708:
801:
410:
205:
553:
987:
915:
1048:
263:
234:
160:
97:
346:
1013:
477:
747:
503:
603:
304:
127:
655:
631:
573:
1312:
68:
1087:
431:
Over a finite field, the above open subset may not contain rational points and in general there is no hyperplanes with smooth intersection with
1272:
1205:
424:
is infinite, this open subset then has infinitely many rational points and there are infinitely many smooth hyperplane sections in
1307:
1264:
1117:
163:
510:
1071:
Theorem of
Bertini has also been generalized to discrete valuation domains or finite fields, or for Ă©tale coverings of
1112:
672:
40:
1259:
764:
658:
373:
168:
130:
52:
48:
523:
952:
880:
1149:
1018:
1107:
239:
210:
136:
73:
820:
435:. However, if we take hypersurfaces of sufficiently big degrees, then the theorem of Bertini holds.
831:
313:
992:
1158:
446:
36:
713:
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1201:
1197:
1254:
1189:
1168:
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824:
815:
The theorem of
Bertini has been generalized in various ways. For example, a result due to
44:
578:
279:
102:
67:
be a smooth quasi-projective variety over an algebraically closed field, embedded in a
1221:
1057:
816:
640:
616:
558:
413:
1301:
1190:
942:
17:
1140:
1144:
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28:
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1237:
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The rank of the fibration in the product is one less than the codimension of
47:. This is the simplest and broadest of the "Bertini theorems" applying to a
803:. This theorem also holds in characteristic p>0 when the linear system
1288:
268:
The theorem of
Bertini states that the set of hyperplanes not containing
669:
of the system. For clarification, this means that given a linear system
55:
of the underlying field, while the extensions require characteristic 0.
575:
and so its projection is contained in a divisor of the complete system
1163:
1015:, as well. The above theorem of Bertini is the special case where
276:
contains an open dense subset of the total system of divisors
35:
is an existence and genericity theorem for smooth connected
363:
is smooth, that is: the property of smoothness is generic.
348:, then these intersections (called hyperplane sections of
761:
in some dense open subset of the dual projective space
1224:(1974), "The transversality of a general translate",
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995:
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883:
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242:
213:
171:
139:
105:
76:
59:
Statement for hyperplane sections of smooth varieties
555:, so that the total space has lesser dimension than
443:
We consider the subfibration of the product variety
1042:
1007:
981:
909:
857:be the variety obtained by letting σ ∈
795:
741:
702:
649:
625:
597:
567:
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497:
471:
404:
340:
298:
257:
228:
199:
154:
121:
91:
51:; simplest because there is no restriction on the
945:and the base field has characteristic zero, then
416:define hyperplanes smooth hyperplane sections of
370:, there is a dense open subset of the dual space
505:the linear system of hyperplanes that intersect
1291:by Steven L. Kleiman, on the life and works of
1078:The theorem is often used for induction steps.
703:{\displaystyle f:X\rightarrow \mathbf {P} ^{n}}
1196:. Boston, MA: Birkhäuser Boston, Inc. p.
8:
917:is either empty or purely of the (expected)
1267:, vol. 52, New York: Springer-Verlag,
796:{\displaystyle (\mathbf {P} ^{n})^{\star }}
405:{\displaystyle (\mathbf {P} ^{n})^{\star }}
200:{\displaystyle (\mathbf {P} ^{n})^{\star }}
1172:
1162:
1034:
1030:
1029:
1020:
994:
970:
960:
954:
898:
888:
882:
865:. Then, there is an open dense subscheme
787:
777:
772:
766:
721:
715:
694:
689:
674:
642:
618:
590:
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580:
560:
548:{\displaystyle X\subset \mathbf {P} ^{n}}
539:
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484:
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386:
381:
375:
315:
291:
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146:
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114:
106:
104:
83:
78:
75:
1289:Bertini and his two fundamental theorems
1099:
982:{\displaystyle Y^{\sigma }\times _{X}Z}
910:{\displaystyle Y^{\sigma }\times _{X}Z}
753:is smooth -- outside the base locus of
1145:"Bertini theorems over finite fields"
39:for smooth projective varieties over
7:
1192:Théorèmes de Bertini et applications
1088:Grothendieck's connectedness theorem
352:) are connected, hence irreducible.
1050:is expressed as the quotient of SL
1043:{\displaystyle X=\mathbb {P} ^{n}}
272:and with smooth intersection with
25:
355:The theorem hence asserts that a
773:
690:
657:-variety, a general member of a
535:
382:
359:hyperplane section not equal to
258:{\displaystyle \mathbf {P} ^{n}}
245:
229:{\displaystyle \mathbf {P} ^{n}}
216:
177:
155:{\displaystyle \mathbf {P} ^{n}}
142:
92:{\displaystyle \mathbf {P} ^{n}}
79:
1188:Jouanolou, Jean-Pierre (1983).
1313:Theorems in algebraic geometry
1060:of upper triangular matrices,
784:
768:
736:
730:
685:
591:
583:
465:
457:
393:
377:
329:
323:
292:
284:
188:
172:
115:
107:
1:
1265:Graduate Texts in Mathematics
873:such that for σ ∈
637:is a smooth quasi-projective
341:{\displaystyle \dim(X)\geq 2}
1174:10.4007/annals.2004.160.1099
1008:{\displaystyle \sigma \in H}
306:. The set itself is open if
1113:Encyclopedia of Mathematics
819:asserts the following (cf.
472:{\displaystyle X\times |H|}
41:algebraically closed fields
1329:
133:of hyperplane divisors in
742:{\displaystyle f^{-1}(H)}
659:linear system of divisors
49:linear system of divisors
949:may be taken such that
665:is smooth away from the
633:of characteristic 0, if
613:Over any infinite field
366:Over an arbitrary field
162:. Recall that it is the
1130:Hartshorne, Ch. III.10.
757:-- for all hyperplanes
1226:Compositio Mathematica
1044:
1009:
983:
911:
797:
743:
704:
651:
627:
599:
569:
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499:
498:{\displaystyle x\in X}
473:
406:
342:
300:
259:
230:
201:
156:
123:
93:
1150:Annals of Mathematics
1045:
1010:
984:
912:
798:
744:
705:
652:
628:
600:
570:
550:
500:
474:
407:
343:
301:
260:
236:and is isomorphic to
231:
202:
157:
124:
94:
1308:Geometry of divisors
1064:is a subvariety and
1019:
993:
953:
881:
841:, and two varieties
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579:
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314:
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211:
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103:
74:
18:Bertini's lemma
933:. If, in addition,
823:): for a connected
598:{\displaystyle |H|}
299:{\displaystyle |H|}
122:{\displaystyle |H|}
37:hyperplane sections
1260:Algebraic Geometry
1222:Kleiman, Steven L.
1108:"Bertini theorems"
1040:
1005:
989:is smooth for all
979:
907:
793:
739:
700:
647:
623:
595:
565:
545:
495:
469:
439:Outline of a proof
402:
338:
310:is projective. If
296:
255:
226:
197:
152:
119:
89:
33:theorem of Bertini
1274:978-0-387-90244-9
1255:Hartshorne, Robin
1068:is a hyperplane.
821:Kleiman's theorem
650:{\displaystyle k}
626:{\displaystyle k}
609:General statement
568:{\displaystyle n}
479:with fiber above
16:(Redirected from
1320:
1285:
1241:
1240:
1218:
1212:
1211:
1195:
1185:
1179:
1178:
1176:
1166:
1157:(3): 1099–1127.
1137:
1131:
1128:
1122:
1121:
1104:
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1047:
1046:
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975:
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964:
916:
914:
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903:
902:
893:
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807:is unramified.
802:
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799:
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749:of a hyperplane
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69:projective space
43:, introduced by
21:
1328:
1327:
1323:
1322:
1321:
1319:
1318:
1317:
1298:
1297:
1293:Eugenio Bertini
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825:algebraic group
813:
811:Generalizations
783:
771:
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712:
711:
710:, the preimage
688:
671:
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522:
521:
481:
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414:rational points
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380:
372:
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312:
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278:
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243:
238:
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131:complete system
101:
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45:Eugenio Bertini
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1083:
1080:
1058:Borel subgroup
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1027:
1024:
1004:
1001:
998:
978:
973:
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963:
959:
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891:
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817:Steven Kleiman
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113:
109:
86:
81:
60:
57:
53:characteristic
24:
14:
13:
10:
9:
6:
4:
3:
2:
1325:
1314:
1311:
1309:
1306:
1305:
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1290:
1287:
1284:
1280:
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1270:
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1262:
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1252:
1251:
1247:
1239:
1235:
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1227:
1223:
1217:
1214:
1209:
1207:0-8176-3164-X
1203:
1199:
1194:
1193:
1184:
1181:
1175:
1170:
1165:
1160:
1156:
1152:
1151:
1146:
1142:
1141:Poonen, Bjorn
1136:
1133:
1127:
1124:
1119:
1115:
1114:
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1103:
1100:
1093:
1089:
1086:
1085:
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1079:
1076:
1074:
1069:
1067:
1063:
1059:
1054:
1035:
1025:
1022:
1002:
999:
996:
976:
971:
967:
961:
957:
948:
944:
940:
936:
932:
928:
924:
920:
904:
899:
895:
889:
885:
876:
872:
868:
864:
860:
856:
852:
848:
844:
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837:
835:
829:
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806:
788:
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733:
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664:
660:
644:
636:
620:
608:
606:
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562:
540:
530:
527:
518:
516:
512:
511:transversally
508:
492:
489:
486:
461:
453:
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438:
436:
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419:
415:
397:
387:
369:
364:
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317:
309:
288:
275:
271:
266:
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182:
165:
147:
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111:
84:
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66:
58:
56:
54:
50:
46:
42:
38:
34:
30:
19:
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1229:
1225:
1216:
1191:
1183:
1164:math/0204002
1154:
1148:
1135:
1126:
1111:
1102:
1077:
1072:
1070:
1065:
1061:
1052:
946:
938:
934:
930:
929:− dim
926:
922:
874:
870:
866:
862:
858:
854:
850:
846:
842:
838:
833:
832:homogeneous
827:
814:
804:
758:
754:
750:
662:
634:
612:
519:
514:
506:
442:
432:
430:
425:
421:
417:
367:
365:
360:
356:
354:
349:
307:
273:
269:
267:
64:
62:
32:
26:
1232:: 287–297,
849:mapping to
129:denote the
29:mathematics
1302:Categories
1248:References
830:, and any
667:base locus
164:dual space
1238:0010-437X
1118:EMS Press
1000:∈
997:σ
968:×
962:σ
919:dimension
896:×
890:σ
789:⋆
723:−
686:→
531:⊂
490:∈
454:×
398:⋆
333:≥
321:
193:⋆
1257:(1977),
1143:(2004).
1082:See also
836:-variety
1283:0463157
1120:, 2001
1056:by the
861:act on
420:. When
357:general
1281:
1271:
1236:
1204:
943:smooth
925:+ dim
853:, let
412:whose
99:. Let
31:, the
1159:arXiv
1094:Notes
1269:ISBN
1234:ISSN
1202:ISBN
941:are
937:and
921:dim
845:and
509:non-
63:Let
1169:doi
1155:160
869:of
661:on
513:at
318:dim
207:of
27:In
1304::
1279:MR
1277:,
1263:,
1230:28
1228:,
1200:.
1198:89
1167:.
1153:.
1147:.
1116:,
1110:,
1075:.
877:,
605:.
517:.
428:.
265:.
1210:.
1177:.
1171::
1161::
1073:X
1066:Y
1062:Z
1053:n
1036:n
1031:P
1026:=
1023:X
1003:H
977:Z
972:X
958:Y
947:H
939:Z
935:Y
931:X
927:Z
923:Y
905:Z
900:X
886:Y
875:H
871:G
867:H
863:Y
859:G
855:Y
851:X
847:Z
843:Y
839:X
834:G
828:G
805:f
785:)
779:n
774:P
769:(
759:H
755:f
751:H
737:)
734:H
731:(
726:1
719:f
696:n
691:P
683:X
680::
677:f
663:X
645:k
635:X
621:k
592:|
588:H
584:|
563:n
541:n
536:P
528:X
515:x
507:X
493:X
487:x
466:|
462:H
458:|
451:X
433:X
426:X
422:k
418:X
394:)
388:n
383:P
378:(
368:k
361:X
350:X
336:2
330:)
327:X
324:(
308:X
293:|
289:H
285:|
274:X
270:X
251:n
246:P
222:n
217:P
189:)
183:n
178:P
173:(
148:n
143:P
116:|
112:H
108:|
85:n
80:P
65:X
20:)
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