22:
1684:
The terminology in this article matches the terminology in the text of Fried and Jarden, who adopt Weil's nomenclature for varieties. The second edition reference here also contains a subsection providing a dictionary between this nomenclature and the more modern one of
1752:
Mumford only spends one section of the book on arithmetic concerns like the field of definition, but in it covers in full generality many scheme-theoretic results stated in this article.
1576:)), the variety consists of a single point and so the action of the absolute Galois group cannot distinguish whether the ideal of vanishing polynomials was generated by
1709:
Kunz deals strictly with affine and projective varieties and schemes but to some extent covers the relationship between Weil's definitions for varieties and
51:
1537:). In the affine case, this means the action of the absolute Galois group on the zero-locus is sufficient to recover the subset of
932:
530:
1518:
959:
594:
is itself a field of definition. This justifies saying that any variety possesses a unique, minimal field of definition.
1195:
936:
1743:
1700:
1675:
73:
309:
44:
1727:
1659:
973:
1247:. The reason for this discrepancy is that the scheme-theoretic definitions only keep track of the polynomial set
1767:
1180:
1157:
981:
1772:
234:
1506:
1364:
230:
358:
34:
1470:
1012:
678:
38:
30:
789:
1710:
427:
258:
55:
1348:
914:
808:
on the complex projective line swaps points with the same longitude but opposite latitudes.
801:
565:
133:
8:
885:
805:
102:
944:
753:
is a variety; it is absolutely irreducible because it consists of a single point. But
455:
152:
of a field is denoted by adding a superscript of "alg", e.g. the algebraic closure of
1739:
1696:
1671:
581:
545:
149:
95:
1731:
1663:
892:
interchanges opposite points of the sphere. The complex projective line cannot be
246:
182:
917:
is that such definitions are intrinsic and free of embeddings into ambient affine
297:-algebraic set that is irreducible, i.e. is not the union of two strictly smaller
940:
190:
1235:
are also discrepant, e.g. the (scheme-theoretic) minimal field of definition of
913:
One advantage of defining varieties over arbitrary fields through the theory of
1761:
1719:
1427:
1165:
270:
587:
553:
1161:
1114:
202:
194:
186:
1175:
One disadvantage of the scheme-theoretic definition is that a scheme over
804:
as being the equator on the
Riemann sphere, the coordinate-wise action of
87:
106:
1148:
Analogously to the definitions for affine and projective varieties, a
884:
with the
Riemann sphere using this map, the coordinate-wise action of
590:
proved that the intersection of all fields of definition of a variety
1667:
1735:
316:-algebraic sets whose defining polynomials' coefficients belong to
1251:. In this example, one way to avoid these problems is to use the
224:
904:, points fixed by complex conjugation, while the latter does not.
1461:) on the latter scheme: the sections of the structure sheaf of
1172:; furthermore, every variety has a minimal field of definition.
113:
can belong. Given polynomials, with coefficients in a field
1038:-algebraic set is an irreducible scheme; in this case, the
1693:
Introduction to
Commutative Algebra and Algebraic Geometry
1544:
In general, this information is not sufficient to recover
117:, it may not be obvious whether there is a smaller field
1342:
1034:-variety is absolutely irreducible if the associated
1243:, while in the first definition it would have been
840:is also a variety with minimal field of definition
800:as its minimal field of definition.) Viewing the
132:The issue of field of definition is of concern in
1653:
848:-isomorphism from the complex projective line to
1759:
431:, i.e. is not the union of two strictly smaller
43:but its sources remain unclear because it lacks
908:
390:) can be identical; in fact, the zero-locus in
225:Definitions for affine and projective varieties
478:-algebraic set for infinitely many subfields
323:One reason for considering the zero-locus in
625:-algebraic set but neither a variety nor a
257:, and by insisting that all polynomials be
796:-variety. (In fact, it is a variety with
229:Results and definitions stated below, for
74:Learn how and when to remove this message
1654:Fried, Michael D.; Moshe Jarden (2005).
1549:
1231:)-valued point. The two definitions of
1117:in the category of reduced schemes over
1718:
1061:-algebraic set is a reduced scheme. A
629:-variety, since it is the union of the
1760:
1198:(1,1,1) is a solution to the equation
819:defined by the homogeneous polynomial
633:-varieties defined by the polynomials
181:represent, respectively, the field of
1724:The Red Book of Varieties and Schemes
1541:consisting of vanishing polynomials.
1375:) of a subset of the polynomial ring
1003:-algebraic set associated to a given
576:. In other words those extensions of
282:) of a subset of the polynomial ring
121:, and other polynomials defined over
1690:
980:-algebraic sets regarded as schemes
15:
1343:Action of the absolute Galois group
1310:-algebraic set is the union of the
1152:-variety is a variety defined over
13:
1647:
1453:) together with the action of Gal(
572:is also linearly independent over
14:
1784:
1437:can be recovered from the scheme
105:to which the coefficients of the
712:) in the polynomial ring (
20:
844:. The following map defines a
1609:raised to some other power of
1339:)) and its complex conjugate.
1:
1046:. An absolutely irreducible
991:To every algebraic extension
556:. That means every subset of
414:is not algebraically closed.
1426:is a variety defined over a
909:Scheme-theoretic definitions
435:-algebraic sets. A variety
339:) is that, for two distinct
101:is essentially the smallest
7:
1713:'s definitions for schemes.
1695:. Birkhäuser. p. 256.
597:
139:
10:
1789:
1473:of the structure sheaf of
1168:is a regular extension of
774:is also the zero-locus of
525:is a variety defined over
474:-algebraic set is also an
253: − 1 over
1509:are constant on each Gal(
1415:) via its action on Spec(
1077:such that there exists a
144:Throughout this article,
1640:isomorphically onto a σ(
1013:fiber product of schemes
679:transcendental extension
29:This article includes a
1632:, an automorphism σ of
1190:is not an extension of
900:because the former has
790:complex projective line
446:if every polynomial in
402:is the zero-locus of a
233:, can be translated to
58:more precise citations.
1306:)), whose associated
1219:but the corresponding
1141:-variety defined over
510:-variety defined over
428:absolutely irreducible
148:denotes a field. The
1552:of the zero-locus of
1391:-algebraic set), Gal(
1367:on the zero-locus in
1349:absolute Galois group
1249:up to change of basis
1042:-variety is called a
421:-variety is called a
274:is the zero-locus in
125:, which still define
1768:Diophantine geometry
1730:. pp. 198–203.
1691:Kunz, Ernst (1985).
1399:) naturally acts on
1194:. For example, the
802:real projective line
757:is not defined over
566:linearly independent
462:) of polynomials in
301:-algebraic sets. A
235:projective varieties
134:diophantine geometry
1233:field of definition
1109:) is isomorphic to
1063:field of definition
965:-algebraic set. A
886:complex conjugation
806:complex conjugation
529:if and only if the
488:field of definition
398:) of any subset of
92:field of definition
1773:Algebraic geometry
1465:on an open subset
1379:. In general, if
1332: - (1+i)
1212: - (1+i)
1057:if the associated
603:The zero-locus of
552:, in the sense of
456:linear combination
31:list of references
1616:For any subfield
1598:, or, indeed, by
1383:is a scheme over
880:)). Identifying
690:, the polynomial
582:linearly disjoint
546:regular extension
450:that vanishes on
150:algebraic closure
96:algebraic variety
84:
83:
76:
1780:
1749:
1706:
1681:
1656:Field Arithmetic
1469:are exactly the
860:) → (2
792:is a projective
738:)-algebraic set
517:Equivalently, a
343:-algebraic sets
310:regular function
247:projective space
231:affine varieties
183:rational numbers
79:
72:
68:
65:
59:
54:this article by
45:inline citations
24:
23:
16:
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1668:10.1007/b138352
1662:. p. 780.
1650:
1648:Further reading
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1179:cannot have an
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1132:
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1007:-algebraic set
911:
896:-isomorphic to
839:
832:
825:
811:The projective
780:
765:
748:
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707:
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689:
672:
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639:
621:-variety and a
616:
609:
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466:that vanish on
381:
366:
356:
349:
237:, by replacing
227:
191:complex numbers
189:, the field of
185:, the field of
180:
160:. The symbols
142:
80:
69:
63:
60:
49:
35:related reading
25:
21:
12:
11:
5:
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1776:
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1736:10.1007/b62130
1720:Mumford, David
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1322:
1314:-variety Spec(
1303:
1297:
1293: - 2
1290:
1284:
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1270:
1263:
1255:-variety Spec(
1216:
1209:
1205: + i
1202:
1196:rational point
1122:
1094:
1069:is a subfield
1019:
929:-algebraic set
910:
907:
906:
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494:is a subfield
379:
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39:external links
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1065:of a variety
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1052:defined over
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490:of a variety
489:
485:
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457:
453:
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441:defined over
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359:intersections
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271:algebraic set
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249:of dimension
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64:November 2021
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50:Please help
42:
1644:)-variety.
1533: Spec(
1501: Spec(
1485: Spec(
1449: Spec(
1411: Spec(
1026: Spec(
960:irreducible
941:finite type
902:real points
742:defined by
406:element of
259:homogeneous
201:elements.
197:containing
107:polynomials
88:mathematics
56:introducing
1762:Categories
1548:. In the
1363:naturally
1089:such that
939:scheme of
872:, -i(
617:is both a
588:André Weil
564:) that is
502:such that
331:) and not
193:, and the
1636:will map
1628:-variety
1223:-variety
1085:-variety
970:-morphism
933:separated
815:-variety
770:), since
725:)). The
521:-variety
425:if it is
306:-morphism
109:defining
1728:Springer
1722:(1999).
1685:schemes.
1660:Springer
1624:and any
1507:residues
1505:) whose
1471:sections
1387:(e.g. a
1280:- 2
1273:+ 2
1137:) is an
1113:and the
976:between
974:morphism
956:-variety
921:-space.
701:equals (
598:Examples
312:between
291:-variety
140:Notation
1594:-
1559:-
1550:example
1525: ×
1493: ×
1477: ×
1441: ×
1403: ×
1266:+
1164:at the
1160:of the
1156:if the
1044:variety
1018: ×
1011:is the
937:reduced
915:schemes
864:,
833:+
826:+
749:-
697:-
610:+
454:is the
423:variety
245:) with
203:Affine
52:improve
1742:
1699:
1674:
1030:). A
999:, the
958:is an
506:is an
458:(over
404:single
375:) and
357:, the
207:-space
172:, and
94:of an
90:, the
1587:, by
1527:Spec(
1519:orbit
1495:Spec(
1489:) on
1479:Spec(
1443:Spec(
1422:When
1405:Spec(
1359:) of
1158:stalk
1133:Spec(
1123:Spec(
1105:Spec(
1095:Spec(
1020:Spec(
984:Spec(
972:is a
951:. A
945:Spec(
943:over
931:is a
664:With
568:over
544:is a
540:) of
486:. A
470:. A
308:is a
293:is a
286:. A
103:field
37:, or
1740:ISBN
1697:ISBN
1672:ISBN
1563:in (
1365:acts
1351:Gal(
1347:The
982:over
935:and
788:The
677:) a
647:and
580:are
554:Weil
350:and
1732:doi
1664:doi
1620:of
1521:in
1419:).
1239:is
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888:on
852:: (
681:of
548:of
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482:of
439:is
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