equal to the
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1500:Delta invariant
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1354:Abel–Jacobi map
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1305:Torelli theorem
1275:Dessin d'enfant
1255:Belyi's theorem
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1227:PlĂĽcker formula
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1149:Hurwitz surface
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1005:Analytic theory
997:Elliptic curves
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972:Projective line
959:Rational curves
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821:Fulton, William
805:
795:Springer-Verlag
783:Eisenbud, David
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1451:Prym variety
1425:Stable curve
1415:Hodge bundle
1405:ELSV formula
1363:
1207:Fermat curve
1164:Plane curves
1127:Higher genus
1102:Applications
1027:Modular form
906:
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696:Satter Prize
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1480:singularity
1326:Polar curve
847:Harris, Joe
720:Harris, Joe
337:, one has:
40:on a curve
18:mathematics
1552:Categories
1321:Dual curve
949:Topics in
813:1066.14001
705:References
547:Mark Green
114:of points
65:formal sum
1434:Morphisms
1182:Bitangent
913:EMS Press
768:0080-4614
628:arbitrary
516:⌋
503:−
493:⌊
475:curve of
453:−
441:ℓ
432:−
375:≤
366:−
354:ℓ
269:−
257:ℓ
216:ℓ
134:defining
82:∑
48:Statement
871:(1977).
849:(1994).
823:(1974).
785:(2005).
722:(1985).
535:gonality
177:at least
1505:Tacnode
1490:Crunode
624:generic
569:of the
473:generic
320:is the
301:special
126:in the
54:divisor
28: (
1485:Acnode
1398:Moduli
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857:
831:
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776:109316
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766:
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237:. The
772:JSTOR
634:Notes
477:genus
402:is a
63:is a
56:on a
32:) on
1495:Cusp
881:ISBN
855:ISBN
829:ISBN
799:ISBN
764:ISSN
728:ISBN
413:The
406:and
30:1878
809:Zbl
756:doi
752:169
594:+ 2
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130:of
118:on
16:In
1554::
911:,
905:,
875:.
845:;
807:.
797:.
789:.
770:,
762:,
750:,
718:;
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