223:
has higher order singularities then these are counted as multiple double points according to an analysis of the nature of the singularity. For example an ordinary triple point is counted as 3 double points. Again, complex points and points at infinity are included in these counts. The corrected form
1202:
Curves are classified into types according to their Plücker invariants. The Plücker equations together with the restriction that the Plücker invariants must all be natural numbers greatly limits the number of possible types for curves of a given degree. Curves which are projectively equivalent have
40:, common to both the curve and its dual, is connected to the other invariants by similar formulae. These formulae, and the fact that each of the invariants must be a positive integer, place quite strict limitations on their possible values.
495:
inflections. If the cubic degenerates and gets a double point, then 6 points converge to the singular point and only 3 inflection remain along the singular curve. If the cubic degenerates and gets a cusp then only one inflection remains.
902:
686:
591:
793:
378:
1065:
295:
1185:
450:
1118:
115:
with multiplicities properly counted. (This includes complex points and points at infinity since the curves are taken to be subsets of the complex projective plane.) Similarly,
490:
976:
189:
691:
The four equations given so far are, in fact, dependent, so any three may be used to derive the remaining one. From them, given any three of the six invariants,
1203:
the same type, though curves of the same type are not, in general, projectively equivalent. Curves of degree 2, conic sections, have a single type given by
910:
Altogether there are four independent equations in 7 unknowns, and with them any three of these invariants can be used to compute the remaining four.
804:
597:
505:
28:, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of
136:
720:
310:
982:
230:
1124:
1703:
408:
1071:
1698:
922:
is non-singular, or equivalently δ and κ are 0, so the remaining invariants can be computed in terms of
1693:
1728:
461:
932:
145:
49:
205:, let δ be the number that are ordinary, i.e. that have distinct tangents (these are also called
53:
8:
1710:
1670:
1191:
37:
300:
Similarly, let δ be the number of ordinary double points, and κ the number of cusps of
216:
25:
396:
703:
48:
A curve in this context is defined by a non-degenerate algebraic equation in the
29:
1664:
1344:
1722:
198:
128:
68:. In the correspondence between the projective plane and its dual, points on
17:
87:
The first two invariants covered by the Plücker formulas are the degree
897:{\displaystyle g={1 \over 2}(d^{*}-1)(d^{*}-2)-\delta ^{*}-\kappa ^{*}}
65:
33:
388:
1335:
Curves of types (ii) and (iii) are the rational cubics and are call
681:{\displaystyle \kappa =3d^{*}(d^{*}-2)-6\delta ^{*}-8\kappa ^{*}.\,}
127:
that are lines through a given point on the plane; so for example a
1218:
For curves of degree 3 there are three possible types, given by:
1194:
is of genus 3 and has 28 bitangents and 24 points of inflection.
207:
120:
212:
1350:
For curves of degree 4 there are 10 possible types, given by:
499:
Note that the first two Plücker equations have dual versions:
1343:
respectively. Curves of type (i) are the nonsingular cubics (
586:{\displaystyle d=d^{*}(d^{*}-1)-2\delta ^{*}-3\kappa ^{*},\,}
383:
The geometric interpretation of an ordinary double point of
43:
788:{\displaystyle g={1 \over 2}(d-1)(d-2)-\delta -\kappa .}
373:{\displaystyle \kappa ^{*}=3d(d-2)-6\delta -8\kappa .\,}
1060:{\displaystyle \delta ^{*}={1 \over 2}d(d-2)(d-3)(d+3)}
387:
is a line that is tangent to the curve at two points (
1127:
1074:
985:
935:
807:
723:
600:
508:
464:
411:
313:
233:
148:
402:Consider for instance, the case of a smooth cubic:
699:, δ, δ, κ, κ, the remaining three can be computed.
1179:
1112:
1059:
970:
896:
787:
680:
585:
484:
444:
372:
289:
183:
52:. Lines in this plane correspond to points in the
290:{\displaystyle d^{*}=d(d-1)-2\delta -3\kappa .\,}
56:and the lines tangent to a given algebraic curve
1720:
391:) and the geometric interpretation of a cusp of
194:but this must be corrected for singular curves.
111:is the number of times a given line intersects
219:, i.e. having a single tangent (spinodes). If
918:An important special case is when the curve
60:correspond to points in an algebraic curve
304:. Then the second Plücker equation states
1109:
967:
710:, classically known as the deficiency of
677:
582:
481:
369:
286:
180:
139:, the first Plücker equation states that
1691:
1180:{\displaystyle g={1 \over 2}(d-1)(d-2).}
445:{\displaystyle d=3,\ \delta =\kappa =0}
1721:
1662:
913:
44:Plücker invariants and basic equations
1712:A Treatise on the Higher Plane Curves
1113:{\displaystyle \kappa ^{*}=3d(d-2)\,}
32:to corresponding invariants of their
926:only. In this case the results are:
455:The above formula shows that it has
224:is of the first Plücker equation is
798:This is equal to the dual quantity
215:, and let κ be the number that are
13:
14:
1740:
1190:So, for example, a non-singular
131:has degree and class both 2. If
485:{\displaystyle \kappa ^{*}=9\,}
1677:
1656:
1197:
1171:
1159:
1156:
1144:
1106:
1094:
1054:
1042:
1039:
1027:
1024:
1012:
971:{\displaystyle d^{*}=d(d-1)\,}
964:
952:
865:
846:
843:
824:
767:
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639:
620:
544:
525:
345:
333:
262:
250:
184:{\displaystyle d^{*}=d(d-1)\,}
177:
165:
1:
1649:
72:correspond to lines tangent
7:
1699:Encyclopedia of Mathematics
907:and is a positive integer.
36:. The invariant called the
10:
1745:
1692:Shokurov, V. V. (2001) ,
99:, classically called the
50:complex projective plane
1663:Hilton, Harold (1920).
399:(stationary tangent).
80:can be identified with
1709:Salmon, George (1879)
1666:Plane Algebraic Curves
1181:
1114:
1061:
972:
898:
789:
682:
587:
486:
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374:
291:
185:
1182:
1115:
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54:dual projective plane
1125:
1072:
983:
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805:
721:
714:, can be defined as
598:
506:
462:
409:
311:
231:
146:
1192:quartic plane curve
914:Non-singular curves
397:point of inflection
1694:"Plücker formulas"
1669:. Oxford. p.
1177:
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968:
894:
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678:
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181:
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1007:
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426:
119:is the number of
107:. Geometrically,
76:, so the dual of
1736:
1729:Algebraic curves
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30:algebraic curves
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1345:elliptic curves
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213:isolated points
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95:and the degree
46:
22:Plücker formula
12:
11:
5:
1742:
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1717:
1716:
1715:pp. 64ff.
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389:double tangent
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26:Julius Plücker
24:, named after
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3:
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1683:Hilton p. 264
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702:Finally, the
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199:double points
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141:
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138:
137:singularities
134:
130:
129:conic section
126:
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91:of the curve
90:
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1249:
1232:
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1212:
1211:=2, δ=δ=κ=κ=
1208:
1204:
1201:
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57:
47:
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15:
1198:Curve types
64:called the
34:dual curves
18:mathematics
1650:References
66:dual curve
1704:EMS Press
1166:−
1151:−
1101:−
1081:∗
1077:κ
1034:−
1019:−
992:∗
988:δ
959:−
942:∗
890:∗
886:κ
882:−
877:∗
873:δ
869:−
860:−
855:∗
838:−
833:∗
780:κ
777:−
774:δ
771:−
762:−
747:−
670:∗
666:κ
659:−
654:∗
650:δ
643:−
634:−
629:∗
616:∗
602:κ
575:∗
571:κ
564:−
559:∗
555:δ
548:−
539:−
534:∗
521:∗
471:∗
467:κ
434:κ
428:δ
364:κ
358:−
355:δ
349:−
340:−
320:∗
316:κ
281:κ
275:−
272:δ
266:−
257:−
240:∗
211:) or are
172:−
155:∗
1723:Category
1341:cuspidal
121:tangents
197:Of the
135:has no
1570:(viii)
425:
1544:(vii)
1440:(iii)
1337:nodal
1308:(iii)
704:genus
395:is a
217:cusps
208:nodes
101:class
38:genus
1596:(ix)
1518:(vi)
1466:(iv)
1414:(ii)
1356:Type
1339:and
1282:(ii)
1224:Type
1215:=0.
20:, a
1671:201
1622:(x)
1492:(v)
1388:(i)
1347:).
1256:(i)
706:of
201:of
123:to
103:of
16:In
1725::
1702:,
1696:,
1643:0
1617:0
1591:0
1565:0
1539:1
1513:1
1510:10
1487:1
1484:12
1461:2
1458:16
1452:10
1435:2
1432:18
1426:16
1420:10
1409:3
1406:24
1400:28
1394:12
1329:0
1303:0
1277:1
695:,
84:.
1673:.
1640:0
1637:3
1634:1
1631:0
1628:3
1625:4
1614:2
1611:2
1608:1
1605:1
1602:4
1599:4
1588:4
1585:1
1582:2
1579:2
1576:5
1573:4
1562:6
1559:0
1556:4
1553:3
1550:6
1547:4
1536:8
1533:2
1530:1
1527:0
1524:6
1521:4
1507:1
1504:4
1501:1
1498:7
1495:4
1481:0
1478:8
1475:2
1472:8
1469:4
1455:1
1449:0
1446:9
1443:4
1429:0
1423:1
1417:4
1403:0
1397:0
1391:4
1382:g
1378:κ
1375:κ
1372:δ
1369:δ
1365:d
1360:d
1326:1
1323:1
1320:0
1317:0
1314:3
1311:3
1300:3
1297:0
1294:0
1291:1
1288:4
1285:3
1274:9
1271:0
1268:0
1265:0
1262:6
1259:3
1250:g
1246:κ
1243:κ
1240:δ
1237:δ
1233:d
1228:d
1213:g
1209:d
1207:=
1205:d
1175:.
1172:)
1169:2
1163:d
1160:(
1157:)
1154:1
1148:d
1145:(
1140:2
1137:1
1132:=
1129:g
1107:)
1104:2
1098:d
1095:(
1092:d
1089:3
1086:=
1055:)
1052:3
1049:+
1046:d
1043:(
1040:)
1037:3
1031:d
1028:(
1025:)
1022:2
1016:d
1013:(
1010:d
1005:2
1002:1
997:=
965:)
962:1
956:d
953:(
950:d
947:=
938:d
924:d
920:C
866:)
863:2
851:d
847:(
844:)
841:1
829:d
825:(
820:2
817:1
812:=
809:g
783:.
768:)
765:2
759:d
756:(
753:)
750:1
744:d
741:(
736:2
733:1
728:=
725:g
712:C
708:C
697:d
693:d
675:.
662:8
646:6
640:)
637:2
625:d
621:(
612:d
608:3
605:=
580:,
567:3
551:2
545:)
542:1
530:d
526:(
517:d
513:=
510:d
479:9
476:=
440:0
437:=
431:=
422:,
419:3
416:=
413:d
393:C
385:C
367:.
361:8
352:6
346:)
343:2
337:d
334:(
331:d
328:3
325:=
302:C
284:.
278:3
269:2
263:)
260:1
254:d
251:(
248:d
245:=
236:d
221:C
203:C
178:)
175:1
169:d
166:(
163:d
160:=
151:d
133:C
125:C
117:d
113:C
109:d
105:C
97:d
93:C
89:d
82:C
78:C
74:C
70:C
62:C
58:C
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