33:
176:
887:, 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
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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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and the lines are pairwise disjoint, except at points of the curve itself. In fact, the union of the
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vanish identically when using the explicit parameterization above; that is, substitute
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17:
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994:
with no four coplanar, there is a unique twisted cubic passing through them.
1682:
1667:
1022:
1002:
1662:
767:
is generated by these three homogeneous polynomials of degree 2.
1111:
26:
174:
1037:
with the property that every length four subscheme spans
232:{\displaystyle \nu :\mathbf {P} ^{1}\to \mathbf {P} ^{3}}
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779:It is the set-theoretic complete intersection of
1059:The projection from a point on a secant line of
775:The twisted cubic has the following properties:
54:but its sources remain unclear because it lacks
1048:onto a plane from a point on a tangent line of
1123:
455:{\displaystyle \nu :x\mapsto (x,x^{2},x^{3})}
139:, the twisted cubic is a simple example of a
8:
392:of projective space, the map is simply the
1532:
1190:
1130:
1116:
1108:
1033:is three-dimensional. Further, any smooth
151:. It is the three-dimensional case of the
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85:Learn how and when to remove this message
531:, defined as the intersection of three
465:That is, it is the closure by a single
182:The twisted cubic is most easily given
1553:Clifford's theorem on special divisors
587:defined by the vanishing of the three
7:
880:{\displaystyle Z(YW-Z^{2})-W(XW-YZ)}
744:It may be checked that these three
123:. It is a fundamental example of a
1711:Vector bundles on algebraic curves
1645:Weber's theorem (Algebraic curves)
1242:Hasse's theorem on elliptic curves
1232:Counting points on elliptic curves
1091:Algebraic Geometry, A First Course
583:, the twisted cubic is the closed
127:. It is essentially unique, up to
25:
219:
204:
31:
1333:Hurwitz's automorphisms theorem
1070:The projection from a point on
1029:lines of any non-planar smooth
517:{\displaystyle (x,x^{2},x^{3})}
1558:Gonality of an algebraic curve
1469:Differential of the first kind
933:{\displaystyle (YW-Z^{2})^{2}}
921:
898:
874:
856:
847:
825:
685:{\displaystyle F_{1}=YW-Z^{2}}
635:{\displaystyle F_{0}=XZ-Y^{2}}
566:
542:
511:
479:
449:
417:
414:
378:{\displaystyle \nu :\mapsto .}
369:
311:
308:
305:
293:
264:
252:
214:
135:twisted cubic, therefore). In
1:
1701:Birkhoff–Grothendieck theorem
1411:Nagata's conjecture on curves
1282:Schoof–Elkies–Atkin algorithm
1156:Five points determine a conic
1093:, New York: Springer-Verlag,
535:. In homogeneous coordinates
1272:Supersingular elliptic curve
734:{\displaystyle F_{2}=XW-YZ.}
1479:Riemann's existence theorem
1406:Hilbert's sixteenth problem
1298:Elliptic curve cryptography
1211:Fundamental pair of periods
1752:
1609:Moduli of algebraic curves
129:projective transformation
1376:Cayley–Bacharach theorem
1303:Elliptic curve primality
969:{\displaystyle YW-Z^{2}}
808:{\displaystyle XZ-Y^{2}}
186:as the image of the map
143:that is not linear or a
40:This article includes a
1635:Riemann–Hurwitz formula
1599:Gromov–Witten invariant
1459:Compact Riemann surface
1247:Mazur's torsion theorem
589:homogeneous polynomials
527:The twisted cubic is a
163:of degree three on the
69:more precise citations.
1252:Modular elliptic curve
970:
934:
881:
809:
735:
686:
636:
573:
518:
456:
379:
271:
244:homogeneous coordinate
233:
179:
1166:Rational normal curve
1013:) of a twisted cubic
971:
935:
882:
810:
763:of the twisted cubic
736:
687:
637:
574:
519:
457:
380:
272:
242:which assigns to the
234:
178:
153:rational normal curve
149:complete intersection
1706:Stable vector bundle
1578:Weil reciprocity law
1568:Riemann–Roch theorem
1548:Brill–Noether theory
1484:Riemann–Roch theorem
1401:Genus–degree formula
1262:Mordell–Weil theorem
1237:Division polynomials
1089:Harris, Joe (1992),
990:Given six points in
944:
895:
819:
783:
697:
647:
597:
539:
476:
402:
284:
249:
193:
1529:Structure of curves
1421:Quartic plane curve
1343:Hyperelliptic curve
1323:De Franchis theorem
1267:Nagell–Lutz theorem
979:Any four points on
759:More strongly, the
1536:Divisors on curves
1328:Faltings's theorem
1277:Schoof's algorithm
1257:Modularity theorem
1044:The projection of
966:
930:
877:
805:
731:
682:
632:
569:
529:projective variety
514:
452:
375:
267:
229:
180:
141:projective variety
137:algebraic geometry
118:projective 3-space
42:list of references
1723:
1722:
1719:
1718:
1630:Hasse–Witt matrix
1573:Weierstrass point
1520:Smooth completion
1489:TeichmĂĽller space
1391:Cubic plane curve
1311:
1310:
1225:Arithmetic theory
1206:Elliptic integral
1201:Elliptic function
1035:algebraic variety
761:homogeneous ideal
467:point at infinity
95:
94:
87:
16:(Redirected from
1743:
1736:Algebraic curves
1563:Jacobian variety
1533:
1436:Riemann surfaces
1426:Real plane curve
1386:Cramer's paradox
1366:BĂ©zout's theorem
1191:
1140:algebraic curves
1132:
1125:
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1103:
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572:{\displaystyle }
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390:coordinate patch
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270:{\displaystyle }
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147:, in fact not a
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65:this article by
56:inline citations
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1678:Delta invariant
1649:
1618:
1582:
1543:Abel–Jacobi map
1524:
1498:
1494:Torelli theorem
1464:Dessin d'enfant
1444:Belyi's theorem
1430:
1416:PlĂĽcker formula
1347:
1338:Hurwitz surface
1307:
1286:
1220:
1194:Analytic theory
1186:Elliptic curves
1180:
1161:Projective line
1148:Rational curves
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1085:
1031:algebraic curve
956:
942:
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746:quadratic forms
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46:related reading
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1171:Riemann sphere
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1054:cuspidal cubic
1042:
1011:secant variety
995:
988:
977:
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959:
955:
952:
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940:is in it, but
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184:parametrically
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107:rational curve
93:
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50:external links
39:
37:
30:
24:
18:Twisted cubics
14:
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1658:
1656:
1655:Singularities
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1505:
1503:Constructions
1501:
1495:
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1487:
1485:
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1477:
1475:
1474:Klein quartic
1472:
1470:
1467:
1465:
1462:
1460:
1457:
1455:
1454:Bolza surface
1452:
1450:
1449:Bring's curve
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1382:
1381:Conic section
1379:
1377:
1374:
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1364:
1362:
1361:AF+BG theorem
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1176:Twisted cubic
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1100:0-387-97716-3
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1082:
1077:
1076:conic section
1073:
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768:
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762:
757:
756:, and so on.
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105:is a smooth,
104:
103:twisted cubic
100:
89:
86:
78:
75:February 2022
68:
64:
58:
57:
51:
47:
43:
38:
29:
28:
19:
1640:Prym variety
1614:Stable curve
1604:Hodge bundle
1594:ELSV formula
1396:Fermat curve
1353:Plane curves
1316:Higher genus
1291:Applications
1216:Modular form
1175:
1090:
1071:
1060:
1049:
1045:
1038:
1018:
1014:
1007:secant lines
991:
984:
980:
774:
764:
758:
753:
749:
743:
580:
526:
471:affine curve
464:
394:moment curve
387:
241:
181:
161:Veronese map
145:hypersurface
132:
120:
109:
102:
96:
81:
72:
61:Please help
53:
1515:Polar curve
99:mathematics
67:introducing
1510:Dual curve
1138:Topics in
1083:References
771:Properties
277:the value
171:Definition
125:skew curve
1623:Morphisms
1371:Bitangent
1074:yields a
1063:yields a
1052:yields a
954:−
908:−
891:, since
866:−
851:−
835:−
793:−
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415:↦
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116:three in
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533:quadrics
1683:Tacnode
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1003:tangent
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1663:Acnode
1587:Moduli
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661:=
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628:2
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614:X
611:=
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581:P
567:]
564:W
561::
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546:X
543:[
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370:]
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357::
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341::
338:T
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320:3
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294:[
291::
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256:S
253:[
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200::
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20:)
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