22:
165:
876:, but not a scheme-theoretic or ideal-theoretic complete intersection; meaning to say that the ideal of the variety cannot be generated by only 2 polynomials; a minimum of 3 are needed. (An attempt to use only two polynomials make the resulting ideal not
226:
449:
874:
511:
927:
679:
629:
372:
728:
963:
802:
566:
264:
1118:
51:
1541:
181:
1633:
1270:
1230:
1111:
1710:
1321:
1220:
1700:
1087:
73:
44:
1399:
1104:
390:
1030:
has the property that the tangent and secant lines are pairwise disjoint, except at points of the variety itself.
1546:
1467:
1457:
1394:
102:
1144:
1364:
1260:
521:
807:
1623:
1587:
1286:
1199:
1597:
1235:
1010:
and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the
1735:
1643:
464:
117:
34:
883:
635:
585:
272:
1556:
1536:
1472:
1389:
1291:
1250:
38:
30:
685:
1447:
1255:
149:
1240:
577:
232:
55:
1354:
932:
771:
1618:
1316:
1265:
1154:
141:
137:
1705:
1566:
1225:
573:
1477:
8:
1531:
1409:
1374:
1331:
1311:
145:
1672:
1452:
1432:
1245:
987:
737:
vanish identically when using the explicit parameterization above; that is, substitute
517:
129:
125:
1404:
527:
237:
1561:
1508:
1379:
1194:
1189:
1083:
1023:
749:
455:
1551:
1437:
1414:
378:
106:
1677:
1482:
1424:
1326:
1149:
1128:
1019:
153:
1651:
1349:
1174:
1159:
1136:
1042:
999:
734:
172:
95:
1729:
1692:
1462:
1442:
1369:
1096:
1064:
1053:
877:
983:
with no four coplanar, there is a unique twisted cubic passing through them.
1628:
1602:
1592:
1582:
1384:
1204:
459:
382:
133:
1503:
1341:
1015:
995:
87:
1498:
113:
164:
1359:
1682:
1667:
1011:
991:
756:
is generated by these three homogeneous polynomials of degree 2.
1662:
1026:
with the property that every length four subscheme spans
221:{\displaystyle \nu :\mathbf {P} ^{1}\to \mathbf {P} ^{3}}
935:
886:
810:
774:
688:
638:
588:
530:
467:
393:
275:
240:
184:
957:
921:
868:
796:
722:
673:
623:
560:
505:
443:
366:
258:
220:
768:It is the set-theoretic complete intersection of
1727:
1048:The projection from a point on a secant line of
764:The twisted cubic has the following properties:
43:but its sources remain unclear because it lacks
1037:onto a plane from a point on a tangent line of
1126:
1112:
444:{\displaystyle \nu :x\mapsto (x,x^{2},x^{3})}
128:, the twisted cubic is a simple example of a
381:of projective space, the map is simply the
1119:
1105:
1022:is three-dimensional. Further, any smooth
140:. It is the three-dimensional case of the
74:Learn how and when to remove this message
520:, defined as the intersection of three
454:That is, it is the closure by a single
171:The twisted cubic is most easily given
1728:
1542:Clifford's theorem on special divisors
1077:
576:defined by the vanishing of the three
1100:
869:{\displaystyle Z(YW-Z^{2})-W(XW-YZ)}
15:
733:It may be checked that these three
112:. It is a fundamental example of a
13:
1711:Vector bundles on algebraic curves
1634:Weber's theorem (Algebraic curves)
1231:Hasse's theorem on elliptic curves
1221:Counting points on elliptic curves
1080:Algebraic Geometry, A First Course
572:, the twisted cubic is the closed
163:
116:. It is essentially unique, up to
14:
1747:
208:
193:
20:
1322:Hurwitz's automorphisms theorem
1059:The projection from a point on
1018:lines of any non-planar smooth
506:{\displaystyle (x,x^{2},x^{3})}
1547:Gonality of an algebraic curve
1458:Differential of the first kind
922:{\displaystyle (YW-Z^{2})^{2}}
910:
887:
863:
845:
836:
814:
674:{\displaystyle F_{1}=YW-Z^{2}}
624:{\displaystyle F_{0}=XZ-Y^{2}}
555:
531:
500:
468:
438:
406:
403:
367:{\displaystyle \nu :\mapsto .}
358:
300:
297:
294:
282:
253:
241:
203:
124:twisted cubic, therefore). In
1:
1701:Birkhoff–Grothendieck theorem
1400:Nagata's conjecture on curves
1271:Schoof–Elkies–Atkin algorithm
1145:Five points determine a conic
1082:, New York: Springer-Verlag,
1071:
759:
524:. In homogeneous coordinates
159:
1261:Supersingular elliptic curve
723:{\displaystyle F_{2}=XW-YZ.}
7:
1468:Riemann's existence theorem
1395:Hilbert's sixteenth problem
1287:Elliptic curve cryptography
1200:Fundamental pair of periods
10:
1752:
1598:Moduli of algebraic curves
1691:
1642:
1611:
1575:
1524:
1517:
1491:
1423:
1340:
1304:
1279:
1213:
1182:
1173:
1135:
118:projective transformation
1365:Cayley–Bacharach theorem
1292:Elliptic curve primality
958:{\displaystyle YW-Z^{2}}
797:{\displaystyle XZ-Y^{2}}
175:as the image of the map
132:that is not linear or a
29:This article includes a
1624:Riemann–Hurwitz formula
1588:Gromov–Witten invariant
1448:Compact Riemann surface
1236:Mazur's torsion theorem
578:homogeneous polynomials
516:The twisted cubic is a
152:of degree three on the
58:more precise citations.
1241:Modular elliptic curve
959:
923:
870:
798:
724:
675:
625:
562:
507:
445:
368:
260:
233:homogeneous coordinate
222:
168:
1155:Rational normal curve
1002:) of a twisted cubic
960:
924:
871:
799:
752:of the twisted cubic
725:
676:
626:
563:
508:
446:
369:
261:
231:which assigns to the
223:
167:
142:rational normal curve
138:complete intersection
1706:Stable vector bundle
1567:Weil reciprocity law
1557:Riemann–Roch theorem
1537:Brill–Noether theory
1473:Riemann–Roch theorem
1390:Genus–degree formula
1251:Mordell–Weil theorem
1226:Division polynomials
1078:Harris, Joe (1992),
979:Given six points in
933:
884:
808:
772:
686:
636:
586:
528:
465:
391:
273:
238:
182:
1518:Structure of curves
1410:Quartic plane curve
1332:Hyperelliptic curve
1312:De Franchis theorem
1256:Nagell–Lutz theorem
968:Any four points on
748:More strongly, the
1525:Divisors on curves
1317:Faltings's theorem
1266:Schoof's algorithm
1246:Modularity theorem
1033:The projection of
955:
919:
866:
794:
720:
671:
621:
558:
518:projective variety
503:
441:
364:
256:
218:
169:
130:projective variety
126:algebraic geometry
107:projective 3-space
31:list of references
1723:
1722:
1719:
1718:
1619:Hasse–Witt matrix
1562:Weierstrass point
1509:Smooth completion
1478:TeichmĂĽller space
1380:Cubic plane curve
1300:
1299:
1214:Arithmetic theory
1195:Elliptic integral
1190:Elliptic function
1024:algebraic variety
750:homogeneous ideal
456:point at infinity
84:
83:
76:
1743:
1736:Algebraic curves
1552:Jacobian variety
1522:
1521:
1425:Riemann surfaces
1415:Real plane curve
1375:Cramer's paradox
1355:BĂ©zout's theorem
1180:
1179:
1129:algebraic curves
1121:
1114:
1107:
1098:
1097:
1092:
964:
962:
961:
956:
954:
953:
928:
926:
925:
920:
918:
917:
908:
907:
875:
873:
872:
867:
835:
834:
803:
801:
800:
795:
793:
792:
729:
727:
726:
721:
698:
697:
680:
678:
677:
672:
670:
669:
648:
647:
630:
628:
627:
622:
620:
619:
598:
597:
567:
565:
564:
561:{\displaystyle }
559:
512:
510:
509:
504:
499:
498:
486:
485:
450:
448:
447:
442:
437:
436:
424:
423:
379:coordinate patch
373:
371:
370:
365:
357:
356:
344:
343:
325:
324:
312:
311:
265:
263:
262:
259:{\displaystyle }
257:
227:
225:
224:
219:
217:
216:
211:
202:
201:
196:
136:, in fact not a
79:
72:
68:
65:
59:
54:this article by
45:inline citations
24:
23:
16:
1751:
1750:
1746:
1745:
1744:
1742:
1741:
1740:
1726:
1725:
1724:
1715:
1687:
1678:Delta invariant
1656:
1638:
1607:
1571:
1532:Abel–Jacobi map
1513:
1487:
1483:Torelli theorem
1453:Dessin d'enfant
1433:Belyi's theorem
1419:
1405:PlĂĽcker formula
1336:
1327:Hurwitz surface
1296:
1275:
1209:
1183:Analytic theory
1175:Elliptic curves
1169:
1150:Projective line
1137:Rational curves
1131:
1125:
1090:
1074:
1020:algebraic curve
949:
945:
934:
931:
930:
913:
909:
903:
899:
885:
882:
881:
830:
826:
809:
806:
805:
788:
784:
773:
770:
769:
762:
735:quadratic forms
693:
689:
687:
684:
683:
665:
661:
643:
639:
637:
634:
633:
615:
611:
593:
589:
587:
584:
583:
529:
526:
525:
494:
490:
481:
477:
466:
463:
462:
432:
428:
419:
415:
392:
389:
388:
352:
348:
339:
335:
320:
316:
307:
303:
274:
271:
270:
239:
236:
235:
212:
207:
206:
197:
192:
191:
183:
180:
179:
162:
154:projective line
80:
69:
63:
60:
49:
35:related reading
25:
21:
12:
11:
5:
1749:
1739:
1738:
1721:
1720:
1717:
1716:
1714:
1713:
1708:
1703:
1697:
1695:
1693:Vector bundles
1689:
1688:
1686:
1685:
1680:
1675:
1670:
1665:
1660:
1654:
1648:
1646:
1640:
1639:
1637:
1636:
1631:
1626:
1621:
1615:
1613:
1609:
1608:
1606:
1605:
1600:
1595:
1590:
1585:
1579:
1577:
1573:
1572:
1570:
1569:
1564:
1559:
1554:
1549:
1544:
1539:
1534:
1528:
1526:
1519:
1515:
1514:
1512:
1511:
1506:
1501:
1495:
1493:
1489:
1488:
1486:
1485:
1480:
1475:
1470:
1465:
1460:
1455:
1450:
1445:
1440:
1435:
1429:
1427:
1421:
1420:
1418:
1417:
1412:
1407:
1402:
1397:
1392:
1387:
1382:
1377:
1372:
1367:
1362:
1357:
1352:
1346:
1344:
1338:
1337:
1335:
1334:
1329:
1324:
1319:
1314:
1308:
1306:
1302:
1301:
1298:
1297:
1295:
1294:
1289:
1283:
1281:
1277:
1276:
1274:
1273:
1268:
1263:
1258:
1253:
1248:
1243:
1238:
1233:
1228:
1223:
1217:
1215:
1211:
1210:
1208:
1207:
1202:
1197:
1192:
1186:
1184:
1177:
1171:
1170:
1168:
1167:
1162:
1160:Riemann sphere
1157:
1152:
1147:
1141:
1139:
1133:
1132:
1124:
1123:
1116:
1109:
1101:
1095:
1094:
1088:
1073:
1070:
1069:
1068:
1057:
1046:
1043:cuspidal cubic
1031:
1000:secant variety
984:
977:
966:
952:
948:
944:
941:
938:
929:is in it, but
916:
912:
906:
902:
898:
895:
892:
889:
865:
862:
859:
856:
853:
850:
847:
844:
841:
838:
833:
829:
825:
822:
819:
816:
813:
791:
787:
783:
780:
777:
761:
758:
731:
730:
719:
716:
713:
710:
707:
704:
701:
696:
692:
681:
668:
664:
660:
657:
654:
651:
646:
642:
631:
618:
614:
610:
607:
604:
601:
596:
592:
557:
554:
551:
548:
545:
542:
539:
536:
533:
502:
497:
493:
489:
484:
480:
476:
473:
470:
452:
451:
440:
435:
431:
427:
422:
418:
414:
411:
408:
405:
402:
399:
396:
375:
374:
363:
360:
355:
351:
347:
342:
338:
334:
331:
328:
323:
319:
315:
310:
306:
302:
299:
296:
293:
290:
287:
284:
281:
278:
255:
252:
249:
246:
243:
229:
228:
215:
210:
205:
200:
195:
190:
187:
173:parametrically
161:
158:
96:rational curve
82:
81:
39:external links
28:
26:
19:
9:
6:
4:
3:
2:
1748:
1737:
1734:
1733:
1731:
1712:
1709:
1707:
1704:
1702:
1699:
1698:
1696:
1694:
1690:
1684:
1681:
1679:
1676:
1674:
1671:
1669:
1666:
1664:
1661:
1659:
1657:
1650:
1649:
1647:
1645:
1644:Singularities
1641:
1635:
1632:
1630:
1627:
1625:
1622:
1620:
1617:
1616:
1614:
1610:
1604:
1601:
1599:
1596:
1594:
1591:
1589:
1586:
1584:
1581:
1580:
1578:
1574:
1568:
1565:
1563:
1560:
1558:
1555:
1553:
1550:
1548:
1545:
1543:
1540:
1538:
1535:
1533:
1530:
1529:
1527:
1523:
1520:
1516:
1510:
1507:
1505:
1502:
1500:
1497:
1496:
1494:
1492:Constructions
1490:
1484:
1481:
1479:
1476:
1474:
1471:
1469:
1466:
1464:
1463:Klein quartic
1461:
1459:
1456:
1454:
1451:
1449:
1446:
1444:
1443:Bolza surface
1441:
1439:
1438:Bring's curve
1436:
1434:
1431:
1430:
1428:
1426:
1422:
1416:
1413:
1411:
1408:
1406:
1403:
1401:
1398:
1396:
1393:
1391:
1388:
1386:
1383:
1381:
1378:
1376:
1373:
1371:
1370:Conic section
1368:
1366:
1363:
1361:
1358:
1356:
1353:
1351:
1350:AF+BG theorem
1348:
1347:
1345:
1343:
1339:
1333:
1330:
1328:
1325:
1323:
1320:
1318:
1315:
1313:
1310:
1309:
1307:
1303:
1293:
1290:
1288:
1285:
1284:
1282:
1278:
1272:
1269:
1267:
1264:
1262:
1259:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1224:
1222:
1219:
1218:
1216:
1212:
1206:
1203:
1201:
1198:
1196:
1193:
1191:
1188:
1187:
1185:
1181:
1178:
1176:
1172:
1166:
1165:Twisted cubic
1163:
1161:
1158:
1156:
1153:
1151:
1148:
1146:
1143:
1142:
1140:
1138:
1134:
1130:
1122:
1117:
1115:
1110:
1108:
1103:
1102:
1099:
1091:
1089:0-387-97716-3
1085:
1081:
1076:
1075:
1066:
1065:conic section
1062:
1058:
1055:
1051:
1047:
1044:
1040:
1036:
1032:
1029:
1025:
1021:
1017:
1013:
1009:
1005:
1001:
997:
993:
989:
985:
982:
978:
975:
971:
967:
950:
946:
942:
939:
936:
914:
904:
900:
896:
893:
890:
879:
860:
857:
854:
851:
848:
842:
839:
831:
827:
823:
820:
817:
811:
789:
785:
781:
778:
775:
767:
766:
765:
757:
755:
751:
746:
745:, and so on.
744:
740:
736:
717:
714:
711:
708:
705:
702:
699:
694:
690:
682:
666:
662:
658:
655:
652:
649:
644:
640:
632:
616:
612:
608:
605:
602:
599:
594:
590:
582:
581:
580:
579:
575:
571:
552:
549:
546:
543:
540:
537:
534:
523:
519:
514:
495:
491:
487:
482:
478:
474:
471:
461:
457:
433:
429:
425:
420:
416:
412:
409:
400:
397:
394:
387:
386:
385:
384:
380:
361:
353:
349:
345:
340:
336:
332:
329:
326:
321:
317:
313:
308:
304:
291:
288:
285:
279:
276:
269:
268:
267:
250:
247:
244:
234:
213:
198:
188:
185:
178:
177:
176:
174:
166:
157:
155:
151:
147:
144:, and is the
143:
139:
135:
131:
127:
123:
119:
115:
111:
108:
104:
100:
97:
94:is a smooth,
93:
92:twisted cubic
89:
78:
75:
67:
64:February 2022
57:
53:
47:
46:
40:
36:
32:
27:
18:
17:
1652:
1629:Prym variety
1603:Stable curve
1593:Hodge bundle
1583:ELSV formula
1385:Fermat curve
1342:Plane curves
1305:Higher genus
1280:Applications
1205:Modular form
1164:
1079:
1060:
1049:
1038:
1034:
1027:
1007:
1003:
996:secant lines
980:
973:
969:
763:
753:
747:
742:
738:
732:
569:
515:
460:affine curve
453:
383:moment curve
376:
230:
170:
150:Veronese map
134:hypersurface
121:
109:
98:
91:
85:
70:
61:
50:Please help
42:
1658:singularity
1504:Polar curve
88:mathematics
56:introducing
1499:Dual curve
1127:Topics in
1072:References
760:Properties
266:the value
160:Definition
114:skew curve
1612:Morphisms
1360:Bitangent
1063:yields a
1052:yields a
1041:yields a
943:−
897:−
880:, since
855:−
840:−
824:−
782:−
709:−
659:−
609:−
574:subscheme
404:↦
395:ν
298:↦
277:ν
204:→
186:ν
105:three in
1730:Category
1006:fill up
965:is not).
522:quadrics
1683:Tacnode
1668:Crunode
1012:tangent
992:tangent
990:of the
878:radical
458:of the
377:In one
52:improve
1663:Acnode
1576:Moduli
1086:
1056:cubic.
1016:secant
103:degree
1054:nodal
998:(the
988:union
972:span
148:of a
146:image
37:, or
1673:Cusp
1084:ISBN
1014:and
994:and
986:The
804:and
741:for
90:, a
568:on
122:the
101:of
86:In
1732::
513:.
156:.
41:,
33:,
1655:k
1653:A
1120:e
1113:t
1106:v
1093:.
1067:.
1061:C
1050:C
1045:.
1039:C
1035:C
1028:P
1008:P
1004:C
981:P
976:.
974:P
970:C
951:2
947:Z
940:W
937:Y
915:2
911:)
905:2
901:Z
894:W
891:Y
888:(
864:)
861:Z
858:Y
852:W
849:X
846:(
843:W
837:)
832:2
828:Z
821:W
818:Y
815:(
812:Z
790:2
786:Y
779:Z
776:X
754:C
743:X
739:x
718:.
715:Z
712:Y
706:W
703:X
700:=
695:2
691:F
667:2
663:Z
656:W
653:Y
650:=
645:1
641:F
617:2
613:Y
606:Z
603:X
600:=
595:0
591:F
570:P
556:]
553:W
550::
547:Z
544::
541:Y
538::
535:X
532:[
501:)
496:3
492:x
488:,
483:2
479:x
475:,
472:x
469:(
439:)
434:3
430:x
426:,
421:2
417:x
413:,
410:x
407:(
401:x
398::
362:.
359:]
354:3
350:T
346::
341:2
337:T
333:S
330::
327:T
322:2
318:S
314::
309:3
305:S
301:[
295:]
292:T
289::
286:S
283:[
280::
254:]
251:T
248::
245:S
242:[
214:3
209:P
199:1
194:P
189::
120:(
110:P
99:C
77:)
71:(
66:)
62:(
48:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
