419:(more precisely, by thinking of φ as a matrix of homogeneous polynomials, the count of entries of that homogeneous degree incremented by the gradings acquired inductively from the right). In other words, weights in all the free modules may be inferred from the resolution, and the graded Betti numbers count the number of generators of a given weight in a given module of the resolution. The properties of these invariants of
537:, but not conversely: the property of projective normality is not independent of the projective embedding, as is shown by the example of a rational quartic curve in three dimensions. Another equivalent condition is in terms of the
767:
275:
the number of equations can be taken as the codimension; but the general projective variety has no defining set of equations that is so transparent. Detailed studies, for example of
662:
from a projective space of higher dimension, except in the trivial way of lying in a proper linear subspace. Projective normality may similarly be translated, by using enough
865:
772:
is considered as graded module over the homogeneous coordinate ring of the projective space, and a minimal free resolution taken. Condition
280:
505:; it is therefore small when the shifts increase only by increments of 1 as we move to the left in the resolution (linear syzygies only).
451:
137:
underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the
173:. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero
702:
129:. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the
974:
212:
95:
538:
439:
154:
546:
526:
208:
corresponds to the empty set, since not all homogeneous coordinates can vanish at a point of projective space.
431:
130:
283:, show the geometric interest of systematic techniques to handle these cases. The subject also grew out of
670:
566:
427:
272:
252:
354:
232:
924:, Transactions of the American Mathematical Society, Vol. 352, No. 6 (Jun., 2000), pp. 2569–2579.
834:
829:
284:
259:. In a classical perspective, such generators are simply the equations one writes down to define
28:
861:
817:
659:
570:
186:
174:
138:
83:
46:
39:
423:
in a given projective embedding poses active research questions, even in the case of curves.
895:
678:
663:
292:
178:
50:
562:
276:
240:
170:
113:
937:, Journal of the American Mathematical Society, Vol. 13, No. 3 (Jul., 2000), pp. 651–664.
853:
534:
435:
389:
215:
gives a bijective correspondence between projective varieties and homogeneous ideals
968:
954:
950:
385:
248:
236:
182:
58:
17:
569:. Alternatively one can think of the dual of the tautological line bundle as the
268:
134:
689:
as embedded is projectively normal. Projective normality is the first condition
426:
There are examples where the minimal free resolution is known explicitly. For a
311:
162:
565:, it is projectively normal if and only if each such linear system is a
646:. A non-singular variety is projectively normal if and only if it is
696:
of a sequence of conditions defined by Green and
Lazarsfeld. For this
458:
defining the projective variety. In terms of the imputed "shifts"
381:
185:
these cases may be dealt with on the same footing by means of the
576:(1) on projective space, and use it to twist the structure sheaf
235:
techniques to algebraic geometry, it has been traditional since
291:
is supposed to become an algorithmic process (now handled by
654:≥ 1. Linear normality may also be expressed geometrically:
438:
in projective space the resolution may be constructed as a
392:
in any free resolution. Since this complex is intrinsic to
658:
as projective variety cannot be obtained by an isomorphic
762:{\displaystyle \bigoplus _{d=0}^{\infty }H^{0}(V,L^{d})}
251:
over the polynomial ring. This yields information about
669:
Looking at the issue from the point of view of a given
705:
454:
may be read off the minimum resolution of the ideal
255:, namely relations between generators of the ideal
761:
298:There are for general reasons free resolutions of
239:(though modern terminology is different) to apply
783:graded Betti numbers, requiring they vanish when
287:in its classical form, in which reduction modulo
666:to reduce it to conditions of linear normality.
353:is the irrelevant ideal. As a consequence of
8:
791:+ 1. For curves Green showed that condition
673:giving rise to the projective embedding of
380:is well-defined in a strong sense: unique
750:
731:
721:
710:
704:
271:there need only be one equation, and for
310:if the image in each module morphism of
846:
153:is assumed to be a variety, and so an
133:, for a given choice of basis (in the
7:
281:equations defining abelian varieties
722:
366:to a minimal set of generators in
25:
961:Vol. II (1960), pp. 168–172.
922:On Syzygies of Abelian Varieties
357:, φ then takes a given basis in
621:) map surjectively to those of
517:in its projective embedding is
816:= 0 was a classical result of
756:
737:
529:. This condition implies that
452:Castelnuovo–Mumford regularity
442:of Eagon–Northcott complexes.
1:
935:Syzygies of Abelian Varieties
549:on projective space, and its
306:. A resolution is defined as
181:. From the point of view of
557:= 1, 2, 3, ... ; when
545:cut out by the dual of the
33:homogeneous coordinate ring
991:
613:if the global sections of
339:in the resolution lies in
213:projective Nullstellensatz
96:algebraically closed field
642:is 1-normal it is called
589:times, obtaining a sheaf
585:any number of times, say
539:linear system of divisors
484:, it is the maximum over
155:irreducible algebraic set
858:The Geometry of Syzygies
547:tautological line bundle
227:Resolutions and syzygies
432:Eagon–Northcott complex
406:as the number of grade-
378:minimal free resolution
131:homogeneous coordinates
920:See e.g. Elena Rubei,
800:is satisfied when deg(
763:
726:
677:, such a line bundle (
671:very ample line bundle
567:complete linear system
302:as graded module over
273:complete intersections
161:can be chosen to be a
779:applied to the first
764:
706:
428:rational normal curve
396:, one may define the
199:generated by all the
57:is by definition the
53:of a given dimension
703:
509:Projective normality
398:graded Betti numbers
18:Projective normality
975:Algebraic varieties
959:Commutative Algebra
933:Giuseppe Pareschi,
911:Hartshorne, p. 159.
519:projectively normal
410:images coming from
388:and occurring as a
233:homological algebra
900:Algebraic Geometry
835:Hilbert polynomial
830:Projective variety
759:
683:normally generated
285:elimination theory
247:, considered as a
231:In application of
175:nilpotent elements
29:algebraic geometry
866:978-0-387-22215-8
818:Guido Castelnuovo
664:Veronese mappings
660:linear projection
571:Serre twist sheaf
527:integrally closed
376:. The concept of
187:Proj construction
139:symmetric algebra
84:homogeneous ideal
40:algebraic variety
16:(Redirected from
982:
938:
931:
925:
918:
912:
909:
903:
896:Robin Hartshorne
893:
887:
886:Eisenbud, Ch. 4.
884:
878:
877:Eisenbud, Ch. 6.
875:
869:
851:
768:
766:
765:
760:
755:
754:
736:
735:
725:
720:
681:) is said to be
679:invertible sheaf
650:-normal for all
355:Nakayama's lemma
277:canonical curves
241:free resolutions
194:irrelevant ideal
179:divisors of zero
102:is defined, and
51:projective space
21:
990:
989:
985:
984:
983:
981:
980:
979:
965:
964:
947:
942:
941:
932:
928:
919:
915:
910:
906:
894:
890:
885:
881:
876:
872:
852:
848:
843:
826:
799:
778:
746:
727:
701:
700:
695:
644:linearly normal
634:), for a given
629:
597:
584:
553:-th powers for
511:
500:
483:
470:
448:
436:elliptic curves
418:
405:
386:chain complexes
384:isomorphism of
375:
365:
348:
335:
325:
229:
219:not containing
207:
171:integral domain
147:
128:
114:polynomial ring
71: /
23:
22:
15:
12:
11:
5:
988:
986:
978:
977:
967:
966:
963:
962:
946:
943:
940:
939:
926:
913:
904:
902:(1977), p. 23.
888:
879:
870:
854:David Eisenbud
845:
844:
842:
839:
838:
837:
832:
825:
822:
795:
776:
770:
769:
758:
753:
749:
745:
742:
739:
734:
730:
724:
719:
716:
713:
709:
693:
625:
593:
580:
535:normal variety
510:
507:
492:
479:
462:
447:
444:
414:
401:
390:direct summand
370:
361:
343:
337:
336:
330:
321:
295:in practice).
228:
225:
203:
146:
143:
124:
120:+ 1 variables
110:
109:
76:
75:
24:
14:
13:
10:
9:
6:
4:
3:
2:
987:
976:
973:
972:
970:
960:
956:
955:Pierre Samuel
952:
951:Oscar Zariski
949:
948:
944:
936:
930:
927:
923:
917:
914:
908:
905:
901:
897:
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883:
880:
874:
871:
867:
863:
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840:
836:
833:
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828:
827:
823:
821:
819:
815:
811:
807:
803:
798:
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790:
786:
782:
775:
751:
747:
743:
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732:
728:
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714:
711:
707:
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697:
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688:
684:
680:
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661:
657:
653:
649:
645:
641:
637:
633:
628:
624:
620:
616:
612:
610:
605:
601:
596:
592:
588:
583:
579:
575:
572:
568:
564:
560:
556:
552:
548:
544:
540:
536:
532:
528:
524:
520:
516:
508:
506:
504:
499:
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487:
482:
478:
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469:
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453:
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443:
441:
437:
433:
429:
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404:
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391:
387:
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379:
373:
369:
364:
360:
356:
352:
346:
342:
333:
329:
324:
320:
316:
315:
314:
313:
309:
305:
301:
296:
294:
293:Gröbner bases
290:
286:
282:
278:
274:
270:
266:
262:
258:
254:
250:
249:graded module
246:
242:
238:
237:David Hilbert
234:
226:
224:
222:
218:
214:
209:
206:
202:
198:
195:
190:
188:
184:
183:scheme theory
180:
176:
172:
168:
164:
160:
156:
152:
144:
142:
140:
136:
132:
127:
123:
119:
115:
108:
105:
104:
103:
101:
97:
93:
89:
85:
81:
74:
70:
66:
63:
62:
61:
60:
59:quotient ring
56:
52:
48:
44:
41:
37:
34:
30:
19:
958:
934:
929:
921:
916:
907:
899:
891:
882:
873:
857:
849:
813:
812:, which for
809:
805:
801:
796:
792:
788:
784:
780:
773:
771:
690:
686:
682:
674:
668:
655:
651:
647:
643:
639:
635:
631:
626:
622:
618:
614:
608:
607:
603:
599:
594:
590:
586:
581:
577:
573:
563:non-singular
558:
554:
550:
542:
530:
522:
518:
514:
513:The variety
512:
502:
497:
493:
489:
485:
480:
476:
472:
467:
463:
459:
455:
449:
440:mapping cone
425:
420:
415:
411:
407:
402:
397:
393:
377:
371:
367:
362:
358:
350:
344:
340:
338:
331:
327:
322:
318:
312:free modules
307:
303:
299:
297:
288:
269:hypersurface
264:
260:
256:
244:
230:
220:
216:
210:
204:
200:
196:
193:
191:
166:
158:
157:, the ideal
150:
148:
135:vector space
125:
121:
117:
111:
106:
99:
91:
87:
79:
77:
72:
68:
64:
54:
42:
35:
32:
26:
868:), pp. 5–8.
475:-th module
163:prime ideal
145:Formulation
98:over which
45:given as a
945:References
606:is called
446:Regularity
177:and other
47:subvariety
860:, (2005,
723:∞
708:⨁
638:, and if
602:). Then
430:it is an
165:, and so
86:defining
969:Category
824:See also
279:and the
253:syzygies
611:-normal
488:of the
471:in the
308:minimal
112:is the
94:is the
82:is the
864:
808:+ 1 +
434:. For
349:where
169:is an
149:Since
78:where
38:of an
31:, the
841:Notes
804:) ≥ 2
787:>
533:is a
382:up to
267:is a
263:. If
953:and
862:ISBN
450:The
403:i, j
347:− 1,
211:The
192:The
685:if
561:is
541:on
525:is
521:if
374:− 1
334:− 1
243:of
116:in
49:of
27:In
971::
957:,
898:,
856:,
820:.
501:−
496:,
466:,
341:JF
326:→
317:φ:
223:.
189:.
141:.
90:,
67:=
814:p
810:p
806:g
802:L
797:p
793:N
789:i
785:j
781:p
777:p
774:N
757:)
752:d
748:L
744:,
741:V
738:(
733:0
729:H
718:0
715:=
712:d
694:0
691:N
687:V
675:V
656:V
652:k
648:k
640:V
636:k
632:k
630:(
627:V
623:O
619:k
617:(
615:O
609:k
604:V
600:k
598:(
595:V
591:O
587:k
582:V
578:O
574:O
559:V
555:d
551:d
543:V
531:V
523:R
515:V
503:i
498:j
494:i
490:a
486:i
481:i
477:F
473:i
468:j
464:i
460:a
456:I
421:V
416:i
412:F
408:j
400:β
394:R
372:i
368:F
363:i
359:F
351:J
345:i
332:i
328:F
323:i
319:F
304:K
300:R
289:I
265:V
261:V
257:I
245:R
221:J
217:I
205:i
201:X
197:J
167:R
159:I
151:V
126:i
122:X
118:N
107:K
100:V
92:K
88:V
80:I
73:I
69:K
65:R
55:N
43:V
36:R
20:)
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