42:
Zariski's main theorem can be stated in several ways which at first sight seem to be quite different, but are in fact deeply related. Some of the variations that have been called
Zariski's main theorem are as follows:
145:
366:
294:
246:
757:
202:
35:), is a statement about the structure of birational morphisms stating roughly that there is only one branch at any normal point of a variety. It is the special case of
721:
1041:Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Troisième partie
591:
has 2 irreducible components meeting at a point. As predicted by
Hartshorne's form of the main theorem, the total transform is connected and of dimension at least 1.
176:
1027:
Eléments de géométrie algébrique (rédigés avec la collaboration de Jean
Dieudonné) : III. Étude cohomologique des faisceaux cohérents, Première partie
1089:, Lecture Notes in Mathematics, vol. 1358 (expanded, Includes Michigan Lectures (1974) on Curves and their Jacobians ed.), Berlin, New York:
975:
1304:
980:
933:
The property of being normal is stronger than the property of being unibranch: for example, a cusp of a plane curve is unibranch but not normal.
575:
was irreducible. It is easy to find examples where the total transform is reducible by blowing up other points on the transform. For example, if
1164:
1106:
1065:
50:
The total transform of a normal fundamental point of a birational map has positive dimension. This is essentially
Zariski's original version.
36:
675:
are quasi-finite. Grothendieck proved that under the hypothesis of separatedness all quasi-finite morphisms are compositions of such
663:
In EGA IV, Grothendieck observed that the last statement could be deduced from a more general theorem about the structure of
1014:
667:, and the latter is often referred to as the "Zariski's main theorem in the form of Grothendieck". It is well known that
490:
is a birational projective morphism between noetherian integral schemes, then the inverse image of every normal point of
970:
631:
In EGA III, Grothendieck calls the following statement which does not involve connectedness a "Main theorem" of
Zariski
651:
is a quasi-projective morphism of
Noetherian schemes then the set of points that are isolated in their fiber is open in
1009:
990:
763:
The relation between this theorem about quasi-finite morphisms and Théorème 4.4.3 of EGA III quoted above is that if
775:
is a projective morphism of varieties, then the set of points that are isolated in their fiber is quasifinite over
118:
1219:
Zariski, Oscar (1949), "A simple analytical proof of a fundamental property of birational transformations.",
655:. Moreover the induced scheme of this set is isomorphic to an open subset of a scheme that is finite over
323:
251:
1035:
1021:
898:
are equal. This is essentially
Zariski's formulation of his main theorem in terms of commutative rings.
664:
57:
68:
Several results in commutative algebra imply the geometric forms of
Zariski's main theorem, including:
1228:
1039:
1025:
1004:
61:
985:
950:
207:
80:
1246:
1200:
730:
724:
47:
A birational morphism with finite fibers to a normal variety is an isomorphism to an open subset.
20:
790:
reformulated his main theorem in terms of commutative algebra as a statement about local rings.
181:
1272:
1160:
1102:
1061:
694:
1262:
1236:
1190:
1152:
1094:
1049:
474:, Corollary III.11.4) calls the following connectedness statement "Zariski's Main theorem":
1258:
1212:
1174:
1116:
1075:
779:. Then structure theorem for quasi-finite morphisms applies and yields the desired result.
154:
1254:
1208:
1170:
1148:
1112:
1090:
1071:
1057:
672:
76:, which is a variation of the statement that the transform of a normal point is connected.
1181:
Zariski, Oscar (1943), "Foundations of a general theory of birational correspondences.",
1232:
1267:
1140:
668:
53:
The total transform of a normal point under a proper birational morphism is connected.
1298:
1082:
688:
28:
622:
consists of two points, which is not connected and does not have positive dimension.
938:
727:, quasi-finite, finitely presented morphism then there is a factorization into
470:
Here are some variants of this theorem stated using more recent terminology.
911:
587:
and then blowing up another point on this transform, the total transform of
73:
1276:
1241:
759:, where the first map is an open immersion and the second one is finite.
87:
The original result was labelled as the "MAIN THEOREM" in
Zariski (
1204:
1156:
1250:
945:
A formal power series version of
Zariski's main theorem says that if
1195:
1044:, Publications Mathématiques de l'IHÉS, vol. 28, pp. 43–48
1030:, Publications Mathématiques de l'IHÉS, vol. 11, pp. 5–167
1289:
1098:
1147:, Lecture Notes in Mathematics, vol. 169, Berlin, New York:
794:, Théorème 4.4.7) generalized Zariski's formulation as follows:
598:
is not normal and the conclusion of the main theorem fails, take
56:
A generalization due to Grothendieck describes the structure of
83:. This is a strong form of the statement that it is unibranch.
906:
A topological version of Zariski's main theorem says that if
451:
is not defined whose local ring is normal, and the transform
568:
as predicted by Zariski's original form of his main theorem.
518:
is a birational transformation of projective varieties with
914:; in other words there are arbitrarily small neighborhoods
522:
normal, then the total transform of a fundamental point of
802:
is an algebra of finite type over a local Noetherian ring
1290:
Is there an intuitive reason for Zariski's main theorem?
564:
is a projective space, which has dimension greater than
886:
are integral and have the same field of fractions, and
910:
is a (closed) point of a normal complex variety it is
626:
953:; in other words the completion of the local ring at
890:
is integrally closed, then this theorem implies that
733:
697:
326:
254:
210:
184:
157:
121:
94:
16:
Theorem of algebraic geometry and commutative algebra
540:
is a smooth variety of dimension greater than 1 and
1123:Peskine, Christian (1966), "Une généralisation du
782:
751:
715:
606:to be given by identifying two distinct points on
360:
288:
240:
196:
170:
139:
79:The local ring of a normal point of a variety is
627:Zariski's main theorem for quasifinite morphisms
447:is a subvariety of the set where the inverse of
936:
498:The following consequence of it (Theorem V.5.2,
151:) such that the projection on the first factor
103:be a birational mapping of algebraic varieties
95:Zariski's main theorem for birational morphisms
901:
419:— each irreducible component of the transform
8:
922:such that the set of non-singular points of
791:
783:Zariski's main theorem for commutative rings
676:
632:
375:The original statement of the theorem in (
949:is a normal point of a variety then it is
614:to be the image of these two points. Then
471:
1266:
1240:
1194:
732:
696:
602:′ to be a smooth variety, and take
571:In the previous example the transform of
526:is connected and of dimension at least 1.
411:′ then — under the assumption that
387:is an irreducible fundamental variety on
349:
344:
331:
325:
277:
272:
259:
253:
220:
215:
209:
183:
162:
156:
120:
902:Zariski's main theorem: topological form
560:, and the component of the transform of
140:{\displaystyle \Gamma \subset V\times W}
958:
927:
787:
579:′ is given by blowing up a point
544:′ is given by blowing up a point
376:
178:induces an isomorphism between an open
88:
39:when the two varieties are birational.
32:
1087:The red book of varieties and schemes
361:{\displaystyle p_{2}\circ p_{1}^{-1}}
289:{\displaystyle p_{2}\circ p_{1}^{-1}}
7:
981:Grothendieck's connectedness theorem
618:is not normal, and the transform of
976:Fulton–Hansen connectedness theorem
122:
115:is defined by a closed subvariety
14:
814:which is minimal among ideals of
842:′ (whose inverse image in
971:Deligne's connectedness theorem
435:is essentially a morphism from
407:has no fundamental elements on
391:of a birational correspondence
316:, and the image of a subset of
37:Zariski's connectedness theorem
1305:Theorems in algebraic geometry
743:
737:
707:
235:
229:
1:
1221:Proc. Natl. Acad. Sci. U.S.A.
957:is a normal integral domain (
850:) such that the localization
838:′ with a maximal ideal
502:) also goes under this name:
241:{\displaystyle p_{1}^{-1}(U)}
423:is of higher dimension than
1010:Encyclopedia of Mathematics
991:Theorem on formal functions
752:{\displaystyle X\to Z\to Y}
455:means the inverse image of
1321:
197:{\displaystyle U\subset V}
1145:Anneaux locaux henséliens
830:, then there is a finite
937:Zariski's main theorem:
818:whose inverse image in
716:{\displaystyle f:X\to Y}
459:under the morphism from
1183:Trans. Amer. Math. Soc.
1036:Grothendieck, Alexandre
1022:Grothendieck, Alexandre
1003:Danilov, V.I. (2001) ,
300:too. The complement of
72:A normal local ring is
810:is a maximal ideal of
753:
717:
665:quasi-finite morphisms
379:, p. 522) reads:
362:
290:
242:
198:
172:
141:
58:quasi-finite morphisms
25:Zariski's main theorem
859:is isomorphic to the
822:is the maximal ideal
754:
718:
691:separated scheme and
594:For an example where
415:is locally normal at
363:
296:is an isomorphism on
291:
243:
199:
173:
171:{\displaystyle p_{1}}
142:
1242:10.1073/pnas.35.1.62
1129:Bull. Sci. Math. (2)
1056:, Berlin, New York:
731:
695:
679:, Théorème 8.12.6):
443:that is birational,
324:
252:
208:
182:
155:
119:
1233:1949PNAS...35...62Z
986:Stein factorization
951:analytically normal
635:, Théorème 4.4.3):
357:
314:indeterminacy locus
310:fundamental variety
285:
228:
81:analytically normal
1157:10.1007/BFb0069571
1054:Algebraic Geometry
792:Grothendieck (1961
749:
713:
677:Grothendieck (1966
633:Grothendieck (1961
610:′, and take
358:
340:
286:
268:
238:
211:
194:
168:
137:
21:algebraic geometry
1166:978-3-540-05283-8
1108:978-3-540-63293-1
1067:978-0-387-90244-9
1050:Hartshorne, Robin
1005:"Zariski theorem"
383:MAIN THEOREM: If
29:Oscar Zariski
1312:
1279:
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673:finite morphisms
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1196:10.2307/1990215
1180:
1167:
1149:Springer-Verlag
1141:Raynaud, Michel
1139:
1122:
1109:
1091:Springer-Verlag
1081:
1068:
1058:Springer-Verlag
1048:
1034:
1020:
1002:
999:
967:
943:
904:
878:If in addition
873:
858:
785:
729:
728:
693:
692:
669:open immersions
629:
533:
403:′ and if
370:total transform
327:
322:
321:
255:
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206:
205:
180:
179:
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117:
116:
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12:
11:
5:
1318:
1316:
1308:
1307:
1297:
1296:
1293:
1292:
1285:
1284:External links
1282:
1281:
1280:
1216:
1189:(3): 490–542,
1178:
1165:
1137:
1120:
1107:
1099:10.1007/b62130
1083:Mumford, David
1079:
1066:
1046:
1032:
1018:
998:
995:
994:
993:
988:
983:
978:
973:
966:
963:
942:
935:
926:is connected (
903:
900:
876:
875:
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854:
788:Zariski (1949)
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111:. Recall that
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556:is normal at
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536:Suppose that
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521:
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1227:(1): 62–66,
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1182:
1144:
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1125:main theorem
1124:
1086:
1053:
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959:Mumford 1999
954:
946:
944:
939:power series
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928:Mumford 1999
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384:
377:Zariski 1943
374:
369:
368:is called a
317:
313:
309:
308:is called a
305:
301:
297:
148:
112:
108:
104:
100:
98:
86:
67:
41:
27:, proved by
24:
18:
463:′ to
439:′ to
997:References
961:, III.9).
930:, III.9).
1135:: 119–127
1085:(1999) ,
1015:EMS Press
912:unibranch
863:-algebra
834:-algebra
744:→
738:→
725:separated
708:→
351:−
338:∘
279:−
266:∘
222:−
189:⊂
132:×
126:⊂
123:Γ
74:unibranch
1299:Category
1277:16588856
1143:(1970),
1052:(1977),
1038:(1966),
1024:(1961),
965:See also
531:Examples
500:loc.cit.
395:between
1268:1062959
1259:0028056
1229:Bibcode
1213:0008468
1205:1990215
1175:0277519
1117:1748380
1076:0463157
872:′
867:′
647:→
552:. Then
514:→
486:→
372:of it.
62:schemes
31: (
1275:
1265:
1257:
1249:
1211:
1203:
1173:
1163:
1115:
1105:
1074:
1064:
806:, and
320:under
1251:88284
1247:JSTOR
1201:JSTOR
723:is a
687:is a
431:Here
1273:PMID
1161:ISBN
1103:ISBN
1062:ISBN
941:form
894:and
882:and
671:and
399:and
204:and
107:and
99:Let
89:1943
33:1943
1263:PMC
1237:doi
1191:doi
1153:doi
1095:doi
918:of
846:is
826:of
798:If
683:if
639:If
583:on
548:on
506:If
478:If
312:or
304:in
91:).
60:of
19:In
1301::
1271:,
1261:,
1255:MR
1253:,
1245:,
1235:,
1225:35
1223:,
1209:MR
1207:,
1199:,
1187:53
1185:,
1171:MR
1169:,
1159:,
1151:,
1133:90
1131:,
1113:MR
1111:,
1101:,
1093:,
1072:MR
1070:,
1060:,
1013:,
1007:,
467:.
23:,
1239::
1231::
1193::
1155::
1097::
955:x
947:x
924:U
920:x
916:U
908:x
896:B
892:A
888:A
884:B
880:A
874:.
870:m
865:A
861:A
856:n
852:B
848:m
844:A
840:m
836:A
832:A
828:A
824:m
820:A
816:B
812:B
808:n
804:A
800:B
777:Y
773:Y
771:→
769:X
767::
765:f
747:Y
741:Z
735:X
711:Y
705:X
702::
699:f
685:Y
659:.
657:Y
653:X
649:Y
645:X
643::
641:f
620:W
616:W
612:W
608:V
604:V
600:V
596:W
589:W
585:V
581:W
577:V
573:W
566:W
562:W
558:W
554:V
550:V
546:W
542:V
538:V
524:f
520:Y
516:Y
512:X
510::
508:f
492:Y
488:Y
484:X
482::
480:f
465:V
461:V
457:W
453:T
449:T
445:W
441:V
437:V
433:T
427:.
425:W
421:T
417:W
413:V
409:V
405:T
401:V
397:V
393:T
389:V
385:W
354:1
346:1
342:p
333:2
329:p
318:V
306:V
302:U
298:U
282:1
274:1
270:p
261:2
257:p
236:)
233:U
230:(
225:1
217:1
213:p
192:V
186:U
164:1
160:p
149:f
135:W
129:V
113:f
109:W
105:V
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