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:
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511:
927:
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1118:
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has the property that the tangent and secant lines are pairwise disjoint, except at points of the variety itself.
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102:
1144:
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807:
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and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the
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117:
34:
883:
635:
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272:
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1389:
1291:
1250:
38:
30:
685:
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149:
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577:
232:
55:
1354:
932:
771:
1618:
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1154:
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8:
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987:
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vanish identically when using the explicit parameterization above; that is, substitute
517:
129:
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527:
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1023:
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734:
172:
95:
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with no four coplanar, there is a unique twisted cubic passing through them.
1628:
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1204:
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382:
133:
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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}}
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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:
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438:
406:
403:
367:{\displaystyle \nu :\mapsto .}
358:
300:
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282:
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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:
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1611:
1575:
1524:
1517:
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1213:
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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:
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368:
260:
233:homogeneous coordinate
222:
168:
1155:Rational normal curve
1002:) of a twisted cubic
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799:
752:of the twisted cubic
725:
676:
626:
563:
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446:
369:
261:
231:which assigns to the
223:
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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
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884:
808:
772:
686:
636:
586:
528:
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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
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561:{\displaystyle }
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379:coordinate patch
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259:{\displaystyle }
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136:, in fact not a
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1687:
1678:Delta invariant
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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:
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1020:algebraic curve
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735:quadratic forms
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35:related reading
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1693:Vector bundles
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1160:Riemann sphere
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1043:cuspidal cubic
1031:
1000:secant variety
984:
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952:
948:
944:
941:
938:
929:is in it, but
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173:parametrically
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96:rational curve
82:
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39:external links
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1647:
1645:
1644:Singularities
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1492:Constructions
1490:
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1471:
1469:
1466:
1464:
1463:Klein quartic
1461:
1459:
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1454:
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1449:
1446:
1444:
1443:Bolza surface
1441:
1439:
1438:Bring's curve
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1370:Conic section
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1350:AF+BG theorem
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1165:Twisted cubic
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1103:
1102:
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1091:
1089:0-387-97716-3
1085:
1081:
1076:
1075:
1066:
1065:conic section
1062:
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745:, and so on.
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94:is a smooth,
93:
92:twisted cubic
89:
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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
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105:three in
1730:Category
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522:quadrics
1683:Tacnode
1668:Crunode
1012:tangent
992:tangent
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1663:Acnode
1576:Moduli
1086:
1056:cubic.
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122:the
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1655:k
1653:A
1120:e
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1093:.
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970:C
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718:.
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700:=
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650:=
645:1
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617:2
613:Y
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603:X
600:=
595:0
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570:P
556:]
553:W
550::
547:Z
544::
541:Y
538::
535:X
532:[
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496:3
492:x
488:,
483:2
479:x
475:,
472:x
469:(
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434:3
430:x
426:,
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417:x
413:,
410:x
407:(
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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:.
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