540:
123:
399:
480:
293:
246:
757:
217:
172:
482:
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
637:
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
951:
560:
states that the
Clifford index for a curve over the complex numbers that is not hyperelliptic should be determined by the extent to which
1573:
630:
1578:
138:
1466:
1103:
1063:
944:
499:
1543:
1154:
1053:
544:
The
Clifford index measures how far the curve is from being hyperelliptic. It may be thought of as a refinement of the
1533:
895:
869:
843:
813:
742:
1232:
937:
80:
1379:
1300:
1290:
1227:
831:
17:
633:
for her solution of the generic case of Green's conjecture in two papers. The case of Green's conjecture for
977:
923:
887:
801:
64:
1197:
1093:
1456:
1420:
1119:
1032:
918:
581:
354:
1430:
1068:
857:
730:
913:
1568:
1476:
435:
249:
1389:
1369:
1305:
1222:
1124:
1083:
752:
36:
1280:
1088:
684:
696:
Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface - Claire Voisin
695:
263:
1073:
1187:
222:
1451:
1149:
1098:
987:
48:
706:
1538:
1399:
1058:
134:
1310:
823:
193:
148:
8:
1364:
1242:
1207:
1164:
1144:
593:
487:
414:
798:
The
Geometry of Syzygies. A second course in commutative algebra and algebraic geometry
1505:
1285:
1265:
1078:
782:
1237:
1394:
1341:
1212:
1027:
1022:
891:
865:
853:
839:
809:
774:
738:
726:
608:
332:
1384:
1270:
1247:
879:
819:
766:
722:
685:
Green's canonical syzygy conjecture for generic curves of odd genus - Claire Voisin
257:
1510:
1315:
1257:
1159:
982:
961:
805:
577:
565:
342:
311:
68:
44:
1484:
1182:
1007:
992:
969:
793:
557:
494:
133:
with integer coefficients. One considers a divisor as a set of constraints on
1562:
1525:
1295:
1275:
1202:
997:
929:
778:
626:
1461:
1435:
1425:
1415:
1217:
1037:
770:
1336:
1174:
28:
421:
linearly equivalent to an integral multiple of a hyperelliptic divisor.
1331:
623:
states that equality always holds. There are numerous partial results.
75:
786:
1192:
548:: in many cases the Clifford index is equal to the gonality minus 2.
545:
1515:
1500:
721:
174:
as the vector space of functions having poles only at points of
1495:
838:. Mathematics Lecture Note Series. W.A. Benjamin. p. 212.
182:
as the coefficient indicates, and having zeros at points of
864:. Wiley Classics Library. Wiley Interscience. p. 251.
568:
has linear syzygies. In detail, one defines the invariant
758:
507:
84:
502:
438:
357:
266:
225:
196:
151:
83:
737:. Grundlehren de mathematischen Wisenschaften 267.
619:) + 1 is a lower bound for the Clifford index, and
535:{\displaystyle \lfloor {\tfrac {g-1}{2}}\rfloor .}
534:
474:
393:
287:
240:
211:
166:
117:
588:in its canonical embedding, as the largest index
1560:
852:
959:
945:
526:
503:
306:, which is the sum of all its coefficients.
118:{\displaystyle \textstyle D=\sum _{P}m_{P}P}
952:
938:
911:
878:
755:(1878), "On the Classification of Loci",
792:
751:
40:
631:Ruth Lyttle Satter Prize in Mathematics
14:
1561:
1375:Clifford's theorem on special divisors
830:
669:
409:is zero or a canonical divisor, or if
33:Clifford's theorem on special divisors
933:
735:Geometry of Algebraic Curves Volume I
660:
551:
190:that multiplicity. The dimension of
432:is then defined as the minimum of
394:{\displaystyle 2(\ell (D)-1)\leq d}
298:The other significant invariant of
24:
1544:Vector bundles on algebraic curves
1467:Weber's theorem (Algebraic curves)
1064:Hasse's theorem on elliptic curves
1054:Counting points on elliptic curves
25:
1590:
905:
862:Principles of Algebraic Geometry
405:and that equality holds only if
186:with negative coefficient, with
1155:Hurwitz's automorphisms theorem
804:. Vol. 229. New York, NY:
475:{\displaystyle d-2(\ell (D)-1)}
1574:Theorems in algebraic geometry
1380:Gonality of an algebraic curve
1291:Differential of the first kind
765:, The Royal Society: 663–681,
700:
689:
678:
651:
469:
460:
454:
448:
382:
373:
367:
361:
276:
270:
235:
229:
206:
200:
161:
155:
13:
1:
1579:Unsolved problems in geometry
1534:Birkhoff–Grothendieck theorem
1233:Nagata's conjecture on curves
1104:Schoof–Elkies–Atkin algorithm
978:Five points determine a conic
888:Graduate Texts in Mathematics
802:Graduate Texts in Mathematics
715:
341:states that for an effective
47:, showing the constraints on
1094:Supersingular elliptic curve
58:
7:
1301:Riemann's existence theorem
1228:Hilbert's sixteenth problem
1120:Elliptic curve cryptography
1033:Fundamental pair of periods
919:Encyclopedia of Mathematics
582:homogeneous coordinate ring
178:with positive coefficient,
10:
1595:
1431:Moduli of algebraic curves
912:Iskovskikh, V.A. (2001) ,
675:Eisenbud (2005) pp. 183-4.
576:) in terms of the minimal
288:{\displaystyle \ell (D)-1}
1524:
1475:
1444:
1408:
1357:
1350:
1324:
1256:
1173:
1137:
1112:
1046:
1015:
1006:
968:
250:linear system of divisors
1198:Cayley–Bacharach theorem
1125:Elliptic curve primality
644:
241:{\displaystyle \ell (D)}
1457:Riemann–Hurwitz formula
1421:Gromov–Witten invariant
1281:Compact Riemann surface
1069:Mazur's torsion theorem
219:is finite, and denoted
37:William K. Clifford
1074:Modular elliptic curve
771:10.1098/rstl.1878.0020
725:; Cornalba, Maurizio;
536:
476:
395:
289:
242:
213:
168:
119:
49:special linear systems
988:Rational normal curve
854:Griffiths, Phillip A.
727:Griffiths, Phillip A.
666:Eisenbud (2005) p.178
641:curves remains open.
537:
477:
396:
290:
256:is the corresponding
243:
214:
169:
135:meromorphic functions
120:
1539:Stable vector bundle
1400:Weil reciprocity law
1390:Riemann–Roch theorem
1370:Brill–Noether theory
1306:Riemann–Roch theorem
1223:Genus–degree formula
1084:Mordell–Weil theorem
1059:Division polynomials
753:Clifford, William K.
500:
436:
355:
309:A divisor is called
264:
223:
212:{\displaystyle L(D)}
194:
167:{\displaystyle L(D)}
149:
81:
1351:Structure of curves
1243:Quartic plane curve
1165:Hyperelliptic curve
1145:De Franchis theorem
1089:Nagell–Lutz theorem
607:is zero. Green and
594:graded Betti number
415:hyperelliptic curve
323: −
1358:Divisors on curves
1150:Faltings's theorem
1099:Schoof's algorithm
1079:Modularity theorem
914:"Clifford theorem"
884:Algebraic Geometry
621:Green's conjecture
552:Green's conjecture
532:
524:
472:
391:
339:Clifford's theorem
285:
238:
209:
164:
115:
114:
100:
1556:
1555:
1552:
1551:
1452:Hasse–Witt matrix
1395:Weierstrass point
1342:Smooth completion
1311:TeichmĂĽller space
1213:Cubic plane curve
1133:
1132:
1047:Arithmetic theory
1028:Elliptic integral
1023:Elliptic function
880:Hartshorne, Robin
723:Arbarello, Enrico
609:Robert Lazarsfeld
523:
333:canonical divisor
91:
16:(Redirected from
1586:
1569:Algebraic curves
1385:Jacobian variety
1355:
1354:
1258:Riemann surfaces
1248:Real plane curve
1208:Cramer's paradox
1188:BĂ©zout's theorem
1013:
1012:
962:algebraic curves
954:
947:
940:
931:
930:
926:
901:
890:. Vol. 52.
875:
849:
836:Algebraic Curves
827:
789:
748:
709:
704:
698:
693:
687:
682:
676:
673:
667:
664:
658:
657:Hartshorne p.296
655:
629:was awarded the
556:A conjecture of
541:
539:
538:
533:
525:
519:
508:
493:is equal to the
481:
479:
478:
473:
400:
398:
397:
392:
327:) > 0, where
294:
292:
291:
286:
258:projective space
247:
245:
244:
239:
218:
216:
215:
210:
173:
171:
170:
165:
124:
122:
121:
116:
110:
109:
99:
45:algebraic curves
21:
1594:
1593:
1589:
1588:
1587:
1585:
1584:
1583:
1559:
1558:
1557:
1548:
1520:
1511:Delta invariant
1489:
1471:
1440:
1404:
1365:Abel–Jacobi map
1346:
1320:
1316:Torelli theorem
1286:Dessin d'enfant
1266:Belyi's theorem
1252:
1238:PlĂĽcker formula
1169:
1160:Hurwitz surface
1129:
1108:
1042:
1016:Analytic theory
1008:Elliptic curves
1002:
983:Projective line
970:Rational curves
964:
958:
908:
898:
872:
846:
832:Fulton, William
816:
806:Springer-Verlag
794:Eisenbud, David
745:
718:
713:
712:
705:
701:
694:
690:
683:
679:
674:
670:
665:
661:
656:
652:
647:
606:
578:free resolution
566:canonical curve
554:
509:
506:
501:
498:
497:
437:
434:
433:
356:
353:
352:
343:special divisor
265:
262:
261:
224:
221:
220:
195:
192:
191:
150:
147:
146:
105:
101:
95:
82:
79:
78:
69:Riemann surface
61:
35:is a result of
23:
22:
15:
12:
11:
5:
1592:
1582:
1581:
1576:
1571:
1554:
1553:
1550:
1549:
1547:
1546:
1541:
1536:
1530:
1528:
1526:Vector bundles
1522:
1521:
1519:
1518:
1513:
1508:
1503:
1498:
1493:
1487:
1481:
1479:
1473:
1472:
1470:
1469:
1464:
1459:
1454:
1448:
1446:
1442:
1441:
1439:
1438:
1433:
1428:
1423:
1418:
1412:
1410:
1406:
1405:
1403:
1402:
1397:
1392:
1387:
1382:
1377:
1372:
1367:
1361:
1359:
1352:
1348:
1347:
1345:
1344:
1339:
1334:
1328:
1326:
1322:
1321:
1319:
1318:
1313:
1308:
1303:
1298:
1293:
1288:
1283:
1278:
1273:
1268:
1262:
1260:
1254:
1253:
1251:
1250:
1245:
1240:
1235:
1230:
1225:
1220:
1215:
1210:
1205:
1200:
1195:
1190:
1185:
1179:
1177:
1171:
1170:
1168:
1167:
1162:
1157:
1152:
1147:
1141:
1139:
1135:
1134:
1131:
1130:
1128:
1127:
1122:
1116:
1114:
1110:
1109:
1107:
1106:
1101:
1096:
1091:
1086:
1081:
1076:
1071:
1066:
1061:
1056:
1050:
1048:
1044:
1043:
1041:
1040:
1035:
1030:
1025:
1019:
1017:
1010:
1004:
1003:
1001:
1000:
995:
993:Riemann sphere
990:
985:
980:
974:
972:
966:
965:
957:
956:
949:
942:
934:
928:
927:
907:
906:External links
904:
903:
902:
896:
876:
870:
850:
844:
828:
814:
790:
749:
743:
717:
714:
711:
710:
699:
688:
677:
668:
659:
649:
648:
646:
643:
597:
592:for which the
553:
550:
531:
528:
522:
518:
515:
512:
505:
495:floor function
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
426:Clifford index
403:
402:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
302:is its degree
284:
281:
278:
275:
272:
269:
237:
234:
231:
228:
208:
205:
202:
199:
180:at most as bad
163:
160:
157:
154:
139:function field
113:
108:
104:
98:
94:
90:
87:
60:
57:
18:Clifford index
9:
6:
4:
3:
2:
1591:
1580:
1577:
1575:
1572:
1570:
1567:
1566:
1564:
1545:
1542:
1540:
1537:
1535:
1532:
1531:
1529:
1527:
1523:
1517:
1514:
1512:
1509:
1507:
1504:
1502:
1499:
1497:
1494:
1492:
1490:
1483:
1482:
1480:
1478:
1477:Singularities
1474:
1468:
1465:
1463:
1460:
1458:
1455:
1453:
1450:
1449:
1447:
1443:
1437:
1434:
1432:
1429:
1427:
1424:
1422:
1419:
1417:
1414:
1413:
1411:
1407:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1368:
1366:
1363:
1362:
1360:
1356:
1353:
1349:
1343:
1340:
1338:
1335:
1333:
1330:
1329:
1327:
1325:Constructions
1323:
1317:
1314:
1312:
1309:
1307:
1304:
1302:
1299:
1297:
1296:Klein quartic
1294:
1292:
1289:
1287:
1284:
1282:
1279:
1277:
1276:Bolza surface
1274:
1272:
1271:Bring's curve
1269:
1267:
1264:
1263:
1261:
1259:
1255:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1224:
1221:
1219:
1216:
1214:
1211:
1209:
1206:
1204:
1203:Conic section
1201:
1199:
1196:
1194:
1191:
1189:
1186:
1184:
1183:AF+BG theorem
1181:
1180:
1178:
1176:
1172:
1166:
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1142:
1140:
1136:
1126:
1123:
1121:
1118:
1117:
1115:
1111:
1105:
1102:
1100:
1097:
1095:
1092:
1090:
1087:
1085:
1082:
1080:
1077:
1075:
1072:
1070:
1067:
1065:
1062:
1060:
1057:
1055:
1052:
1051:
1049:
1045:
1039:
1036:
1034:
1031:
1029:
1026:
1024:
1021:
1020:
1018:
1014:
1011:
1009:
1005:
999:
998:Twisted cubic
996:
994:
991:
989:
986:
984:
981:
979:
976:
975:
973:
971:
967:
963:
955:
950:
948:
943:
941:
936:
935:
932:
925:
921:
920:
915:
910:
909:
899:
897:0-387-90244-9
893:
889:
885:
881:
877:
873:
871:0-471-05059-8
867:
863:
859:
855:
851:
847:
845:0-8053-3080-1
841:
837:
833:
829:
825:
821:
817:
815:0-387-22215-4
811:
807:
803:
799:
795:
791:
788:
784:
780:
776:
772:
768:
764:
760:
759:
754:
750:
746:
744:0-387-90997-4
740:
736:
732:
728:
724:
720:
719:
708:
703:
697:
692:
686:
681:
672:
663:
654:
650:
642:
640:
636:
632:
628:
627:Claire Voisin
624:
622:
618:
614:
610:
604:
600:
595:
591:
587:
583:
579:
575:
571:
567:
563:
559:
549:
547:
542:
529:
520:
516:
513:
510:
496:
492:
489:
485:
466:
463:
457:
451:
445:
442:
439:
431:
427:
422:
420:
416:
412:
408:
388:
385:
379:
376:
370:
364:
358:
351:
350:
349:
347:
344:
340:
336:
334:
330:
326:
322:
318:
314:
313:
307:
305:
301:
296:
282:
279:
273:
267:
260:of dimension
259:
255:
251:
232:
226:
203:
197:
189:
185:
181:
177:
158:
152:
144:
140:
136:
132:
128:
111:
106:
102:
96:
92:
88:
85:
77:
73:
70:
66:
56:
54:
50:
46:
42:
38:
34:
30:
19:
1485:
1462:Prym variety
1436:Stable curve
1426:Hodge bundle
1416:ELSV formula
1374:
1218:Fermat curve
1175:Plane curves
1138:Higher genus
1113:Applications
1038:Modular form
917:
883:
861:
835:
797:
762:
756:
734:
707:Satter Prize
702:
691:
680:
671:
662:
653:
638:
634:
625:
620:
616:
612:
611:showed that
602:
598:
589:
585:
573:
569:
561:
555:
543:
490:
483:
429:
425:
423:
418:
410:
406:
404:
345:
338:
337:
328:
324:
320:
316:
310:
308:
303:
299:
297:
253:
252:attached to
187:
183:
179:
175:
142:
130:
126:
71:
62:
52:
32:
26:
1491:singularity
1337:Polar curve
858:Harris, Joe
731:Harris, Joe
348:, one has:
51:on a curve
29:mathematics
1563:Categories
1332:Dual curve
960:Topics in
824:1066.14001
716:References
558:Mark Green
125:of points
76:formal sum
1445:Morphisms
1193:Bitangent
924:EMS Press
779:0080-4614
639:arbitrary
527:⌋
514:−
504:⌊
486:curve of
464:−
452:ℓ
443:−
386:≤
377:−
365:ℓ
280:−
268:ℓ
227:ℓ
145:defining
93:∑
59:Statement
882:(1977).
860:(1994).
834:(1974).
796:(2005).
733:(1985).
546:gonality
188:at least
1516:Tacnode
1501:Crunode
635:generic
580:of the
484:generic
331:is the
312:special
137:in the
65:divisor
39: (
1496:Acnode
1409:Moduli
894:
868:
842:
822:
812:
787:109316
785:
777:
741:
248:. The
783:JSTOR
645:Notes
488:genus
413:is a
74:is a
67:on a
43:) on
1506:Cusp
892:ISBN
866:ISBN
840:ISBN
810:ISBN
775:ISSN
739:ISBN
424:The
417:and
41:1878
820:Zbl
767:doi
763:169
605:+ 2
584:of
564:as
428:of
315:if
295:.
141:of
129:on
27:In
1565::
922:,
916:,
886:.
856:;
818:.
808:.
800:.
781:,
773:,
761:,
729:;
601:,
335:.
143:C,
63:A
55:.
31:,
1488:k
1486:A
953:e
946:t
939:v
900:.
874:.
848:.
826:.
769::
747:.
617:C
615:(
613:a
603:i
599:i
596:β
590:i
586:C
574:C
572:(
570:a
562:C
530:.
521:2
517:1
511:g
491:g
470:)
467:1
461:)
458:D
455:(
449:(
446:2
440:d
430:C
419:D
411:C
407:D
401:,
389:d
383:)
380:1
374:)
371:D
368:(
362:(
359:2
346:D
329:K
325:D
321:K
319:(
317:â„“
304:d
300:D
283:1
277:)
274:D
271:(
254:D
236:)
233:D
230:(
207:)
204:D
201:(
198:L
184:D
176:D
162:)
159:D
156:(
153:L
131:C
127:P
112:P
107:P
103:m
97:P
89:=
86:D
72:C
53:C
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.