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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
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383:, 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
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If the initial scheme is not irreducible, the normalization is defined to be the disjoint union of the normalizations of the irreducible components.
1301:
123:. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve
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967:
1739:
1704:
528:
1605:{\displaystyle {\text{Proj}}\left(\prod {\frac {k}{(f_{i},g)}}\right)\to {\text{Proj}}\left({\frac {k}{(f_{1}\cdots f_{k},g)}}\right)}
325:.) This is a direct translation, in terms of local rings, of the geometric condition that every finite birational morphism to
171:
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206:+ 1), is not normal. This also follows from the definition of normality, since there is a finite birational morphism from
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is not an irreducible scheme since it has two components. Its normalization is given by the scheme morphism
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951:{\displaystyle {\text{Spec}}(\mathbb {C} /(x)\times \mathbb {C} /(y))\to {\text{Spec}}(\mathbb {C} /(xy))}
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A morphism of varieties is finite if the inverse image of every point is finite and the morphism is
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using the classical topology, that every link is connected. Equivalently, every complex point
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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.
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257:
17:
1402:{\displaystyle {\text{Proj}}\left({\frac {k}{(f_{1}\cdots f_{k},g)}}\right)}
1234:{\displaystyle \mathbb {C} /(xy)\to \mathbb {C} /(y,xy)=\mathbb {C} /(y)}
1094:{\displaystyle \mathbb {C} /(xy)\to \mathbb {C} /(x,xy)=\mathbb {C} /(x)}
1778:
340:
if the linear system giving the embedding is complete. Equivalently,
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603:{\displaystyle C={\text{Spec}}\left({\frac {k}{y^{2}-x^{5}}}\right)}
224:
190:
is connected. For example, it follows that the nodal cubic curve
61:(understood to be irreducible) is normal if and only if the ring
159:)) which is not an isomorphism. By contrast, the affine line
139:
is not normal, because there is a finite birational morphism
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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
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Geometric and algebraic interpretations of normality
1249:Similarly, for homogeneous irreducible polynomials
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950:
795:{\displaystyle X={\text{Spec}}(\mathbb {C} /(xy))}
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364:). This is the meaning of "normal" in the phrases
495:in its fraction field. Then the normalization of
46:if it is normal at every point, meaning that the
438:To define the normalization, first suppose that
1245:Normalization of reducible projective variety
713:{\displaystyle x\mapsto t^{2},y\mapsto t^{5}}
348:is not the linear projection of an embedding
8:
81:over a field is normal if and only if every
1734:, vol. 52, New York: Springer-Verlag,
77:is an integrally closed domain. A variety
1662:(1995). Springer, Berlin. Corollary 13.13
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470:as a union of affine open subsets Spec
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1649:(1995). Springer, Berlin. Theorem 11.5
724:Normalization of axes in affine plane
655:{\displaystyle {\text{Spec}}(k)\to C}
422:a variety over a field, the morphism
410:with an integral birational morphism
332:An older notion is that a subvariety
278:. That is, each of these rings is an
7:
178:has arbitrarily small neighborhoods
103:Normal varieties were introduced by
1687:, vol. 150, Berlin, New York:
1288:{\displaystyle f_{1},\ldots ,f_{k}}
170:has the property, when viewed as a
961:induced from the two quotient maps
25:
435:for schemes of higher dimension.
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450:. Every affine open subset of
1:
1732:Graduate Texts in Mathematics
1685:Graduate Texts in Mathematics
360:is contained in a hyperplane
665:induced from the algebra map
305:is finitely generated as an
1627:Resolution of singularities
1622:Noether normalization lemma
433:resolution of singularities
194:in the figure, defined by
1824:
186:minus the singular set of
1697:10.1007/978-1-4612-5350-1
522:Consider the affine curve
166:A normal complex variety
1412:is given by the morphism
276:integrally closed domain
52:integrally closed domain
518:Normalization of a cusp
379:is normal. Conversely,
336:of projective space is
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370:rational normal scroll
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111:, section III).
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366:rational normal curve
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218:to the same point in
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309:-module is equal to
127:in the affine plane
1660:Commutative Algebra
1647:Commutative Algebra
329:is an isomorphism.
86:birational morphism
50:at the point is an
1808:Algebraic geometry
1727:Algebraic Geometry
1602:
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454:has the form Spec
406:: a normal scheme
319:field of fractions
245:More generally, a
243:
29:algebraic geometry
1741:978-0-387-90244-9
1722:Hartshorne, Robin
1706:978-0-387-94268-1
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391:The normalization
285:, and every ring
88:from any variety
71:regular functions
33:algebraic variety
16:(Redirected from
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1689:Springer-Verlag
1677:Eisenbud, David
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464:integral domain
446:reduced scheme
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338:linearly normal
280:integral domain
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256:if each of its
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1759:Amer. J. Math.
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397:reduced scheme
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377:regular scheme
317:) denotes the
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56:affine variety
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18:Normal variety
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405:
404:normalization
402:has a unique
401:
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390:
388:
386:
382:
381:Zariski (1939
378:
373:
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363:
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324:
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313:. (Here Frac(
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728:For example,
727:
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459:
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387:is regular.
374:
361:
357:
353:
349:
345:
341:
333:
331:
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322:
314:
310:
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302:
301:) such that
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66:
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43:
39:
26:
444:irreducible
258:local rings
131:defined by
98:isomorphism
1797:Categories
1670:References
182:such that
48:local ring
1574:⋯
1540:…
1506:→
1455:…
1433:∏
1371:⋯
1337:…
1270:…
1149:→
1009:→
898:→
858:×
698:↦
679:↦
647:→
581:−
151:maps to (
147:(namely,
1724:(1977),
1679:(1995),
1616:See also
513:Examples
466:. Write
356:(unless
1787:1507376
1779:2371499
1750:0463157
1715:1322960
484:be the
477:. Let
418:. (For
297:⊆ Frac(
107: (
105:Zariski
1785:
1777:
1748:
1738:
1713:
1703:
442:is an
375:Every
274:is an
254:normal
247:scheme
229:Curve
121:proper
96:is an
83:finite
44:normal
37:scheme
1775:JSTOR
1633:Notes
458:with
385:curve
289:with
69:) of
54:. An
31:, an
1736:ISBN
1701:ISBN
1510:Proj
1424:Proj
1307:Proj
902:Spec
817:Spec
746:Spec
625:Spec
540:Spec
395:Any
368:and
241:+ 1)
109:1939
1767:doi
1693:doi
488:of
462:an
352:⊆
321:of
268:X,x
252:is
210:to
92:to
73:on
42:is
35:or
27:In
1799::
1783:MR
1781:,
1773:,
1763:61
1761:,
1746:MR
1744:,
1730:,
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233:=
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1590:g
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1582:k
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1532:0
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1524:[
1521:k
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1219:/
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490:A
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468:X
460:R
456:R
452:X
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400:X
362:P
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75:X
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65:(
63:O
59:X
40:X
20:)
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