504:
302:
353:
378:
589:
219:
319:
658:
608:
639:
236:
200:
677:
223:
627:
334:
215:
85:
541:
58:
140:
327:
28:
8:
579:
132:
42:
365:
20:
654:
635:
623:
107:
96:
584:
196:
148:
62:
46:
514:
The definition of geometric genus is carried over classically to singular curves
499:{\displaystyle {\mathcal {K}}_{C}=\left_{\vert C}={\mathcal {O}}(d-3)_{\vert C}}
195:
In the case of complex varieties, (the complex loci of) non-singular curves are
144:
671:
81:
66:
184:
567:
226:. By the Riemann-Roch theorem, an irreducible plane curve of degree
648:
214:
The notion of genus features prominently in the statement of the
203:. On a nonsingular curve, the canonical line bundle has degree
634:. Wiley Classics Library. Wiley Interscience. p. 494.
570:, the definition is extended by birational invariance.
311:
is the number of singularities when properly counted.
622:
381:
337:
239:
102:
it is the number of linearly independent holomorphic
199:. The algebraic definition of genus agrees with the
61:
complex projective varieties and more generally for
318:is an irreducible (and smooth) hypersurface in the
651:Algebraic curves, algebraic manifolds, and schemes
498:
347:
296:
669:
649:V. I. Danilov; Vyacheslav V. Shokurov (1998).
509:
488:
453:
297:{\displaystyle g={\frac {(d-1)(d-2)}{2}}-s,}
322:cut out by a polynomial equation of degree
154:The geometric genus is the first invariant
416:
220:Riemann–Roch theorem for algebraic curves
114:. This definition, as the dimension of
57:The geometric genus can be defined for
670:
326:, then its normal line bundle is the
13:
466:
433:
408:
385:
340:
14:
689:
190:
84:), that is, the dimension of the
632:Principles of Algebraic Geometry
607:Danilov & Shokurov (1998),
368:, the canonical line bundle of
601:
540:is the geometric genus of the
484:
471:
444:
438:
348:{\displaystyle {\mathcal {O}}}
276:
264:
261:
249:
131:then carries over to any base
91:In other words, for a variety
1:
616:
550:. That is, since the mapping
52:
172:of a sequence of invariants
139:is taken to be the sheaf of
7:
573:
510:Genus of singular varieties
143:and the power is the (top)
10:
694:
16:Basic birational invariant
595:
224:Riemann–Hurwitz formula
86:canonical linear system
590:Invariants of surfaces
500:
349:
298:
501:
350:
299:
149:canonical line bundle
518:, by decreeing that
379:
335:
328:Serre twisting sheaf
237:
230:has geometric genus
216:Riemann–Roch theorem
209: − 2
141:Kähler differentials
29:birational invariant
678:Algebraic varieties
580:Genus (mathematics)
43:algebraic varieties
496:
366:adjunction formula
345:
294:
201:topological notion
21:algebraic geometry
660:978-3-540-63705-9
283:
97:complex dimension
63:complex manifolds
47:complex manifolds
685:
664:
645:
611:
605:
585:Arithmetic genus
562:
558:′ →
549:
536:
517:
505:
503:
502:
497:
495:
494:
470:
469:
460:
459:
451:
447:
437:
436:
427:
426:
425:
424:
419:
412:
411:
395:
394:
389:
388:
371:
363:
356:
354:
352:
351:
346:
344:
343:
325:
320:projective plane
317:
303:
301:
300:
295:
284:
279:
247:
210:
197:Riemann surfaces
182:
171:
138:
127:
113:
105:
101:
94:
79:
73:
40:
693:
692:
688:
687:
686:
684:
683:
682:
668:
667:
661:
642:
619:
614:
606:
602:
598:
576:
554:
544:
530:
522:
515:
512:
487:
483:
465:
464:
452:
432:
431:
420:
415:
414:
413:
407:
406:
405:
404:
400:
399:
390:
384:
383:
382:
380:
377:
376:
369:
357:
339:
338:
336:
333:
332:
330:
323:
315:
248:
246:
238:
235:
234:
204:
193:
181:
173:
170:
163:
155:
136:
118:
111:
110:to be found on
103:
99:
92:
75:
69:
55:
39:
31:
25:geometric genus
17:
12:
11:
5:
691:
681:
680:
666:
665:
659:
646:
640:
618:
615:
613:
612:
599:
597:
594:
593:
592:
587:
582:
575:
572:
564:
563:
538:
537:
526:
511:
508:
507:
506:
493:
490:
486:
482:
479:
476:
473:
468:
463:
458:
455:
450:
446:
443:
440:
435:
430:
423:
418:
410:
403:
398:
393:
387:
342:
305:
304:
293:
290:
287:
282:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
245:
242:
192:
191:Case of curves
189:
177:
168:
159:
145:exterior power
129:
128:
54:
51:
35:
15:
9:
6:
4:
3:
2:
690:
679:
676:
675:
673:
662:
656:
652:
647:
643:
641:0-471-05059-8
637:
633:
629:
625:
621:
620:
610:
604:
600:
591:
588:
586:
583:
581:
578:
577:
571:
569:
561:
557:
553:
552:
551:
547:
543:
542:normalization
534:
529:
525:
521:
520:
519:
491:
480:
477:
474:
461:
456:
448:
441:
428:
421:
401:
396:
391:
375:
374:
373:
367:
361:
329:
321:
312:
310:
291:
288:
285:
280:
273:
270:
267:
258:
255:
252:
243:
240:
233:
232:
231:
229:
225:
222:) and of the
221:
217:
212:
208:
202:
198:
188:
186:
180:
176:
167:
162:
158:
152:
150:
146:
142:
134:
125:
121:
117:
116:
115:
109:
98:
89:
87:
83:
82:Serre duality
78:
72:
68:
64:
60:
50:
48:
44:
38:
34:
30:
26:
22:
653:. Springer.
650:
631:
624:P. Griffiths
603:
565:
559:
555:
545:
539:
532:
527:
523:
513:
372:is given by
364:, so by the
359:
313:
308:
306:
227:
213:
206:
194:
178:
174:
165:
160:
156:
153:
130:
123:
119:
90:
76:
70:
67:Hodge number
59:non-singular
56:
36:
32:
24:
18:
185:plurigenera
183:called the
27:is a basic
617:References
568:birational
218:(see also
88:plus one.
74:(equal to
53:Definition
628:J. Harris
478:−
286:−
271:−
256:−
672:Category
630:(1994).
574:See also
126:,Ω)
548:′
355:
331:
135:, when
65:as the
657:
638:
307:where
147:, the
137:Ω
23:, the
609:p. 53
596:Notes
133:field
108:forms
655:ISBN
636:ISBN
45:and
566:is
314:If
95:of
80:by
41:of
19:In
674::
626:;
211:.
187:.
164:=
151:.
49:.
663:.
644:.
560:C
556:C
546:C
535:)
533:C
531:(
528:g
524:p
516:C
492:C
489:|
485:)
481:3
475:d
472:(
467:O
462:=
457:C
454:|
449:]
445:)
442:d
439:(
434:O
429:+
422:2
417:P
409:K
402:[
397:=
392:C
386:K
370:C
362:)
360:d
358:(
341:O
324:d
316:C
309:s
292:,
289:s
281:2
277:)
274:2
268:d
265:(
262:)
259:1
253:d
250:(
244:=
241:g
228:d
207:g
205:2
179:n
175:P
169:1
166:P
161:g
157:p
124:V
122:(
120:H
112:V
106:-
104:n
100:n
93:V
77:h
71:h
37:g
33:p
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