380:
582:
723:
500:
may or may not be a geometric quotient: but both are categorical quotients, which is unique; in other words, one cannot have both types of quotients (without them being the same).
283:
439:
171:
639:
71:
200:
132:
95:
494:
316:
531:
772:
294:
658:
592:
205:
450:
39:
509:
393:
137:
606:
44:
17:
179:
111:
749:
105:
25:
80:
32:
471:
766:
516:
642:
497:
302:
375:{\displaystyle {\mathcal {O}}_{Y}\simeq \pi _{*}({\mathcal {O}}_{X}^{G})}
310:
577:{\displaystyle \mathbb {A} ^{n+1}\setminus 0\to \mathbb {P} ^{n}}
353:
323:
97:
is surjective, and its fibers are exactly the G-orbits in X.
512:. This is proved in Mumford's geometric invariant theory.
313:. (iii) may also be phrased as an isomorphism of sheaves
661:
609:
534:
474:
396:
319:
208:
182:
140:
114:
83:
47:
717:
633:
576:
488:
433:
374:
277:
194:
165:
126:
89:
65:
445:may be viewed as invariant rational functions on
750:"Introduction to actions of algebraic groups"
8:
718:{\displaystyle X_{(0)}^{s}\to X_{(0)}^{s}/G}
707:
701:
690:
677:
666:
660:
625:
614:
608:
568:
564:
563:
541:
537:
536:
533:
478:
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425:
395:
363:
358:
352:
351:
341:
328:
322:
321:
318:
269:
247:
213:
207:
181:
145:
139:
113:
82:
46:
740:
553:
519:whose fibers are orbits of the group.
7:
515:A geometric quotient is precisely a
278:{\displaystyle \pi ^{\#}:k\to k^{G}}
214:
14:
697:
691:
683:
673:
667:
621:
615:
559:
422:
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347:
266:
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160:
154:
57:
1:
434:{\displaystyle k(Y)=k(X)^{G}}
166:{\displaystyle \pi ^{-1}(U)}
634:{\displaystyle X_{(0)}^{s}}
504:Relation to other quotients
496:is a geometric quotient. A
386:is irreducible, then so is
66:{\displaystyle \pi :X\to Y}
789:
508:A geometric quotient is a
295:geometric invariant theory
285:is an isomorphism. (Here,
195:{\displaystyle U\subset Y}
176:(iii) For any open subset
127:{\displaystyle U\subset Y}
730:is a geometric quotient.
584:is a geometric quotient.
464:is a closed subgroup of
441:: rational functions on
134:is open if and only if
719:
635:
593:linearized line bundle
578:
490:
435:
376:
293:The notion appears in
279:
196:
167:
128:
91:
67:
31:with the action of an
720:
636:
579:
491:
436:
377:
297:. (i), (ii) say that
280:
197:
168:
129:
100:(ii) The topology of
92:
68:
40:morphism of varieties
659:
607:
532:
510:categorical quotient
472:
394:
382:. In particular, if
317:
206:
180:
138:
112:
90:{\displaystyle \pi }
81:
45:
706:
682:
630:
489:{\displaystyle G/H}
451:rational-invariants
368:
289:is the base field.)
773:Algebraic geometry
755:. Definition 1.18.
715:
686:
662:
631:
610:
574:
528:The canonical map
486:
431:
372:
350:
275:
192:
163:
124:
87:
63:
22:geometric quotient
18:algebraic geometry
106:quotient topology
26:algebraic variety
780:
757:
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745:
724:
722:
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711:
705:
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681:
676:
645:with respect to
640:
638:
637:
632:
629:
624:
603:, then, writing
595:on an algebraic
583:
581:
580:
575:
573:
572:
567:
552:
551:
540:
495:
493:
492:
487:
482:
460:For example, if
440:
438:
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432:
430:
429:
381:
379:
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373:
367:
362:
357:
356:
346:
345:
333:
332:
327:
326:
284:
282:
281:
276:
274:
273:
255:
254:
218:
217:
201:
199:
198:
193:
172:
170:
169:
164:
153:
152:
133:
131:
130:
125:
96:
94:
93:
88:
72:
70:
69:
64:
788:
787:
783:
782:
781:
779:
778:
777:
763:
762:
761:
760:
752:
747:
746:
742:
737:
657:
656:
641:for the set of
605:
604:
562:
535:
530:
529:
525:
506:
470:
469:
421:
392:
391:
337:
320:
315:
314:
265:
243:
209:
204:
203:
178:
177:
141:
136:
135:
110:
109:
79:
78:
43:
42:
33:algebraic group
12:
11:
5:
786:
784:
776:
775:
765:
764:
759:
758:
739:
738:
736:
733:
732:
731:
728:
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726:
714:
710:
704:
699:
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693:
689:
685:
680:
675:
672:
669:
665:
651:
650:
649:, the quotient
628:
623:
620:
617:
613:
585:
571:
566:
561:
558:
555:
550:
547:
544:
539:
524:
521:
505:
502:
485:
481:
477:
428:
424:
420:
417:
414:
411:
408:
405:
402:
399:
371:
366:
361:
355:
349:
344:
340:
336:
331:
325:
291:
290:
272:
268:
264:
261:
258:
253:
250:
246:
242:
239:
236:
233:
230:
227:
224:
221:
216:
212:
191:
188:
185:
174:
162:
159:
156:
151:
148:
144:
123:
120:
117:
98:
86:
62:
59:
56:
53:
50:
13:
10:
9:
6:
4:
3:
2:
785:
774:
771:
770:
768:
751:
744:
741:
734:
729:
712:
708:
702:
694:
687:
678:
670:
663:
655:
654:
653:
652:
648:
644:
643:stable points
626:
618:
611:
602:
598:
594:
590:
586:
569:
556:
548:
545:
542:
527:
526:
522:
520:
518:
517:good quotient
513:
511:
503:
501:
499:
483:
479:
475:
467:
463:
458:
456:
452:
448:
444:
426:
418:
412:
409:
403:
397:
389:
385:
364:
359:
342:
338:
334:
329:
312:
308:
304:
300:
296:
288:
270:
259:
251:
248:
244:
237:
228:
222:
219:
210:
189:
186:
183:
175:
157:
149:
146:
142:
121:
118:
115:
107:
103:
99:
84:
76:
75:
74:
60:
54:
51:
48:
41:
37:
34:
30:
27:
23:
19:
743:
646:
600:
596:
588:
514:
507:
498:GIT quotient
465:
461:
459:
454:
446:
442:
387:
383:
306:
298:
292:
286:
101:
77:(i) The map
35:
28:
21:
15:
303:orbit space
108:: a subset
748:Brion, M.
735:References
73:such that
684:→
599:-variety
560:→
554:∖
343:∗
339:π
335:≃
249:−
245:π
235:→
215:#
211:π
187:⊂
147:−
143:π
119:⊂
85:π
58:→
49:π
767:Category
523:Examples
311:topology
173:is open.
468:, then
449:(i.e.,
104:is the
725:
301:is an
24:of an
753:(PDF)
591:is a
38:is a
390:and
20:, a
587:If
457:).
453:of
309:in
305:of
16:In
769::
202:,
713:G
709:/
703:s
698:)
695:0
692:(
688:X
679:s
674:)
671:0
668:(
664:X
647:L
627:s
622:)
619:0
616:(
612:X
601:X
597:G
589:L
570:n
565:P
557:0
549:1
546:+
543:n
538:A
484:H
480:/
476:G
466:G
462:H
455:X
447:X
443:Y
427:G
423:)
419:X
416:(
413:k
410:=
407:)
404:Y
401:(
398:k
388:Y
384:X
370:)
365:G
360:X
354:O
348:(
330:Y
324:O
307:X
299:Y
287:k
271:G
267:]
263:)
260:U
257:(
252:1
241:[
238:k
232:]
229:U
226:[
223:k
220::
190:Y
184:U
161:)
158:U
155:(
150:1
122:Y
116:U
102:Y
61:Y
55:X
52::
36:G
29:X
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