3651:
2964:
1462:
4217:
1810:
3490:
2790:
3091:
2654:
1964:
2670:(1). (In fact, the above isomorphism is part of the usual correspondence between Weil divisors and Cartier divisors.) Finally, the dual of the twisting sheaf corresponds to the tautological line bundle (see below).
2550:
1211:
3474:
1303:
3309:
2372:
2047:
1872:
2782:
4103:
263:
3710:
3211:
326:
1693:
2195:
4096:
1606:
1116:
956:
3158:
2439:
1044:
1002:
3905:
3404:
2264:
365:
3945:
3346:
3240:
394:
3848:
3765:
1635:
1522:
4261:
Editorial note: this definition differs from
Hartshorne in that he does not take dual, but is consistent with the standard practice and the other parts of Knowledge (XXG).
1494:
134:
3646:{\displaystyle 0\to I\to {\mathcal {O}}_{\mathbb {P} ^{n}}{\overset {x_{i}\mapsto y_{i}}{\longrightarrow }}\operatorname {Sym} {\mathcal {O}}_{\mathbb {P} ^{n}}(1)\to 0,}
2141:
1681:
4246:
2111:
2084:
1551:
827:
796:
740:
677:
642:
535:
2959:{\displaystyle \mathbf {Spec} \left({\mathcal {O}}_{\mathbb {P} ^{n}}\right)=\mathbb {A} _{\mathbb {P} ^{n}}^{n+1}=\mathbb {A} ^{n+1}\times _{k}{\mathbb {P} ^{n}}}
769:
709:
2741:
873:
897:
847:
615:
595:
575:
555:
508:
488:
174:
154:
108:
88:
61:
2983:
3767:
this is equivalent to saying that it is a negative line bundle, meaning that minus its Chern class is the de Rham class of the standard Kähler form.
203:
Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as
2577:
1888:
4449:
4391:
4355:
4303:
2459:
1457:{\displaystyle {\begin{cases}\phi :\pi ^{-1}(U)\to U\times X\subseteq G_{n}(\mathbb {R} ^{n+k})\times X\\\phi (V,v)=(V,p(v))\end{cases}}}
1141:
3979:(Chern classes of tautological bundles is the algebraically independent generators of the cohomology ring of the infinite Grassmannian.)
3424:
1815:
where on the left the bracket means homotopy class and on the right is the set of isomorphism classes of real vector bundles of rank
617:, varying continuously. All that can stop the definition of the tautological bundle from this indication, is the difficulty that the
3257:
2321:
1981:
1830:
4212:{\displaystyle {\begin{cases}G_{n}(\mathbb {R} ^{n+k})\to \operatorname {End} (\mathbb {R} ^{n+k})\\V\mapsto p_{V}\end{cases}}}
2746:
213:
192:
since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a
Grassmannian is a
3663:
3167:
4328:
282:
3948:
329:
3716:. In practice both the notions (tautological line bundle and the dual of the twisting sheaf) are used interchangeably.
1805:{\displaystyle {\begin{cases}\to \operatorname {Vect} _{n}^{\mathbb {R} }(X)\\f\mapsto f^{*}(\gamma _{n})\end{cases}}}
4324:
443:
4378:
3713:
277:
2146:
4050:
3855:
1560:
1070:
910:
408:
64:
4467:
3477:
1975:: In turn, one can define a tautological bundle as a universal bundle; suppose there is a natural bijection
3103:
2392:
3947:; in other words the hyperplane bundle is the generator of the Picard group having positive degree (as a
2406:
1011:
969:
3966:
3874:
3374:
2234:
2143:
It is precisely the tautological bundle and, by restriction, one gets the tautological bundles over all
335:
3917:
3322:
3216:
370:
4337:
3822:
2086:
is the direct limit of compact spaces, it is paracompact and so there is a unique vector bundle over
597:, this is already almost the data required for a vector bundle: namely a vector space for each point
197:
4112:
2586:
1897:
1702:
1312:
420:
3745:
3415:
1611:
1502:
4428:, Annals of Mathematics Studies, vol. 76, Princeton, New Jersey: Princeton University Press,
4026:
Over a noncompact but paracompact base, this remains true provided one uses infinite
Grassmannian.
1477:
113:
3720:
452:
3982:
396:. The tautological line bundle and the hyperplane bundle are exactly the two generators of the
4445:
4405:
4387:
4351:
4320:
4299:
3739:
2053:
799:
427:
193:
4373:
4341:
4007:
3868:
2309:
2116:
1656:
447:
is heavily overloaded as it is, in mathematical terminology, and (worse) confusion with the
431:
204:
188:
177:
4433:
4401:
4365:
4313:
4224:
2089:
2062:
1529:
805:
774:
718:
655:
620:
513:
4429:
4397:
4383:
4361:
4309:
3851:
745:
685:
464:
448:
196:
for vector bundles. Because of this, the tautological bundle is important in the study of
67:
3808:
2685:
852:
4295:
4287:
3971:
882:
832:
645:
600:
580:
560:
540:
493:
473:
404:
159:
139:
93:
73:
46:
3803:= 1, the real tautological line bundle is none other than the well-known bundle whose
4461:
4001:
2974:
1497:
36:
4421:
3816:
3804:
3788:
3784:
3727:
2400:
1554:
959:
649:
468:
397:
40:
3086:{\displaystyle L=\mathbf {Spec} \left({\mathcal {O}}_{\mathbb {P} ^{n}}/I\right)}
4417:
3976:
3961:
3951:) and the tautological bundle is its opposite: the generator of negative degree.
3864:
3731:
2302:
876:
416:
269:
28:
1467:
which is clearly a homeomorphism. Hence, the result is a vector bundle of rank
3987:
3368:
712:
17:
4035:
In literature and textbooks, they are both often called canonical generators.
4409:
328:. The hyperplane bundle is the line bundle corresponding to the hyperplane (
4346:
2649:{\displaystyle {\begin{cases}O(H)\simeq O(1)\\f\mapsto fx_{0}\end{cases}}}
1959:{\displaystyle {\begin{cases}f_{E}:X\to G_{n}\\x\mapsto E_{x}\end{cases}}}
3719:
Over a field, its dual line bundle is the line bundle associated to the
644:
are going to intersect. Fixing this up is a routine application of the
879:
is one tautological bundle, and the other, just described, is of rank
3863:
In the case of projective space, where the tautological bundle is a
2545:{\displaystyle \Gamma (U,O(D))=\{f\in K|(f)+D\geq 0{\text{ on }}U\}}
2308:
In algebraic geometry, the hyperplane bundle is the line bundle (as
1206:{\displaystyle G_{n}(\mathbb {R} ^{n+k})\times \mathbb {R} ^{n+k}.}
1118:
as follows. The total space of the bundle is the set of all pairs (
3469:{\displaystyle \mathbf {Spec} (\operatorname {Sym} {\check {E}})}
415:. The sphere bundle of the standard bundle is usually called the
3912:
2666:
is, as usual, viewed as a global section of the twisting sheaf
1683:
It is a universal bundle in the sense: for each compact space
3304:{\displaystyle \mathbb {A} ^{n+1}\times _{k}\mathbb {P} ^{n}}
1138:; it is given the subspace topology of the Cartesian product
3923:
3880:
3670:
3605:
3509:
3015:
2816:
2367:{\displaystyle H=\mathbb {P} ^{n-1}\subset \mathbb {P} ^{n}}
2042:{\displaystyle =\operatorname {Vect} _{n}^{\mathbb {R} }(X)}
679:, that now do not intersect. With this, we have the bundle.
289:
220:
4205:
2642:
1952:
1867:{\displaystyle E\hookrightarrow X\times \mathbb {R} ^{n+k}}
1798:
1474:
The above definition continues to make sense if we replace
1450:
2294:.) The rest is exactly like the tautological line bundle.
2678:
In algebraic geometry, this notion exists over any field
463:
Grassmannians by definition are the parameter spaces for
426:
More generally, there are also tautological bundles on a
2777:{\displaystyle \mathbb {P} ^{n}=\operatorname {Proj} A}
90:, given a point in the Grassmannian corresponding to a
1233:
under π, it is given a structure of a vector space by
258:{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1),}
4227:
4106:
4053:
3920:
3911:
the dual vector bundle) of the hyperplane bundle or
3877:
3825:
3748:
3705:{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1)}
3666:
3493:
3427:
3377:
3356:
is the tautological line bundle as defined before if
3325:
3260:
3219:
3206:{\displaystyle \mathbb {A} _{\mathbb {P} ^{n}}^{n+1}}
3170:
3106:
2986:
2793:
2749:
2688:
2580:
2462:
2409:
2324:
2237:
2149:
2119:
2092:
2065:
1984:
1891:
1833:
1696:
1659:
1614:
1563:
1532:
1505:
1480:
1306:
1144:
1073:
1014:
972:
913:
885:
855:
835:
808:
798:
that are their kernels, when considered as (rays of)
777:
748:
721:
688:
682:
The projective space case is included. By convention
658:
623:
603:
583:
563:
543:
516:
496:
476:
373:
338:
285:
216:
162:
142:
116:
96:
76:
49:
43:
321:{\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(1)}
1269:). Finally, to see local triviality, given a point
407:'s "K-theory", the tautological line bundle over a
4240:
4211:
4090:
3939:
3899:
3842:
3759:
3704:
3645:
3484:of finite rank. Since we have the exact sequence:
3468:
3398:
3340:
3303:
3234:
3205:
3152:
3085:
2958:
2776:
2735:
2648:
2544:
2433:
2366:
2258:
2189:
2135:
2105:
2078:
2041:
1958:
1866:
1804:
1675:
1629:
1600:
1545:
1516:
1488:
1456:
1205:
1110:
1038:
996:
950:
891:
867:
841:
821:
790:
763:
734:
711:may usefully carry the tautological bundle in the
703:
671:
636:
609:
589:
569:
549:
529:
502:
482:
388:
359:
320:
257:
168:
148:
128:
102:
82:
55:
3799:In fact, it is straightforward to show that, for
2215:-space is defined as follows. The total space of
3100:is the ideal sheaf generated by global sections
648:device, so that the bundle projection is from a
441:has dropped out of favour, on the grounds that
4270:
2674:Tautological line bundle in algebraic geometry
2682:. The concrete definition is as follows. Let
1819:. The inverse map is given as follows: since
8:
3660:, as defined above, corresponds to the dual
2539:
2493:
3811:. For a full proof of the above fact, see.
3418:. (cf. Hartshorne, Ch. I, the end of § 4.)
186:The tautological bundle is also called the
2441:one defines the corresponding line bundle
2190:{\displaystyle G_{n}(\mathbb {R} ^{n+k}).}
4345:
4294:, Advanced Book Classics (2nd ed.),
4232:
4226:
4196:
4167:
4163:
4162:
4134:
4130:
4129:
4119:
4107:
4105:
4091:{\displaystyle G_{n}(\mathbb {R} ^{n+k})}
4073:
4069:
4068:
4058:
4052:
3922:
3921:
3919:
3879:
3878:
3876:
3827:
3826:
3824:
3795:≥ 1. This remains true over other fields.
3750:
3749:
3747:
3682:
3678:
3677:
3675:
3669:
3668:
3665:
3617:
3613:
3612:
3610:
3604:
3603:
3587:
3574:
3564:
3555:
3536:
3521:
3517:
3516:
3514:
3508:
3507:
3492:
3452:
3451:
3428:
3426:
3384:
3380:
3379:
3376:
3360:is the field of real or complex numbers.
3332:
3328:
3327:
3324:
3295:
3291:
3290:
3283:
3267:
3263:
3262:
3259:
3226:
3222:
3221:
3218:
3191:
3184:
3180:
3179:
3177:
3173:
3172:
3169:
3144:
3134:
3121:
3111:
3105:
3070:
3061:
3042:
3027:
3023:
3022:
3020:
3014:
3013:
2993:
2985:
2949:
2945:
2944:
2942:
2936:
2920:
2916:
2915:
2899:
2892:
2888:
2887:
2885:
2881:
2880:
2862:
2843:
2828:
2824:
2823:
2821:
2815:
2814:
2794:
2792:
2756:
2752:
2751:
2748:
2724:
2705:
2687:
2633:
2581:
2579:
2531:
2505:
2461:
2422:
2418:
2417:
2408:
2358:
2354:
2353:
2337:
2333:
2332:
2323:
2244:
2240:
2239:
2236:
2169:
2165:
2164:
2154:
2148:
2124:
2118:
2097:
2091:
2070:
2064:
2021:
2020:
2019:
2014:
1998:
1983:
1943:
1923:
1904:
1892:
1890:
1852:
1848:
1847:
1832:
1786:
1773:
1741:
1740:
1739:
1734:
1718:
1697:
1695:
1664:
1658:
1613:
1601:{\displaystyle G_{n}(\mathbb {R} ^{n+k})}
1583:
1579:
1578:
1568:
1562:
1537:
1531:
1526:By definition, the infinite Grassmannian
1507:
1506:
1504:
1482:
1481:
1479:
1377:
1373:
1372:
1362:
1325:
1307:
1305:
1188:
1184:
1183:
1164:
1160:
1159:
1149:
1143:
1111:{\displaystyle G_{n}(\mathbb {R} ^{n+k})}
1093:
1089:
1088:
1078:
1072:
1021:
1017:
1016:
1013:
979:
975:
974:
971:
951:{\displaystyle G_{n}(\mathbb {R} ^{n+k})}
933:
929:
928:
918:
912:
884:
854:
834:
813:
807:
782:
776:
747:
726:
720:
687:
663:
657:
628:
622:
602:
582:
562:
542:
521:
515:
495:
475:
380:
376:
375:
372:
345:
341:
340:
337:
301:
297:
296:
294:
288:
287:
284:
232:
228:
227:
225:
219:
218:
215:
161:
141:
115:
95:
75:
48:
4442:Algebraic Geometry: A Concise Dictionary
2113:that corresponds to the identity map on
1637:Taking the direct limit of the bundles γ
180:the tautological bundle is known as the
4019:
3990:(Thom spaces of tautological bundles γ
3480:corresponding to a locally free sheaf
2559:is the field of rational functions on
3153:{\displaystyle x_{i}y_{j}-x_{j}y_{i}}
2274:. The projection map π is given by π(
7:
4444:, Berlin/Boston: Walter De Gruyter,
4336:, Wiley Classics Library, New York:
4252:, is a homeomorphism onto the image.
1827:is a subbundle of a trivial bundle:
1281:such that the orthogonal projection
2434:{\displaystyle X=\mathbb {P} ^{n},}
1213:The projection map π is given by π(
1057:We define the tautological bundle γ
1039:{\displaystyle \mathbb {R} ^{n+k}.}
997:{\displaystyle \mathbb {R} ^{n+k};}
652:made up of identical copies of the
467:, of a given dimension, in a given
4248:is the orthogonal projection onto
3900:{\displaystyle {\mathcal {O}}(-1)}
3399:{\displaystyle \mathbb {A} ^{n+1}}
3371:of the origin of the affine space
2463:
2395:. This can be seen as follows. If
2259:{\displaystyle \mathbb {R} ^{k+1}}
1621:
360:{\displaystyle \mathbb {P} ^{n-1}}
25:
3940:{\displaystyle {\mathcal {O}}(1)}
3738:. This is an example of an anti-
3242:; moreover, the closed points of
2305:of the tautological line bundle.
1130:of the Grassmannian and a vector
1008:-dimensional vector subspaces of
966:-dimensional vector subspaces in
4330:Principles of algebraic geometry
3726:, whose global sections are the
3438:
3435:
3432:
3429:
3341:{\displaystyle \mathbb {P} ^{n}}
3235:{\displaystyle \mathbb {P} ^{n}}
3003:
3000:
2997:
2994:
2804:
2801:
2798:
2795:
430:of a vector bundle as well as a
389:{\displaystyle \mathbb {P} ^{n}}
3843:{\displaystyle \mathbb {P} (V)}
1687:, there is a natural bijection
1647:gives the tautological bundle γ
1050:= 1, it is the real projective
4189:
4179:
4158:
4149:
4146:
4125:
4098:is given a topology such that
4085:
4064:
3934:
3928:
3894:
3885:
3837:
3831:
3776:The tautological line bundle γ
3699:
3690:
3634:
3631:
3625:
3580:
3566:
3561:
3529:
3503:
3497:
3463:
3457:
3442:
3067:
3035:
2868:
2836:
2730:
2698:
2623:
2613:
2607:
2598:
2592:
2516:
2510:
2506:
2487:
2484:
2478:
2466:
2181:
2160:
2036:
2030:
2004:
1985:
1936:
1916:
1837:
1823:is compact, any vector bundle
1792:
1779:
1766:
1756:
1750:
1727:
1724:
1705:
1618:
1595:
1574:
1444:
1441:
1435:
1423:
1417:
1405:
1389:
1368:
1343:
1340:
1334:
1176:
1155:
1105:
1084:
1004:as a set it is the set of all
945:
924:
771:carry the vector subspaces of
758:
752:
698:
692:
315:
309:
249:
240:
1:
3760:{\displaystyle \mathbb {C} ,}
3656:the tautological line bundle
1630:{\displaystyle k\to \infty .}
1517:{\displaystyle \mathbb {C} .}
110:-dimensional vector subspace
2290:is the dual vector space of
1489:{\displaystyle \mathbb {R} }
129:{\displaystyle W\subseteq V}
455:could scarcely be avoided.
4484:
4271:Milnor & Stasheff 1974
3213:over the same base scheme
742:the dual space, points of
3164:is a closed subscheme of
2219:is the set of all pairs (
1273:in the Grassmannian, let
400:of the projective space.
182:tautological line bundle.
3856:tautological line bundle
3315:is zero or the image of
2286:(so that the fiber over
1126:) consisting of a point
409:complex projective space
4288:Atiyah, Michael Francis
3478:algebraic vector bundle
3363:In more concise terms,
2393:homogeneous coordinates
2312:) corresponding to the
2270:a linear functional on
2227:) consisting of a line
1969:unique up to homotopy.
510:is a Grassmannian, and
176:itself. In the case of
4426:Characteristic Classes
4242:
4213:
4092:
3941:
3907:, the tensor inverse (
3901:
3844:
3761:
3714:Serre's twisting sheaf
3706:
3647:
3470:
3400:
3342:
3305:
3236:
3207:
3154:
3087:
2960:
2778:
2737:
2650:
2546:
2435:
2368:
2260:
2231:through the origin in
2191:
2137:
2136:{\displaystyle G_{n}.}
2107:
2080:
2043:
1960:
1868:
1806:
1677:
1676:{\displaystyle G_{n}.}
1631:
1602:
1547:
1518:
1490:
1458:
1207:
1112:
1040:
998:
952:
893:
869:
843:
823:
792:
765:
736:
705:
673:
638:
611:
591:
571:
551:
531:
504:
484:
390:
361:
322:
278:Serre's twisting sheaf
259:
198:characteristic classes
170:
150:
130:
104:
84:
57:
4440:Rubei, Elena (2014),
4347:10.1002/9781118032527
4338:John Wiley & Sons
4243:
4241:{\displaystyle p_{V}}
4214:
4093:
3967:Stiefel-Whitney class
3942:
3902:
3845:
3762:
3707:
3648:
3471:
3401:
3343:
3306:
3237:
3208:
3155:
3088:
2961:
2784:. Note that we have:
2779:
2738:
2651:
2547:
2436:
2369:
2261:
2211:on a real projective
2192:
2138:
2108:
2106:{\displaystyle G_{n}}
2081:
2079:{\displaystyle G_{n}}
2044:
1961:
1869:
1807:
1678:
1632:
1603:
1548:
1546:{\displaystyle G_{n}}
1519:
1491:
1459:
1208:
1113:
1041:
999:
953:
894:
870:
844:
824:
822:{\displaystyle V^{*}}
793:
791:{\displaystyle V^{*}}
766:
737:
735:{\displaystyle V^{*}}
715:sense. That is, with
706:
674:
672:{\displaystyle V_{g}}
639:
637:{\displaystyle V_{g}}
612:
592:
572:
552:
532:
530:{\displaystyle V_{g}}
505:
485:
391:
362:
323:
260:
171:
151:
131:
105:
85:
58:
4382:, Berlin, New York:
4225:
4104:
4051:
3918:
3875:
3823:
3746:
3664:
3491:
3425:
3375:
3323:
3258:
3217:
3168:
3104:
2984:
2791:
2747:
2686:
2578:
2460:
2407:
2322:
2235:
2147:
2117:
2090:
2063:
1982:
1889:
1831:
1694:
1657:
1612:
1561:
1530:
1503:
1478:
1304:
1293:isomorphically onto
1229:is the pre-image of
1142:
1071:
1012:
970:
911:
883:
853:
833:
806:
775:
764:{\displaystyle P(V)}
746:
719:
704:{\displaystyle P(V)}
686:
656:
621:
601:
581:
561:
541:
514:
494:
474:
459:Intuitive definition
413:standard line bundle
371:
336:
283:
214:
160:
140:
114:
94:
74:
47:
3819:of line bundles on
3416:exceptional divisor
3246:are exactly those (
3202:
2910:
2736:{\displaystyle A=k}
2026:
1746:
875:, the tautological
868:{\displaystyle n+1}
537:is the subspace of
33:tautological bundle
4422:Stasheff, James D.
4379:Algebraic Geometry
4321:Griffiths, Phillip
4273:, §2. Theorem 2.1.
4238:
4209:
4204:
4088:
3937:
3897:
3840:
3757:
3721:hyperplane divisor
3702:
3643:
3466:
3406:, where the locus
3396:
3338:
3301:
3232:
3203:
3171:
3150:
3083:
2956:
2879:
2774:
2733:
2646:
2641:
2542:
2431:
2364:
2314:hyperplane divisor
2256:
2187:
2133:
2103:
2076:
2039:
2010:
1956:
1951:
1882:determines a map
1864:
1802:
1797:
1730:
1673:
1627:
1598:
1543:
1514:
1486:
1454:
1449:
1297:, and then define
1277:be the set of all
1203:
1108:
1036:
994:
948:
889:
865:
839:
819:
800:linear functionals
788:
761:
732:
701:
669:
634:
607:
587:
567:
547:
527:
500:
480:
453:algebraic geometry
386:
357:
318:
255:
166:
146:
126:
100:
80:
53:
4451:978-3-11-031622-3
4393:978-0-387-90244-9
4374:Hartshorne, Robin
4357:978-0-471-05059-9
4305:978-0-201-09394-0
4000:→∞ is called the
3913:Serre twist sheaf
3867:, the associated
3740:ample line bundle
3594:
3460:
3311:such that either
2534:
2206:hyperplane bundle
2200:Hyperplane bundle
2054:paracompact space
903:Formal definition
892:{\displaystyle n}
842:{\displaystyle V}
610:{\displaystyle g}
590:{\displaystyle G}
570:{\displaystyle g}
557:corresponding to
550:{\displaystyle W}
503:{\displaystyle G}
483:{\displaystyle W}
428:projective bundle
274:hyperplane bundle
194:classifying space
169:{\displaystyle W}
149:{\displaystyle W}
136:, the fiber over
103:{\displaystyle k}
83:{\displaystyle V}
56:{\displaystyle k}
39:occurring over a
16:(Redirected from
4475:
4454:
4436:
4412:
4368:
4349:
4335:
4316:
4274:
4268:
4262:
4259:
4253:
4247:
4245:
4244:
4239:
4237:
4236:
4218:
4216:
4215:
4210:
4208:
4207:
4201:
4200:
4178:
4177:
4166:
4145:
4144:
4133:
4124:
4123:
4097:
4095:
4094:
4089:
4084:
4083:
4072:
4063:
4062:
4042:
4036:
4033:
4027:
4024:
4008:Grassmann bundle
3946:
3944:
3943:
3938:
3927:
3926:
3906:
3904:
3903:
3898:
3884:
3883:
3869:invertible sheaf
3849:
3847:
3846:
3841:
3830:
3766:
3764:
3763:
3758:
3753:
3711:
3709:
3708:
3703:
3689:
3688:
3687:
3686:
3681:
3674:
3673:
3652:
3650:
3649:
3644:
3624:
3623:
3622:
3621:
3616:
3609:
3608:
3595:
3593:
3592:
3591:
3579:
3578:
3565:
3560:
3559:
3541:
3540:
3528:
3527:
3526:
3525:
3520:
3513:
3512:
3475:
3473:
3472:
3467:
3462:
3461:
3453:
3441:
3405:
3403:
3402:
3397:
3395:
3394:
3383:
3347:
3345:
3344:
3339:
3337:
3336:
3331:
3310:
3308:
3307:
3302:
3300:
3299:
3294:
3288:
3287:
3278:
3277:
3266:
3241:
3239:
3238:
3233:
3231:
3230:
3225:
3212:
3210:
3209:
3204:
3201:
3190:
3189:
3188:
3183:
3176:
3159:
3157:
3156:
3151:
3149:
3148:
3139:
3138:
3126:
3125:
3116:
3115:
3092:
3090:
3089:
3084:
3082:
3078:
3074:
3066:
3065:
3047:
3046:
3034:
3033:
3032:
3031:
3026:
3019:
3018:
3006:
2965:
2963:
2962:
2957:
2955:
2954:
2953:
2948:
2941:
2940:
2931:
2930:
2919:
2909:
2898:
2897:
2896:
2891:
2884:
2875:
2871:
2867:
2866:
2848:
2847:
2835:
2834:
2833:
2832:
2827:
2820:
2819:
2807:
2783:
2781:
2780:
2775:
2761:
2760:
2755:
2742:
2740:
2739:
2734:
2729:
2728:
2710:
2709:
2655:
2653:
2652:
2647:
2645:
2644:
2638:
2637:
2551:
2549:
2548:
2543:
2535:
2532:
2509:
2440:
2438:
2437:
2432:
2427:
2426:
2421:
2373:
2371:
2370:
2365:
2363:
2362:
2357:
2348:
2347:
2336:
2310:invertible sheaf
2297:In other words,
2265:
2263:
2262:
2257:
2255:
2254:
2243:
2196:
2194:
2193:
2188:
2180:
2179:
2168:
2159:
2158:
2142:
2140:
2139:
2134:
2129:
2128:
2112:
2110:
2109:
2104:
2102:
2101:
2085:
2083:
2082:
2077:
2075:
2074:
2048:
2046:
2045:
2040:
2025:
2024:
2018:
2003:
2002:
1965:
1963:
1962:
1957:
1955:
1954:
1948:
1947:
1928:
1927:
1909:
1908:
1873:
1871:
1870:
1865:
1863:
1862:
1851:
1811:
1809:
1808:
1803:
1801:
1800:
1791:
1790:
1778:
1777:
1745:
1744:
1738:
1723:
1722:
1682:
1680:
1679:
1674:
1669:
1668:
1636:
1634:
1633:
1628:
1607:
1605:
1604:
1599:
1594:
1593:
1582:
1573:
1572:
1552:
1550:
1549:
1544:
1542:
1541:
1523:
1521:
1520:
1515:
1510:
1495:
1493:
1492:
1487:
1485:
1463:
1461:
1460:
1455:
1453:
1452:
1388:
1387:
1376:
1367:
1366:
1333:
1332:
1212:
1210:
1209:
1204:
1199:
1198:
1187:
1175:
1174:
1163:
1154:
1153:
1117:
1115:
1114:
1109:
1104:
1103:
1092:
1083:
1082:
1046:For example, if
1045:
1043:
1042:
1037:
1032:
1031:
1020:
1003:
1001:
1000:
995:
990:
989:
978:
957:
955:
954:
949:
944:
943:
932:
923:
922:
898:
896:
895:
890:
874:
872:
871:
866:
848:
846:
845:
840:
828:
826:
825:
820:
818:
817:
797:
795:
794:
789:
787:
786:
770:
768:
767:
762:
741:
739:
738:
733:
731:
730:
710:
708:
707:
702:
678:
676:
675:
670:
668:
667:
643:
641:
640:
635:
633:
632:
616:
614:
613:
608:
596:
594:
593:
588:
576:
574:
573:
568:
556:
554:
553:
548:
536:
534:
533:
528:
526:
525:
509:
507:
506:
501:
489:
487:
486:
481:
465:linear subspaces
439:canonical bundle
432:Grassmann bundle
395:
393:
392:
387:
385:
384:
379:
366:
364:
363:
358:
356:
355:
344:
327:
325:
324:
319:
308:
307:
306:
305:
300:
293:
292:
264:
262:
261:
256:
239:
238:
237:
236:
231:
224:
223:
205:invertible sheaf
189:universal bundle
178:projective space
175:
173:
172:
167:
156:is the subspace
155:
153:
152:
147:
135:
133:
132:
127:
109:
107:
106:
101:
89:
87:
86:
81:
62:
60:
59:
54:
21:
4483:
4482:
4478:
4477:
4476:
4474:
4473:
4472:
4458:
4457:
4452:
4439:
4418:Milnor, John W.
4416:
4394:
4384:Springer-Verlag
4372:
4358:
4333:
4319:
4306:
4286:
4283:
4278:
4277:
4269:
4265:
4260:
4256:
4228:
4223:
4222:
4203:
4202:
4192:
4183:
4182:
4161:
4128:
4115:
4108:
4102:
4101:
4067:
4054:
4049:
4048:
4043:
4039:
4034:
4030:
4025:
4021:
4016:
3995:
3983:Borel's theorem
3958:
3916:
3915:
3873:
3872:
3871:of sections is
3858:is a generator.
3852:infinite cyclic
3821:
3820:
3785:locally trivial
3782:
3773:
3744:
3743:
3676:
3667:
3662:
3661:
3611:
3602:
3583:
3570:
3569:
3551:
3532:
3515:
3506:
3489:
3488:
3423:
3422:
3378:
3373:
3372:
3326:
3321:
3320:
3289:
3279:
3261:
3256:
3255:
3220:
3215:
3214:
3178:
3166:
3165:
3140:
3130:
3117:
3107:
3102:
3101:
3057:
3038:
3021:
3012:
3011:
3007:
2982:
2981:
2943:
2932:
2914:
2886:
2858:
2839:
2822:
2813:
2812:
2808:
2789:
2788:
2750:
2745:
2744:
2720:
2701:
2684:
2683:
2676:
2665:
2640:
2639:
2629:
2617:
2616:
2582:
2576:
2575:
2458:
2457:
2416:
2405:
2404:
2389:
2383:
2377:given as, say,
2352:
2331:
2320:
2319:
2238:
2233:
2232:
2202:
2163:
2150:
2145:
2144:
2120:
2115:
2114:
2093:
2088:
2087:
2066:
2061:
2060:
1994:
1980:
1979:
1950:
1949:
1939:
1930:
1929:
1919:
1900:
1893:
1887:
1886:
1846:
1829:
1828:
1796:
1795:
1782:
1769:
1760:
1759:
1714:
1698:
1692:
1691:
1660:
1655:
1654:
1652:
1646:
1610:
1609:
1577:
1564:
1559:
1558:
1533:
1528:
1527:
1501:
1500:
1476:
1475:
1448:
1447:
1399:
1398:
1371:
1358:
1321:
1308:
1302:
1301:
1182:
1158:
1145:
1140:
1139:
1087:
1074:
1069:
1068:
1066:
1015:
1010:
1009:
973:
968:
967:
927:
914:
909:
908:
905:
881:
880:
851:
850:
831:
830:
809:
804:
803:
778:
773:
772:
744:
743:
722:
717:
716:
684:
683:
659:
654:
653:
624:
619:
618:
599:
598:
579:
578:
559:
558:
539:
538:
517:
512:
511:
492:
491:
472:
471:
461:
449:canonical class
437:The older term
374:
369:
368:
339:
334:
333:
295:
286:
281:
280:
226:
217:
212:
211:
158:
157:
138:
137:
112:
111:
92:
91:
72:
71:
45:
44:
23:
22:
15:
12:
11:
5:
4481:
4479:
4471:
4470:
4468:Vector bundles
4460:
4459:
4456:
4455:
4450:
4437:
4414:
4392:
4370:
4356:
4325:Harris, Joseph
4317:
4304:
4296:Addison-Wesley
4282:
4279:
4276:
4275:
4263:
4254:
4235:
4231:
4220:
4219:
4206:
4199:
4195:
4191:
4188:
4185:
4184:
4181:
4176:
4173:
4170:
4165:
4160:
4157:
4154:
4151:
4148:
4143:
4140:
4137:
4132:
4127:
4122:
4118:
4114:
4113:
4111:
4087:
4082:
4079:
4076:
4071:
4066:
4061:
4057:
4047:is open since
4037:
4028:
4018:
4017:
4015:
4012:
4011:
4010:
4005:
3991:
3985:
3980:
3974:
3972:Euler sequence
3969:
3964:
3957:
3954:
3953:
3952:
3936:
3933:
3930:
3925:
3896:
3893:
3890:
3887:
3882:
3860:
3859:
3839:
3836:
3833:
3829:
3797:
3796:
3777:
3772:
3769:
3756:
3752:
3701:
3698:
3695:
3692:
3685:
3680:
3672:
3654:
3653:
3642:
3639:
3636:
3633:
3630:
3627:
3620:
3615:
3607:
3601:
3598:
3590:
3586:
3582:
3577:
3573:
3568:
3563:
3558:
3554:
3550:
3547:
3544:
3539:
3535:
3531:
3524:
3519:
3511:
3505:
3502:
3499:
3496:
3465:
3459:
3456:
3450:
3447:
3444:
3440:
3437:
3434:
3431:
3393:
3390:
3387:
3382:
3335:
3330:
3298:
3293:
3286:
3282:
3276:
3273:
3270:
3265:
3229:
3224:
3200:
3197:
3194:
3187:
3182:
3175:
3147:
3143:
3137:
3133:
3129:
3124:
3120:
3114:
3110:
3094:
3093:
3081:
3077:
3073:
3069:
3064:
3060:
3056:
3053:
3050:
3045:
3041:
3037:
3030:
3025:
3017:
3010:
3005:
3002:
2999:
2996:
2992:
2989:
2967:
2966:
2952:
2947:
2939:
2935:
2929:
2926:
2923:
2918:
2913:
2908:
2905:
2902:
2895:
2890:
2883:
2878:
2874:
2870:
2865:
2861:
2857:
2854:
2851:
2846:
2842:
2838:
2831:
2826:
2818:
2811:
2806:
2803:
2800:
2797:
2773:
2770:
2767:
2764:
2759:
2754:
2732:
2727:
2723:
2719:
2716:
2713:
2708:
2704:
2700:
2697:
2694:
2691:
2675:
2672:
2663:
2657:
2656:
2643:
2636:
2632:
2628:
2625:
2622:
2619:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2587:
2585:
2553:
2552:
2541:
2538:
2533: on
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2508:
2504:
2501:
2498:
2495:
2492:
2489:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2465:
2430:
2425:
2420:
2415:
2412:
2401:(Weil) divisor
2387:
2381:
2375:
2374:
2361:
2356:
2351:
2346:
2343:
2340:
2335:
2330:
2327:
2253:
2250:
2247:
2242:
2201:
2198:
2186:
2183:
2178:
2175:
2172:
2167:
2162:
2157:
2153:
2132:
2127:
2123:
2100:
2096:
2073:
2069:
2050:
2049:
2038:
2035:
2032:
2029:
2023:
2017:
2013:
2009:
2006:
2001:
1997:
1993:
1990:
1987:
1967:
1966:
1953:
1946:
1942:
1938:
1935:
1932:
1931:
1926:
1922:
1918:
1915:
1912:
1907:
1903:
1899:
1898:
1896:
1861:
1858:
1855:
1850:
1845:
1842:
1839:
1836:
1813:
1812:
1799:
1794:
1789:
1785:
1781:
1776:
1772:
1768:
1765:
1762:
1761:
1758:
1755:
1752:
1749:
1743:
1737:
1733:
1729:
1726:
1721:
1717:
1713:
1710:
1707:
1704:
1703:
1701:
1672:
1667:
1663:
1648:
1638:
1626:
1623:
1620:
1617:
1597:
1592:
1589:
1586:
1581:
1576:
1571:
1567:
1540:
1536:
1513:
1509:
1484:
1465:
1464:
1451:
1446:
1443:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1400:
1397:
1394:
1391:
1386:
1383:
1380:
1375:
1370:
1365:
1361:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1331:
1328:
1324:
1320:
1317:
1314:
1313:
1311:
1202:
1197:
1194:
1191:
1186:
1181:
1178:
1173:
1170:
1167:
1162:
1157:
1152:
1148:
1107:
1102:
1099:
1096:
1091:
1086:
1081:
1077:
1058:
1035:
1030:
1027:
1024:
1019:
993:
988:
985:
982:
977:
947:
942:
939:
936:
931:
926:
921:
917:
904:
901:
888:
864:
861:
858:
849:has dimension
838:
816:
812:
785:
781:
760:
757:
754:
751:
729:
725:
700:
697:
694:
691:
666:
662:
646:disjoint union
631:
627:
606:
586:
566:
546:
524:
520:
499:
479:
460:
457:
421:Bott generator
411:is called the
405:Michael Atiyah
383:
378:
354:
351:
348:
343:
317:
314:
311:
304:
299:
291:
266:
265:
254:
251:
248:
245:
242:
235:
230:
222:
165:
145:
125:
122:
119:
99:
79:
52:
24:
18:Twisting sheaf
14:
13:
10:
9:
6:
4:
3:
2:
4480:
4469:
4466:
4465:
4463:
4453:
4447:
4443:
4438:
4435:
4431:
4427:
4423:
4419:
4415:
4411:
4407:
4403:
4399:
4395:
4389:
4385:
4381:
4380:
4375:
4371:
4367:
4363:
4359:
4353:
4348:
4343:
4339:
4332:
4331:
4326:
4322:
4318:
4315:
4311:
4307:
4301:
4297:
4293:
4289:
4285:
4284:
4280:
4272:
4267:
4264:
4258:
4255:
4251:
4233:
4229:
4197:
4193:
4186:
4174:
4171:
4168:
4155:
4152:
4141:
4138:
4135:
4120:
4116:
4109:
4100:
4099:
4080:
4077:
4074:
4059:
4055:
4046:
4041:
4038:
4032:
4029:
4023:
4020:
4013:
4009:
4006:
4003:
4002:Thom spectrum
3999:
3994:
3989:
3986:
3984:
3981:
3978:
3975:
3973:
3970:
3968:
3965:
3963:
3960:
3959:
3955:
3950:
3931:
3914:
3910:
3891:
3888:
3870:
3866:
3862:
3861:
3857:
3853:
3834:
3818:
3814:
3813:
3812:
3810:
3806:
3802:
3794:
3790:
3786:
3781:
3775:
3774:
3770:
3768:
3754:
3741:
3737:
3733:
3729:
3725:
3722:
3717:
3715:
3696:
3693:
3683:
3659:
3640:
3637:
3628:
3618:
3599:
3596:
3588:
3584:
3575:
3571:
3556:
3552:
3548:
3545:
3542:
3537:
3533:
3522:
3500:
3494:
3487:
3486:
3485:
3483:
3479:
3454:
3448:
3445:
3419:
3417:
3413:
3409:
3391:
3388:
3385:
3370:
3366:
3361:
3359:
3355:
3351:
3333:
3318:
3314:
3296:
3284:
3280:
3274:
3271:
3268:
3253:
3249:
3245:
3227:
3198:
3195:
3192:
3185:
3163:
3145:
3141:
3135:
3131:
3127:
3122:
3118:
3112:
3108:
3099:
3079:
3075:
3071:
3062:
3058:
3054:
3051:
3048:
3043:
3039:
3028:
3008:
2990:
2987:
2980:
2979:
2978:
2976:
2975:relative Spec
2972:
2950:
2937:
2933:
2927:
2924:
2921:
2911:
2906:
2903:
2900:
2893:
2876:
2872:
2863:
2859:
2855:
2852:
2849:
2844:
2840:
2829:
2809:
2787:
2786:
2785:
2771:
2768:
2765:
2762:
2757:
2725:
2721:
2717:
2714:
2711:
2706:
2702:
2695:
2692:
2689:
2681:
2673:
2671:
2669:
2662:
2634:
2630:
2626:
2620:
2610:
2604:
2601:
2595:
2589:
2583:
2574:
2573:
2572:
2570:
2566:
2562:
2558:
2536:
2528:
2525:
2522:
2519:
2513:
2502:
2499:
2496:
2490:
2481:
2475:
2472:
2469:
2456:
2455:
2454:
2452:
2448:
2444:
2428:
2423:
2413:
2410:
2402:
2398:
2394:
2390:
2380:
2359:
2349:
2344:
2341:
2338:
2328:
2325:
2318:
2317:
2316:
2315:
2311:
2306:
2304:
2300:
2295:
2293:
2289:
2285:
2281:
2277:
2273:
2269:
2251:
2248:
2245:
2230:
2226:
2222:
2218:
2214:
2210:
2207:
2199:
2197:
2184:
2176:
2173:
2170:
2155:
2151:
2130:
2125:
2121:
2098:
2094:
2071:
2067:
2058:
2055:
2033:
2027:
2015:
2011:
2007:
1999:
1995:
1991:
1988:
1978:
1977:
1976:
1974:
1970:
1944:
1940:
1933:
1924:
1920:
1913:
1910:
1905:
1901:
1894:
1885:
1884:
1883:
1881:
1877:
1859:
1856:
1853:
1843:
1840:
1834:
1826:
1822:
1818:
1787:
1783:
1774:
1770:
1763:
1753:
1747:
1735:
1731:
1719:
1715:
1711:
1708:
1699:
1690:
1689:
1688:
1686:
1670:
1665:
1661:
1651:
1645:
1641:
1624:
1615:
1590:
1587:
1584:
1569:
1565:
1556:
1538:
1534:
1524:
1511:
1499:
1498:complex field
1472:
1470:
1438:
1432:
1429:
1426:
1420:
1414:
1411:
1408:
1402:
1395:
1392:
1384:
1381:
1378:
1363:
1359:
1355:
1352:
1349:
1346:
1337:
1329:
1326:
1322:
1318:
1315:
1309:
1300:
1299:
1298:
1296:
1292:
1288:
1284:
1280:
1276:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1232:
1228:
1224:
1220:
1216:
1200:
1195:
1192:
1189:
1179:
1171:
1168:
1165:
1150:
1146:
1137:
1133:
1129:
1125:
1121:
1100:
1097:
1094:
1079:
1075:
1065:
1061:
1055:
1053:
1049:
1033:
1028:
1025:
1022:
1007:
991:
986:
983:
980:
965:
961:
940:
937:
934:
919:
915:
902:
900:
886:
878:
862:
859:
856:
836:
814:
810:
801:
783:
779:
755:
749:
727:
723:
714:
695:
689:
680:
664:
660:
651:
647:
629:
625:
604:
584:
564:
544:
522:
518:
497:
477:
470:
466:
458:
456:
454:
450:
446:
445:
440:
435:
433:
429:
424:
422:
418:
414:
410:
406:
401:
399:
381:
352:
349:
346:
331:
312:
302:
279:
275:
271:
252:
246:
243:
233:
210:
209:
208:
206:
201:
199:
195:
191:
190:
184:
183:
179:
163:
143:
123:
120:
117:
97:
77:
69:
66:
50:
42:
38:
37:vector bundle
34:
30:
19:
4441:
4425:
4377:
4329:
4291:
4266:
4257:
4249:
4044:
4040:
4031:
4022:
3997:
3992:
3908:
3817:Picard group
3809:Möbius strip
3800:
3798:
3792:
3779:
3735:
3728:linear forms
3723:
3718:
3657:
3655:
3481:
3421:In general,
3420:
3411:
3407:
3364:
3362:
3357:
3353:
3349:
3316:
3312:
3251:
3247:
3243:
3161:
3097:
3095:
2977:. Now, put:
2970:
2968:
2679:
2677:
2667:
2660:
2658:
2568:
2564:
2560:
2556:
2554:
2450:
2446:
2442:
2396:
2385:
2378:
2376:
2313:
2307:
2298:
2296:
2291:
2287:
2283:
2279:
2275:
2271:
2267:
2228:
2224:
2220:
2216:
2212:
2208:
2205:
2203:
2056:
2051:
1972:
1971:
1968:
1879:
1875:
1824:
1820:
1816:
1814:
1684:
1649:
1643:
1639:
1555:direct limit
1525:
1473:
1468:
1466:
1294:
1290:
1286:
1282:
1278:
1274:
1270:
1266:
1262:
1258:
1254:
1250:
1246:
1242:
1238:
1234:
1230:
1226:
1222:
1218:
1214:
1135:
1131:
1127:
1123:
1119:
1063:
1059:
1056:
1051:
1047:
1005:
963:
960:Grassmannian
906:
681:
469:vector space
462:
442:
438:
436:
425:
412:
402:
398:Picard group
273:
267:
202:
187:
185:
181:
41:Grassmannian
32:
26:
3977:Chern class
3962:Hopf bundle
3865:line bundle
3805:total space
3732:Chern class
2571:, we have:
2303:dual bundle
877:line bundle
650:total space
417:Hopf bundle
65:dimensional
29:mathematics
4014:References
3988:Thom space
3854:, and the
2384:= 0, when
713:dual space
4190:↦
4156:
4150:→
3889:−
3694:−
3635:→
3600:
3581:↦
3567:⟶
3546:…
3504:→
3498:→
3458:ˇ
3449:
3281:×
3128:−
3052:…
2934:×
2853:…
2769:
2715:…
2624:↦
2602:≃
2563:. Taking
2526:≥
2500:∈
2464:Γ
2350:⊂
2342:−
2028:
1937:↦
1917:→
1874:for some
1844:×
1838:↪
1784:γ
1775:∗
1767:↦
1748:
1728:→
1622:∞
1619:→
1496:with the
1403:ϕ
1393:×
1356:⊆
1350:×
1344:→
1327:−
1323:π
1316:ϕ
1180:×
815:∗
784:∗
728:∗
444:canonical
350:−
244:−
121:⊆
68:subspaces
4462:Category
4424:(1974),
4410:13348052
4376:(1977),
4327:(1994),
4292:K-theory
4290:(1989),
3956:See also
3787:but not
3352:. Thus,
2391:are the
2059:. Since
2052:for any
1054:-space.
4434:0440554
4402:0463157
4366:1288523
4314:1043170
4281:Sources
3949:divisor
3807:is the
3789:trivial
3742:. Over
3476:is the
3414:is the
3410:= 0 in
3369:blow-up
3367:is the
3160:. Then
2301:is the
1878:and so
1553:is the
958:be the
419:. (cf.
330:divisor
272:of the
4448:
4432:
4408:
4400:
4390:
4364:
4354:
4312:
4302:
4221:where
3791:, for
3730:. Its
3096:where
2969:where
2659:where
2567:to be
2555:where
1973:Remark
31:, the
4334:(PDF)
3771:Facts
3254:) of
2449:) on
2399:is a
1289:maps
1285:onto
1257:) = (
1225:. If
1067:over
829:. If
490:. If
207:) is
35:is a
4446:ISBN
4406:OCLC
4388:ISBN
4352:ISBN
4300:ISBN
3815:The
3783:is
3734:is −
2971:Spec
2766:Proj
2743:and
2453:by
2282:) =
2266:and
2204:The
2012:Vect
1732:Vect
1245:) +
1221:) =
907:Let
270:dual
268:the
4342:doi
4153:End
3996:as
3850:is
3778:1,
3712:of
3597:Sym
3446:Sym
3348:is
3319:in
2973:is
2403:on
1653:of
1608:as
1557:of
1134:in
962:of
802:on
577:in
451:in
423:.)
403:In
367:in
276:or
70:of
27:In
4464::
4430:MR
4420:;
4404:,
4398:MR
4396:,
4386:,
4362:MR
4360:,
4350:,
4340:,
4323:;
4310:MR
4308:,
4298:,
4004:.)
3909:ie
3250:,
2278:,
2223:,
1642:,
1471:.
1267:bw
1265:+
1263:av
1261:,
1253:,
1241:,
1217:,
1122:,
1062:,
899:.
434:.
332:)
200:.
4413:.
4369:.
4344::
4250:V
4234:V
4230:p
4198:V
4194:p
4187:V
4180:)
4175:k
4172:+
4169:n
4164:R
4159:(
4147:)
4142:k
4139:+
4136:n
4131:R
4126:(
4121:n
4117:G
4110:{
4086:)
4081:k
4078:+
4075:n
4070:R
4065:(
4060:n
4056:G
4045:U
3998:n
3993:n
3935:)
3932:1
3929:(
3924:O
3895:)
3892:1
3886:(
3881:O
3838:)
3835:V
3832:(
3828:P
3801:k
3793:k
3780:k
3755:,
3751:C
3736:H
3724:H
3700:)
3697:1
3691:(
3684:n
3679:P
3671:O
3658:L
3641:,
3638:0
3632:)
3629:1
3626:(
3619:n
3614:P
3606:O
3589:i
3585:y
3576:i
3572:x
3562:]
3557:n
3553:x
3549:,
3543:,
3538:0
3534:x
3530:[
3523:n
3518:P
3510:O
3501:I
3495:0
3482:E
3464:)
3455:E
3443:(
3439:c
3436:e
3433:p
3430:S
3412:L
3408:x
3392:1
3389:+
3386:n
3381:A
3365:L
3358:k
3354:L
3350:y
3334:n
3329:P
3317:x
3313:x
3297:n
3292:P
3285:k
3275:1
3272:+
3269:n
3264:A
3252:y
3248:x
3244:L
3228:n
3223:P
3199:1
3196:+
3193:n
3186:n
3181:P
3174:A
3162:L
3146:i
3142:y
3136:j
3132:x
3123:j
3119:y
3113:i
3109:x
3098:I
3080:)
3076:I
3072:/
3068:]
3063:n
3059:x
3055:,
3049:,
3044:0
3040:x
3036:[
3029:n
3024:P
3016:O
3009:(
3004:c
3001:e
2998:p
2995:S
2991:=
2988:L
2951:n
2946:P
2938:k
2928:1
2925:+
2922:n
2917:A
2912:=
2907:1
2904:+
2901:n
2894:n
2889:P
2882:A
2877:=
2873:)
2869:]
2864:n
2860:x
2856:,
2850:,
2845:0
2841:x
2837:[
2830:n
2825:P
2817:O
2810:(
2805:c
2802:e
2799:p
2796:S
2772:A
2763:=
2758:n
2753:P
2731:]
2726:n
2722:y
2718:,
2712:,
2707:0
2703:y
2699:[
2696:k
2693:=
2690:A
2680:k
2668:O
2664:0
2661:x
2635:0
2631:x
2627:f
2621:f
2614:)
2611:1
2608:(
2605:O
2599:)
2596:H
2593:(
2590:O
2584:{
2569:H
2565:D
2561:X
2557:K
2540:}
2537:U
2529:0
2523:D
2520:+
2517:)
2514:f
2511:(
2507:|
2503:K
2497:f
2494:{
2491:=
2488:)
2485:)
2482:D
2479:(
2476:O
2473:,
2470:U
2467:(
2451:X
2447:D
2445:(
2443:O
2429:,
2424:n
2419:P
2414:=
2411:X
2397:D
2388:i
2386:x
2382:0
2379:x
2360:n
2355:P
2345:1
2339:n
2334:P
2329:=
2326:H
2299:H
2292:L
2288:L
2284:L
2280:f
2276:L
2272:L
2268:f
2252:1
2249:+
2246:k
2241:R
2229:L
2225:f
2221:L
2217:H
2213:k
2209:H
2185:.
2182:)
2177:k
2174:+
2171:n
2166:R
2161:(
2156:n
2152:G
2131:.
2126:n
2122:G
2099:n
2095:G
2072:n
2068:G
2057:X
2037:)
2034:X
2031:(
2022:R
2016:n
2008:=
2005:]
2000:n
1996:G
1992:,
1989:X
1986:[
1945:x
1941:E
1934:x
1925:n
1921:G
1914:X
1911::
1906:E
1902:f
1895:{
1880:E
1876:k
1860:k
1857:+
1854:n
1849:R
1841:X
1835:E
1825:E
1821:X
1817:n
1793:)
1788:n
1780:(
1771:f
1764:f
1757:)
1754:X
1751:(
1742:R
1736:n
1725:]
1720:n
1716:G
1712:,
1709:X
1706:[
1700:{
1685:X
1671:.
1666:n
1662:G
1650:n
1644:k
1640:n
1625:.
1616:k
1596:)
1591:k
1588:+
1585:n
1580:R
1575:(
1570:n
1566:G
1539:n
1535:G
1512:.
1508:C
1483:R
1469:n
1445:)
1442:)
1439:v
1436:(
1433:p
1430:,
1427:V
1424:(
1421:=
1418:)
1415:v
1412:,
1409:V
1406:(
1396:X
1390:)
1385:k
1382:+
1379:n
1374:R
1369:(
1364:n
1360:G
1353:X
1347:U
1341:)
1338:U
1335:(
1330:1
1319::
1310:{
1295:X
1291:V
1287:X
1283:p
1279:V
1275:U
1271:X
1259:V
1255:w
1251:V
1249:(
1247:b
1243:v
1239:V
1237:(
1235:a
1231:V
1227:F
1223:V
1219:v
1215:V
1201:.
1196:k
1193:+
1190:n
1185:R
1177:)
1172:k
1169:+
1166:n
1161:R
1156:(
1151:n
1147:G
1136:V
1132:v
1128:V
1124:v
1120:V
1106:)
1101:k
1098:+
1095:n
1090:R
1085:(
1080:n
1076:G
1064:k
1060:n
1052:k
1048:n
1034:.
1029:k
1026:+
1023:n
1018:R
1006:n
992:;
987:k
984:+
981:n
976:R
964:n
946:)
941:k
938:+
935:n
930:R
925:(
920:n
916:G
887:n
863:1
860:+
857:n
837:V
811:V
780:V
759:)
756:V
753:(
750:P
724:V
699:)
696:V
693:(
690:P
665:g
661:V
630:g
626:V
605:g
585:G
565:g
545:W
523:g
519:V
498:G
478:W
382:n
377:P
353:1
347:n
342:P
316:)
313:1
310:(
303:n
298:P
290:O
253:,
250:)
247:1
241:(
234:n
229:P
221:O
164:W
144:W
124:V
118:W
98:k
78:V
63:-
51:k
20:)
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