180:
22:
501:
506:
whose left-hand side is the term of degree one of the Taylor expansion. Thus, if this term is zero, the tangent may not be defined in the standard way, either because it does not exist or a special definition must be provided.
39:
567:
280:
655:
In case of a real variety (that is the set of the points with real coordinates of a variety defined by polynomials with real coefficients), the variety is a
86:
745:
58:
349:
659:
near every regular point. But it is important to note that a real variety may be a manifold and have singular points. For example the equation
65:
828:
72:
876:
756:
is non-singular; this implies that the singular point has multiplicity two and the tangent cone is not singular outside its vertex.
105:
54:
820:
712:
where all derivatives exist). Analysis of these singular points can be reduced to the algebraic variety case by considering the
636:. It is always true that almost all points are non-singular, in the sense that the non-singular points form a set that is both
696:
As the notion of singular points is a purely local property, the above definition can be extended to cover the wider class of
43:
898:
903:
770:
79:
868:
32:
815:
775:
685:
790:
519:
322:
184:
780:
577:
123:
681:
but has a singular point at the origin. This may be explained by saying that the curve has two
872:
824:
713:
682:
678:
581:
244:
236:
129:
842:
810:
645:
610:
838:
846:
834:
697:
606:
293:
753:
649:
153:
892:
785:
733:
723:
301:
173:
169:
145:
511:
311:
211:
148:
at the variety may not be regularly defined. In case of varieties defined over the
860:
232:
188:
179:
149:
144:
that is 'special' (so, singular), in the geometric sense that at this point the
119:
21:
648:, as well as for the usual topology, in the case of varieties defined over the
765:
590:
641:
496:{\displaystyle (x-x_{0})F'_{x}(x_{0},y_{0})+(y-y_{0})F'_{y}(x_{0},y_{0})=0,}
656:
637:
219:
156:. A point of an algebraic variety that is not singular is said to be
178:
609:
of the first-order partial derivatives of the polynomials has a
160:. An algebraic variety that has no singular point is said to be
15:
619:
that is lower than the rank at other points of the variety.
522:
352:
247:
748:, certain special singular points were also called
46:. Unsourced material may be challenged and removed.
561:
495:
274:
867:. Annals of Mathematics Studies. Vol. 61.
589:being defined as the common zeros of several
8:
343:of such a curve is defined by the equation
521:
475:
462:
446:
433:
408:
395:
379:
366:
351:
246:
106:Learn how and when to remove this message
865:Singular Points of Complex Hypersurfaces
688:that cut the real branch at the origin.
152:, this notion generalizes the notion of
55:"Singular point of an algebraic variety"
802:
752:. A node is a singular point where the
7:
44:adding citations to reliable sources
726:of the mapping truncated at degree
605:to be a singular point is that the
222:may not be correctly defined there.
692:Singular points of smooth mappings
562:{\displaystyle F(x,y,z,\ldots )=0}
14:
628:that are not singular are called
580:simultaneously vanish. A general
321:The reason for this is that, in
168:. The concept is generalized to
20:
31:needs additional citations for
550:
526:
481:
455:
439:
420:
414:
388:
372:
353:
263:
251:
1:
206:crosses itself at the origin
746:classical algebraic geometry
771:Resolution of singularities
593:, the condition on a point
576:are those at which all the
325:, the tangent at the point
920:
869:Princeton University Press
172:in the modern language of
776:Singular point of a curve
700:mappings (functions from
644:in the variety (for the
275:{\displaystyle F(x,y)=0}
563:
497:
276:
223:
791:Zariski tangent space
564:
498:
323:differential calculus
277:
214:of this curve. It is
185:plane algebraic curve
182:
128:singular point of an
819:. Berlin, New York:
716:of the mapping. The
520:
350:
245:
40:improve this article
899:Algebraic varieties
578:partial derivatives
454:
387:
904:Singularity theory
871:. pp. 12–13.
816:Algebraic Geometry
781:Singularity theory
559:
493:
442:
375:
300:at a point if the
272:
224:
210:. The origin is a
154:local non-flatness
124:algebraic geometry
830:978-0-387-90244-9
811:Hartshorne, Robin
732:and deleting the
683:complex conjugate
679:analytic manifold
582:algebraic variety
510:In general for a
237:implicit equation
218:because a single
130:algebraic variety
116:
115:
108:
90:
911:
883:
882:
857:
851:
850:
807:
731:
721:
711:
705:
676:
646:Zariski topology
627:
618:
604:
598:
588:
568:
566:
565:
560:
502:
500:
499:
494:
480:
479:
467:
466:
450:
438:
437:
413:
412:
400:
399:
383:
371:
370:
342:
317:
309:
291:
281:
279:
278:
273:
209:
205:
143:
137:
111:
104:
100:
97:
91:
89:
48:
24:
16:
919:
918:
914:
913:
912:
910:
909:
908:
889:
888:
887:
886:
879:
859:
858:
854:
831:
821:Springer-Verlag
809:
808:
804:
799:
762:
742:
727:
717:
707:
701:
694:
677:defines a real
660:
650:complex numbers
623:
614:
607:Jacobian matrix
600:
594:
584:
574:singular points
518:
517:
471:
458:
429:
404:
391:
362:
348:
347:
340:
333:
326:
318:at this point.
315:
305:
294:smooth function
287:
243:
242:
229:
207:
192:
139:
133:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
917:
915:
907:
906:
901:
891:
890:
885:
884:
877:
852:
829:
823:. p. 33.
801:
800:
798:
795:
794:
793:
788:
783:
778:
773:
768:
761:
758:
754:Hessian matrix
741:
738:
722:th jet is the
693:
690:
570:
569:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
504:
503:
492:
489:
486:
483:
478:
474:
470:
465:
461:
457:
453:
449:
445:
441:
436:
432:
428:
425:
422:
419:
416:
411:
407:
403:
398:
394:
390:
386:
382:
378:
374:
369:
365:
361:
358:
355:
338:
331:
296:is said to be
284:
283:
271:
268:
265:
262:
259:
256:
253:
250:
235:defined by an
228:
225:
191:) of equation
170:smooth schemes
114:
113:
96:September 2008
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
916:
905:
902:
900:
897:
896:
894:
880:
878:0-691-08065-8
874:
870:
866:
862:
856:
853:
848:
844:
840:
836:
832:
826:
822:
818:
817:
812:
806:
803:
796:
792:
789:
787:
786:Smooth scheme
784:
782:
779:
777:
774:
772:
769:
767:
764:
763:
759:
757:
755:
751:
747:
739:
737:
735:
734:constant term
730:
725:
724:Taylor series
720:
715:
710:
704:
699:
691:
689:
687:
684:
680:
674:
670:
667:
663:
658:
653:
651:
647:
643:
639:
635:
631:
626:
620:
617:
612:
608:
603:
597:
592:
587:
583:
579:
575:
556:
553:
547:
544:
541:
538:
535:
532:
529:
523:
516:
515:
514:
513:
508:
490:
487:
484:
476:
472:
468:
463:
459:
451:
447:
443:
434:
430:
426:
423:
417:
409:
405:
401:
396:
392:
384:
380:
376:
367:
363:
359:
356:
346:
345:
344:
337:
330:
324:
319:
313:
308:
303:
302:Taylor series
299:
295:
290:
269:
266:
260:
257:
254:
248:
241:
240:
239:
238:
234:
226:
221:
217:
213:
203:
199:
195:
190:
186:
181:
177:
175:
174:scheme theory
171:
167:
163:
159:
155:
151:
147:
146:tangent space
142:
136:
132:
131:
125:
121:
110:
107:
99:
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
864:
861:Milnor, John
855:
814:
805:
749:
743:
728:
718:
708:
702:
695:
672:
668:
665:
661:
654:
633:
630:non-singular
629:
624:
621:
615:
601:
595:
585:
573:
571:
512:hypersurface
509:
505:
335:
328:
320:
306:
297:
288:
285:
230:
215:
212:double point
201:
197:
193:
165:
162:non-singular
161:
157:
140:
134:
127:
120:mathematical
117:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
591:polynomials
233:plane curve
189:cubic curve
138:is a point
893:Categories
847:0367.14001
797:References
766:Milnor map
622:Points of
227:Definition
66:newspapers
548:…
427:−
360:−
314:at least
122:field of
863:(1969).
813:(1977).
760:See also
686:branches
657:manifold
452:′
385:′
298:singular
216:singular
204:+ 1) = 0
839:0463157
634:regular
220:tangent
158:regular
118:In the
80:scholar
875:
845:
837:
827:
698:smooth
286:where
208:(0, 0)
166:smooth
82:
75:
68:
61:
53:
750:nodes
740:Nodes
642:dense
312:order
292:is a
150:reals
87:JSTOR
73:books
873:ISBN
825:ISBN
714:jets
640:and
638:open
611:rank
572:the
310:has
183:The
126:, a
59:news
843:Zbl
744:In
706:to
675:= 0
664:+ 2
652:).
632:or
613:at
599:of
304:of
187:(a
164:or
42:by
895::
841:.
835:MR
833:.
736:.
671:−
334:,
231:A
196:−
176:.
881:.
849:.
729:k
719:k
709:R
703:M
673:x
669:y
666:x
662:y
625:V
616:P
602:V
596:P
586:V
557:0
554:=
551:)
545:,
542:z
539:,
536:y
533:,
530:x
527:(
524:F
491:,
488:0
485:=
482:)
477:0
473:y
469:,
464:0
460:x
456:(
448:y
444:F
440:)
435:0
431:y
424:y
421:(
418:+
415:)
410:0
406:y
402:,
397:0
393:x
389:(
381:x
377:F
373:)
368:0
364:x
357:x
354:(
341:)
339:0
336:y
332:0
329:x
327:(
316:2
307:F
289:F
282:,
270:0
267:=
264:)
261:y
258:,
255:x
252:(
249:F
202:x
200:(
198:x
194:y
141:P
135:V
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
