31:
178:
206:
117:
446:
called 'double lines'. This is because a double line intersects every line in the plane, since lines in the projective plane intersect, with multiplicity two because it is doubled, and thus satisfies the same intersection condition (intersection of multiplicity two) as a nondegenerate conic that is
191:
is one of the earliest examples of enumerative geometry. This problem asks for the number and construction of circles that are tangent to three given circles, points or lines. In general, the problem for three given circles has eight solutions, which can be seen as 2, each tangency condition
192:
imposing a quadratic condition on the space of circles. However, for special arrangements of the given circles, the number of solutions may also be any integer from 0 (no solutions) to six; there is no arrangement for which there are seven solutions to
Apollonius' problem.
462:. From 32, 31 must be subtracted and attributed to the Veronese, to leave the correct answer (from the point of view of geometry), namely 1. This process of attributing intersections to 'degenerate' cases is a typical geometric introduction of a '
371:
NaĂŻve application of dimension counting and BĂ©zout's theorem yields incorrect results, as the following example shows. In response to these problems, algebraic geometers introduced vague "
351:
value in broader areas. The specific needs of enumerative geometry were not addressed until some further attention was paid to them in the 1960s and 1970s (as pointed out for example by
559:
522:
771:
649:
745:
699:
719:
669:
619:
599:
579:
472:
was to overcome the apparently arbitrary nature of these interventions; this aspect goes beyond the foundational question of the
Schubert calculus itself.
363:
as part of his foundational programme 1942–6, and again subsequently), but this did not exhaust the proper domain of enumerative questions.
949:; de la Ossa, Xenia; Green, Paul; Parks, Linda (1991). "A pair of Calabi-Yau manifolds as an exactly soluble superconformal field theory".
1045:
975:
Kleiman, S.; Strømme, S. A.; Xambó, S. (1987), "Sketch of a verification of
Schubert's number 5819539783680 of twisted cubics",
992:
227:
138:
851:
5819539783680 The number of twisted cubic curves tangent to 12 given quadric surfaces in general position in 3-space (
1020:
929:
901:
253:
164:
74:
52:
235:
146:
45:
316:
917:
469:
231:
142:
840:
666841088 The number of quadric surfaces tangent to 9 given quadric surfaces in general position in 3-space (
458:
says 5 general quadrics in 5-space will intersect in 32 = 2 points. But the relevant quadrics here are not in
395:
339:
Enumerative geometry saw spectacular development towards the end of the nineteenth century, at the hands of
1086:
1091:
312:
817:
415:
531:
494:
402:, as passing through a given point imposes a linear condition. Similarly, tangency to a given line
216:
127:
39:
399:
391:
220:
131:
56:
275:
789:
750:
628:
406:(tangency is intersection with multiplicity two) is one quadratic condition, so determined a
188:
181:
781:
Some of the historically important examples of enumerations in algebraic geometry include:
724:
1030:
1002:
677:
284:
463:
8:
356:
328:
100:
20:
294:
99:
concerned with counting numbers of solutions to geometric questions, mainly by means of
1065:
704:
654:
604:
584:
581:
a positive integer, then there are only a finite number of rational curves with degree
564:
320:
308:
304:
270:
96:
1016:
1010:
988:
962:
925:
897:
828:
811:
489:
344:
280:
455:
1061:
1057:
980:
958:
701:
from the string theoretical viewpoint gives numbers of degree d rational curves on
459:
423:
387:
383:
340:
1026:
998:
946:
481:
821:
485:
352:
1015:, Reprint of the 1879 original (in German), Berlin-New York: Springer-Verlag,
979:, Lecture Notes in Math., vol. 1266, Berlin: Springer, pp. 156–180,
266:
A number of tools, ranging from the elementary to the more advanced, include:
1080:
804:
800:
796:
379:
300:
177:
747:. Prior to this, algebraic geometers could calculate these numbers only for
360:
372:
855:, p.184) (S. Kleiman, S. A. Strømme & S. Xambó
348:
303:
of curves, maps and other geometric objects, sometimes via the theory of
88:
1069:
984:
419:
288:
1043:
674:
In 1991 the paper about mirror symmetry on the quintic threefold in
205:
116:
833:
4407296 The number of conics tangent to 8 general quadric surfaces
407:
366:
16:
Branch of algebraic geometry concerned with counting solutions
945:
788:
8 The number of circles tangent to 3 general circles (the
293:
The connection of counting intersections with cohomology is
785:
2 The number of lines meeting 4 general lines in space
375:", which were only rigorously justified decades later.
1009:
Schubert, Hermann (1979) , Kleiman, Steven L. (ed.),
753:
727:
707:
680:
657:
631:
607:
587:
567:
534:
497:
974:
856:
1044:Bashelor, Andrew; Ksir, Amy; Traves, Will (2008).
765:
739:
713:
693:
663:
643:
613:
593:
573:
553:
516:
347:, which has proved of fundamental geometrical and
418:consisting of all such quadrics is not without a
390:of dimension 5, taking their six coefficients as
1078:
625:This conjecture has been resolved in the case
367:Fudge factors and Hilbert's fifteenth problem
327:Enumerative geometry is very closely tied to
827:609250 The number of conics on a general
234:. Unsourced material may be challenged and
145:. Unsourced material may be challenged and
1046:"Enumerative Algebraic Geometry of Conics"
422:. In fact each such quadric contains the
254:Learn how and when to remove this message
165:Learn how and when to remove this message
75:Learn how and when to remove this message
1008:
876:
852:
841:
176:
38:This article includes a list of general
810:2875 The number of lines on a general
1079:
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834:
524:and reached the following conjecture.
484:studied the counting of the number of
343:. He introduced it for the purpose of
795:27 The number of lines on a smooth
475:
334:
232:adding citations to reliable sources
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143:adding citations to reliable sources
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24:
382:tangent to five given lines in the
13:
977:Space curves (Rocca di Papa, 1985)
44:it lacks sufficient corresponding
14:
1103:
1037:
894:Foundations of Algebraic Geometry
1012:Kalkül der abzählenden Geometrie
891:
879:Kalkül der abzählenden Geometrie
818:conics tangent to 5 plane conics
561:be a general quintic threefold,
426:, which parametrizes the conics
359:had been rigorously defined (by
204:
115:
29:
651:, but is still open for higher
319:gave a significant progress in
1062:10.1080/00029890.2008.11920584
938:
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870:
554:{\displaystyle X\subset P^{4}}
517:{\displaystyle X\subset P^{4}}
1:
863:
396:five points determine a conic
963:10.1016/0550-3213(91)90292-6
195:
7:
776:
470:Hilbert's fifteenth problem
10:
1108:
386:. The conics constitute a
106:
18:
416:linear system of divisors
378:As an example, count the
313:Gromov–Witten invariants
766:{\displaystyle d\leq 5}
644:{\displaystyle d\leq 9}
400:general linear position
398:, if the points are in
392:homogeneous coordinates
59:more precise citations.
767:
741:
740:{\displaystyle d>0}
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285:characteristic classes
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877:Schubert, H. (1879).
820:in general position (
790:problem of Apollonius
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694:{\displaystyle P^{4}}
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283:, and more generally
189:problem of Apollonius
182:Circles of Apollonius
180:
816:3264 The number of
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357:Intersection numbers
228:improve this section
139:improve this section
93:enumerative geometry
1087:Intersection theory
1050:Amer. Math. Monthly
922:Intersection Theory
329:intersection theory
101:intersection theory
21:Intersection theory
1092:Algebraic geometry
985:10.1007/BFb0078183
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476:Clemens conjecture
321:Clemens conjecture
309:quantum cohomology
305:quantum cohomology
271:Dimension counting
185:
97:algebraic geometry
994:978-3-540-18020-3
951:Nuclear Physics B
881:(published 1979).
829:quintic threefold
812:quintic threefold
714:{\displaystyle X}
664:{\displaystyle d}
614:{\displaystyle X}
594:{\displaystyle d}
574:{\displaystyle d}
490:quintic threefold
345:Schubert calculus
335:Schubert calculus
281:Schubert calculus
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384:projective plane
341:Hermann Schubert
295:Poincaré duality
276:BĂ©zout's theorem
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307:. The study of
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892:Weil, Andre.
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853:Schubert 1879
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842:Schubert 1879
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797:cubic surface
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301:moduli spaces
299:The study of
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244:February 2023
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213:This section
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155:February 2023
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27:
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1056:(8): 701–7.
1053:
1049:
1011:
976:
957:(1): 21–74.
954:
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940:
921:
912:
893:
887:
878:
872:
835:Fulton (1984
780:
673:
624:
479:
468:
464:fudge factor
454:The general
453:
448:
445:
439:
435:
431:
411:
403:
377:
370:
338:
326:
265:
250:
241:
226:Please help
214:
186:
161:
152:
137:Please help
125:
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71:
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43:
846:Fulton 1984
349:topological
89:mathematics
57:introducing
1081:Categories
864:References
844:, p.106) (
482:H. Clemens
420:base locus
361:André Weil
289:cohomology
40:references
19:See also:
848:, p. 193)
837:, p. 193)
758:≤
636:≤
539:⊂
502:⊂
215:does not
196:Key tools
126:does not
1070:27642583
777:Examples
721:for all
480:In 1984
1031:0555576
1003:0908713
822:Chasles
449:tangent
408:quadric
236:removed
221:sources
147:removed
132:sources
107:History
53:improve
1068:
1029:
1019:
1001:
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928:
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805:Cayley
801:Salmon
394:, and
42:, but
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488:on a
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1017:ISBN
989:ISBN
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857:1987
803:and
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528:Let
315:and
219:any
217:cite
187:The
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287:in
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