215:
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is finite, which is stronger than "integral".) The normalization of a scheme of dimension 1 is regular, and the normalization of a scheme of dimension 2 has only isolated singularities. Normalization is not usually used for
1396:
1228:
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597:
1407:
789:
707:
649:
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800:
372:, theorem 11) showed that every normal variety is regular outside a subset of codimension at least 2, and a similar result is true for schemes. So, for example, every normal
498:
If the initial scheme is not irreducible, the normalization is defined to be the disjoint union of the normalizations of the irreducible components.
1290:
112:. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve
1096:
956:
517:
1594:{\displaystyle {\text{Proj}}\left(\prod {\frac {k}{(f_{i},g)}}\right)\to {\text{Proj}}\left({\frac {k}{(f_{1}\cdots f_{k},g)}}\right)}
314:.) This is a direct translation, in terms of local rings, of the geometric condition that every finite birational morphism to
1728:
1693:
195:+ 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from
723:
660:
432:
160:
1720:
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is not an irreducible scheme since it has two components. Its normalization is given by the scheme morphism
608:
940:{\displaystyle {\text{Spec}}(\mathbb {C} /(x)\times \mathbb {C} /(y))\to {\text{Spec}}(\mathbb {C} /(xy))}
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1241:
421:
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1715:
108:
A morphism of varieties is finite if the inverse image of every point is finite and the morphism is
264:
40:
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163:
using the classical topology, that every link is connected. Equivalently, every complex point
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36:
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25:
8:
74:
21:
1746:
Zariski, Oscar (1939), "Some
Results in the Arithmetic Theory of Algebraic Varieties.",
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with the cusp singularity at the origin. Its normalization can be given by the map
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is normal: it cannot be simplified any further by finite birational morphisms.
44:
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93:
1391:{\displaystyle {\text{Proj}}\left({\frac {k}{(f_{1}\cdots f_{k},g)}}\right)}
1223:{\displaystyle \mathbb {C} /(xy)\to \mathbb {C} /(y,xy)=\mathbb {C} /(y)}
1083:{\displaystyle \mathbb {C} /(xy)\to \mathbb {C} /(x,xy)=\mathbb {C} /(x)}
86:
1767:
246:
103:
329:
if the linear system giving the embedding is complete. Equivalently,
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592:{\displaystyle C={\text{Spec}}\left({\frac {k}{y^{2}-x^{5}}}\right)}
214:
179:
is connected. For example, it follows that the nodal cubic curve
50:(understood to be irreducible) is normal if and only if the ring
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148:)) which is not an isomorphism. By contrast, the affine line
128:
is not normal, because there is a finite birational morphism
1670:
Commutative algebra. With a view toward algebraic geometry.
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is defined by gluing together the affine schemes Spec
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which is not an isomorphism; it sends two points of
104:
Geometric and algebraic interpretations of normality
1238:Similarly, for homogeneous irreducible polynomials
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1222:
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939:
784:{\displaystyle X={\text{Spec}}(\mathbb {C} /(xy))}
783:
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353:). This is the meaning of "normal" in the phrases
484:in its fraction field. Then the normalization of
35:if it is normal at every point, meaning that the
1783:
427:To define the normalization, first suppose that
1234:Normalization of reducible projective variety
702:{\displaystyle x\mapsto t^{2},y\mapsto t^{5}}
337:is not the linear projection of an embedding
70:over a field is normal if and only if every
1723:, vol. 52, New York: Springer-Verlag,
1709:
506:
66:is an integrally closed domain. A variety
1651:(1995). Springer, Berlin. Corollary 13.13
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813:
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459:as a union of affine open subsets Spec
369:
97:
1784:
1638:(1995). Springer, Berlin. Theorem 11.5
713:Normalization of axes in affine plane
644:{\displaystyle {\text{Spec}}(k)\to C}
411:a variety over a field, the morphism
399:with an integral birational morphism
321:An older notion is that a subvariety
267:. That is, each of these rings is an
379:
167:has arbitrarily small neighborhoods
92:Normal varieties were introduced by
1676:, vol. 150, Berlin, New York:
1277:{\displaystyle f_{1},\ldots ,f_{k}}
159:has the property, when viewed as a
13:
950:induced from the two quotient maps
14:
1808:
424:for schemes of higher dimension.
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1284:in a UFD, the normalization of
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439:. Every affine open subset of
1:
1721:Graduate Texts in Mathematics
1674:Graduate Texts in Mathematics
1658:
349:is contained in a hyperplane
654:induced from the algebra map
294:is finitely generated as an
7:
1616:Resolution of singularities
1611:Noether normalization lemma
1604:
501:
422:resolution of singularities
183:in the figure, defined by
10:
1813:
175:minus the singular set of
1686:10.1007/978-1-4612-5350-1
511:Consider the affine curve
155:A normal complex variety
1621:
1401:is given by the morphism
265:integrally closed domain
41:integrally closed domain
507:Normalization of a cusp
368:is normal. Conversely,
325:of projective space is
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359:rational normal scroll
231:
100:, section III).
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355:rational normal curve
217:
207:to the same point in
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298:-module is equal to
116:in the affine plane
1649:Commutative Algebra
1636:Commutative Algebra
318:is an isomorphism.
75:birational morphism
39:at the point is an
1797:Algebraic geometry
1716:Algebraic Geometry
1591:
1388:
1274:
1220:
1080:
937:
781:
699:
641:
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443:has the form Spec
395:: a normal scheme
308:field of fractions
234:More generally, a
232:
18:algebraic geometry
1730:978-0-387-90244-9
1711:Hartshorne, Robin
1695:978-0-387-94268-1
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380:The normalization
274:, and every ring
77:from any variety
60:regular functions
22:algebraic variety
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475:integral closure
161:stratified space
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1760:10.2307/2371499
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1678:Springer-Verlag
1666:Eisenbud, David
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453:integral domain
435:reduced scheme
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327:linearly normal
269:integral domain
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245:if each of its
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1754:(2): 249–294,
1748:Amer. J. Math.
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386:reduced scheme
381:
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366:regular scheme
306:) denotes the
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45:affine variety
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1792:Scheme theory
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1647:Eisenbud, D.
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402:
398:
394:
393:normalization
391:has a unique
390:
387:
377:
375:
371:
370:Zariski (1939
367:
362:
360:
356:
352:
348:
344:
340:
336:
332:
328:
324:
319:
317:
313:
309:
305:
302:. (Here Frac(
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1742:, p. 91
1714:
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949:
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717:For example,
716:
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448:
444:
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383:
376:is regular.
363:
350:
346:
342:
338:
334:
330:
322:
320:
315:
311:
303:
299:
295:
291:
290:) such that
287:
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107:
91:
82:
78:
67:
63:
55:
51:
47:
32:
28:
15:
433:irreducible
247:local rings
120:defined by
87:isomorphism
1786:Categories
1659:References
171:such that
37:local ring
1563:⋯
1529:…
1495:→
1444:…
1422:∏
1360:⋯
1326:…
1259:…
1138:→
998:→
887:→
847:×
687:↦
668:↦
636:→
570:−
140:maps to (
136:(namely,
1713:(1977),
1668:(1995),
1605:See also
502:Examples
455:. Write
345:(unless
1776:1507376
1768:2371499
1739:0463157
1704:1322960
473:be the
466:. Let
407:. (For
286:⊆ Frac(
96: (
94:Zariski
1774:
1766:
1737:
1727:
1702:
1692:
431:is an
364:Every
263:is an
243:normal
236:scheme
218:Curve
110:proper
85:is an
72:finite
33:normal
26:scheme
1764:JSTOR
1622:Notes
447:with
374:curve
278:with
58:) of
43:. An
20:, an
1725:ISBN
1690:ISBN
1499:Proj
1413:Proj
1296:Proj
891:Spec
806:Spec
735:Spec
614:Spec
529:Spec
384:Any
357:and
230:+ 1)
98:1939
1756:doi
1682:doi
477:of
451:an
341:⊆
310:of
257:X,x
241:is
199:to
81:to
62:on
31:is
24:or
16:In
1788::
1772:MR
1770:,
1762:,
1752:61
1750:,
1735:MR
1733:,
1719:,
1700:MR
1698:,
1688:,
1680:,
1672:,
495:.
415:→
403:→
361:.
333:⊆
282:⊆
222:=
211:.
187:=
144:,
132:→
124:=
89:.
1758::
1684::
1588:)
1582:)
1579:g
1576:,
1571:k
1567:f
1558:1
1554:f
1550:(
1545:]
1540:n
1536:x
1532:,
1526:,
1521:0
1517:x
1513:[
1510:k
1504:(
1491:)
1484:)
1481:g
1478:,
1473:i
1469:f
1465:(
1460:]
1455:n
1451:x
1447:,
1439:0
1435:x
1431:[
1428:k
1418:(
1385:)
1379:)
1376:g
1373:,
1368:k
1364:f
1355:1
1351:f
1347:(
1342:]
1337:n
1333:x
1329:,
1323:,
1318:0
1314:x
1310:[
1307:k
1301:(
1270:k
1266:f
1262:,
1256:,
1251:1
1247:f
1218:)
1215:y
1212:(
1208:/
1204:]
1201:y
1198:,
1195:x
1192:[
1188:C
1184:=
1181:)
1178:y
1175:x
1172:,
1169:y
1166:(
1162:/
1158:]
1155:y
1152:,
1149:x
1146:[
1142:C
1135:)
1132:y
1129:x
1126:(
1122:/
1118:]
1115:y
1112:,
1109:x
1106:[
1102:C
1078:)
1075:x
1072:(
1068:/
1064:]
1061:y
1058:,
1055:x
1052:[
1048:C
1044:=
1041:)
1038:y
1035:x
1032:,
1029:x
1026:(
1022:/
1018:]
1015:y
1012:,
1009:x
1006:[
1002:C
995:)
992:y
989:x
986:(
982:/
978:]
975:y
972:,
969:x
966:[
962:C
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929:y
926:x
923:(
919:/
915:]
912:y
909:,
906:x
903:[
899:C
895:(
884:)
881:)
878:y
875:(
871:/
867:]
864:y
861:,
858:x
855:[
851:C
844:)
841:x
838:(
834:/
830:]
827:y
824:,
821:x
818:[
814:C
810:(
779:)
776:)
773:y
770:x
767:(
763:/
759:]
756:y
753:,
750:x
747:[
743:C
739:(
731:=
728:X
695:5
691:t
684:y
681:,
676:2
672:t
665:x
639:C
633:)
630:]
627:t
624:[
621:k
618:(
586:)
578:5
574:x
565:2
561:y
555:]
552:y
549:,
546:x
543:[
540:k
534:(
525:=
522:C
493:i
490:B
486:X
482:i
479:A
471:i
468:B
464:i
461:A
457:X
449:R
445:R
441:X
437:X
429:X
417:X
413:Y
409:X
405:X
401:Y
397:Y
389:X
351:P
347:X
343:P
339:X
335:P
331:X
323:X
316:X
312:R
304:R
300:R
296:R
292:S
288:R
284:S
280:R
276:S
272:R
253:O
239:X
228:x
226:(
224:x
220:y
209:X
205:A
201:X
197:A
193:x
191:(
189:x
185:y
181:X
177:X
173:U
169:U
165:x
157:X
150:A
146:t
142:t
138:t
134:X
130:A
126:y
122:x
118:A
114:X
83:X
79:Y
68:X
64:X
56:X
54:(
52:O
48:X
29:X
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