661:
is regular in the first sense if and only if it is so in the second sense. Also, it is not immediately clear whether regularity depends on a choice of affine charts (it does not.) This kind of a consistency issue, however, disappears if one adopts the formal definition. Formally, an (abstract)
2202:. Since morphisms of varieties are obtained by gluing morphisms of affine varieties in the same way morphisms of schemes are obtained by gluing morphisms of affine schemes, it follows that the category of varieties is a full subcategory of the category of schemes over
7416:
6141:
7566:
4851:
6635:
7044:
7669:
1196:
1859:
6911:
6765:
4013:
4235:
2034:
2557:
3821:
3465:
3113:
2671:
2094:
2189:
7188:. Then the issue here is whether the "regular-ness" can be patched together; this answer is yes and that can be seen from the construction of the structure sheaf of an affine variety as described at
5153:
2343:
5515:
4961:
252:
4641:
3694:
3358:
770:
7287:
3913:
6225:
6476:
5621:
892:
6962:
6043:
4306:
7092:
5662:
4533:
3548:
1302:
313:
2933:
5812:
5441:
5385:
3005:
2849:
7438:
4662:
1233:
2465:
6002:
1013:
169:
140:
6518:
5279:
5243:
1658:
1571:
1347:
4442:
2397:
205:
6802:
5859:
4137:
1955:
971:
7282:
7243:
5953:
5914:
5759:
5325:
5287:: The above does not say a morphism from a projective variety to a projective space is given by a single set of polynomials (unlike the affine case). For example, let
2730:
522:
2277:
1097:
435:
3247:
1260:
343:
817:
6974:
3298:
1718:
1631:
921:
542:
7571:
1104:
1432:
or more abstractly the ring of global sections of the structure sheaf) is a fundamental object in affine algebraic geometry. The only regular function on a
1785:
6807:
2470:
6643:
3707:
1437:
6246:" is a key in the proof, see Eisenbud, Ch. 14 of "Commutative algebra with a view toward algebraic geometry." In fact, the proof there shows that if
2562:
3934:
4149:
3566:
2282:
8043:
8015:
7907:
7879:
7209:
are quasi-projective, then the proof can be given. The non-quasi-projective case strongly depends on one's definition of an abstract variety
1963:
3607:
3636:
3414:
3303:
66:
of algebraic varieties. Because regular and biregular are very restrictive conditions – there are no non-constant regular functions on
7922:
6258:
3040:
7935:
7999:
7927:
7871:
6235:
4320:. Hence, the above construction determines a contravariant-equivalence between the category of algebraic varieties over a field
2042:
8073:
2115:
5051:
7979:
7861:
7125:
4479:, but the distinction is usually ignored in practice.) In particular, a regular map into the complex numbers is just a usual
3494:
2879:
5446:
4869:
4395:
213:
7411:{\displaystyle g\circ \phi ^{a}=g(\phi ({\overline {y_{1}}}),\dots ,\phi ({\overline {y_{m}}}))=\phi ({\overline {g}})=0}
4580:
8063:
4028:
6482:
699:
8068:
7944:
7156:
3826:
6164:
6409:
5524:
4394:), a rational function is regular if and only if it has no poles of codimension one. This is an algebraic analog of
830:
7889:
1900:
6916:
6136:{\displaystyle e(x)=\max\{\dim Z\mid Z{\text{ an irreducible component of }}f^{-1}(f(x)){\text{ containing }}x\}.}
6243:
4475:. (There is actually a slight technical difference: a regular map is a meromorphic map whose singular points are
4468:
4461:
1425:
7749:
5212:
1535:
4267:
7056:
5626:
4497:
7561:{\displaystyle {\phi ^{a}}^{\#}(g)=g(\phi ({\overline {y_{1}}}),\dots ,\phi ({\overline {y_{m}}}))=\phi (g)}
6513:
4846:{\displaystyle (a_{0}:\dots :a_{m})=(1:a_{1}/a_{0}:\dots :a_{m}/a_{0})\sim (a_{1}/a_{0},\dots ,a_{m}/a_{0})}
1265:
260:
7748:
Proof: it's enough to consider the case when the variety is affine and then use the fact that a
Noetherian
7960:
The Red Book of
Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians
5770:
5390:
5334:
2954:
2735:
93:; a morphism between algebraic varieties is precisely a morphism of the underlying locally ringed spaces.
1204:
6630:{\displaystyle \operatorname {H} ^{p}(Y,R^{q}f_{*}f^{*}F)\Rightarrow \operatorname {H} ^{p+q}(X,f^{*}F)}
4476:
2417:
5960:
3606:) The image will either be a single point, or the whole projective line (this is a consequence of the
979:
524:
is the same as the restriction of a polynomial map whose components satisfy the defining equations of
145:
116:
7161:
7150:
6400:
6151:
5257:
5221:
4480:
4408:
between the underlying topological spaces need not be an isomorphism (a counterexample is given by a
1636:
1549:
1307:
678:
90:
4414:
2348:
666:. When this definition is used, a morphism of varieties is just a morphism of locally ringed spaces.
178:
8025:
6777:
5829:
4085:
3577:
2221:
1873:
1865:
445:
39:
7050:
4324:
and dominant rational maps between them and the category of finitely generated field extension of
1927:
926:
7136:
4539:
4409:
1661:
1433:
671:
67:
35:
7796:
7260:
7221:
5919:
5880:
5725:
8039:
8011:
7975:
7931:
7903:
7875:
7146:
6373:
5294:
2703:
1906:, then, working with only the closed points, the above coincides with the definition given at
1468:
495:
83:
2249:
8003:
7989:
7967:
7917:
6501:
6385:
6344:
6312:
6254:, then the dimension equality in 2. of the theorem holds in general (not just generically).
6239:
4992:
is a fraction of homogeneous elements of the same degree in the homogeneous coordinate ring
3931:
The image of a morphism of varieties need not be open nor closed (for example, the image of
2195:
1441:
1021:
560:
359:
172:
79:
7993:
7039:{\displaystyle \operatorname {deg} (f^{*}L)=\operatorname {deg} (f)\operatorname {deg} (L)}
3118:
1238:
321:
8035:
7963:
7899:
7141:
6281:
5000:. We can arrange the fractions so that they all have the same homogeneous denominator say
4472:
3595:
3580:
1896:
1872:. All morphisms between affine schemes are of this type and gluing such morphisms gives a
1429:
1421:
346:
28:
7664:{\displaystyle f^{\#a}=({\overline {y_{1}}}\circ f,\dots ,{\overline {y_{m}}}\circ f)=f.}
778:
1191:{\displaystyle \phi ^{a}=(\phi ({\overline {y_{1}}}),\dots ,\phi ({\overline {y_{m}}}))}
7189:
6389:
4453:
4391:
4387:
3925:
3252:
1687:
1600:
897:
527:
208:
8057:
7955:
6251:
4405:
4253:
4249:
2097:
1854:{\displaystyle \phi ^{a}:X\to Y,\,{\mathfrak {p}}\mapsto \phi ^{-1}({\mathfrak {p}})}
1765:
670:
The composition of regular maps is again regular; thus, algebraic varieties form the
6906:{\displaystyle R^{q}f_{*}(f^{*}F)=R^{q}f_{*}{\mathcal {O}}_{X}\otimes L^{\otimes n}}
5694:
be a dominating (i.e., having dense image) morphism of algebraic varieties, and let
1740:. This characterization is sometimes taken as the definition of a regular function.
6760:{\displaystyle \chi (f^{*}F)=\sum _{q=0}^{\infty }(-1)^{q}\chi (R^{q}f_{*}f^{*}F).}
4309:
4241:
4024:
663:
110:
71:
2943:. It is regular at (0, 1) despite the expression since, as a rational function on
677:
Regular maps between affine varieties correspond contravariantly in one-to-one to
8029:
7948:
7893:
7865:
17:
1869:
63:
51:
2194:
This fact means that the category of affine varieties can be identified with a
8007:
7180:
Here is the argument showing the definitions coincide. Clearly, we can assume
4566:= 0 for simplicity. Then, by continuity, there is an open affine neighborhood
3483:
given by σ(x : y) = (y : x); in particular, σ exchanges 0 and ∞. If
75:
43:
1538:; i.e., an open subvariety of a projective variety, then the function field
693:
is a morphism of affine varieties, then it defines the algebra homomorphism
4655:
are the homogeneous coordinates. Note the target space is the affine space
4008:{\displaystyle \mathbf {A} ^{2}\to \mathbf {A} ^{2},\,(x,y)\mapsto (x,xy)}
2210:
4230:{\displaystyle k(Y)=\varinjlim k\hookrightarrow k(X),\,g\mapsto g\circ f}
7752:
is the intersection of all the localizations at height-one prime ideals.
2732:
is bijective. But the corresponding ring homomorphism is the inclusion
2029:{\displaystyle {\mathfrak {m}}_{f(x)}=\phi ^{-1}({\mathfrak {m}}_{x})}
3025:
is an algebraic variety since it is an open subset of a variety. If
2552:{\displaystyle f:\mathbf {A} ^{1}\to X,\,t\mapsto (t^{2}-1,t^{3}-t)}
641:
It is not immediately obvious that the two definitions coincide: if
7971:
6485:
for a ramified covering shows the "étale" here cannot be omitted.)
4399:
6292:
is the degree of the finite field extension of the function field
3816:{\displaystyle p^{\#}:k\to k=k\otimes _{k}k,\,f\mapsto f\otimes 1}
662:
algebraic variety is defined to be a particular kind of a locally
3928:
with respect to
Zariski topologies on the source and the target.
3460:{\displaystyle \mathbf {P} ^{1}=\mathbf {A} ^{1}\cup \{\infty \}}
3384:
is in fact a polynomial. Hence, the ring of regular functions on
5195:, then, by the above procedure, one can pick a different set of
3108:{\displaystyle D_{\mathbf {A} ^{2}}(x)=\mathbf {A} ^{2}-\{x=0\}}
7962:. Lecture Notes in Mathematics. Vol. 1358 (2nd ed.).
27:"Biregular" redirects here. For the graph theory concept, see
6284:
surjective morphism between algebraic varieties over a field
42:
is a function between the varieties that is given locally by
7073:
6943:
6876:
4015:
is neither open nor closed). However, one can still say: if
3590:
in the function field may all be realised as morphisms from
2666:{\displaystyle f^{\#}:k\to k,\,g\mapsto g(t^{2}-1,t^{3}-t),}
1436:
is constant (this can be viewed as an algebraic analogue of
4379:
and, conversely, such a morphism as a rational function on
2089:{\displaystyle {\mathfrak {m}}_{x},{\mathfrak {m}}_{f(x)}}
7683:
be the coordinate ring of such an affine neighborhood of
6234:
In
Mumford's red book, the theorem is proved by means of
5187:
simultaneously. If they vanish simultaneously at a point
2184:{\displaystyle {\mathfrak {m}}_{x}=\{g\in k\mid g(x)=0\}}
973:
is an algebra homomorphism, then it induces the morphism
5148:{\displaystyle f(x)=(f_{0}(x):f_{1}(x):\dots :f_{m}(x))}
3704:
are affine, then the corresponding ring homomorphism is
2338:{\displaystyle f:X\to \mathbf {A} ^{1},\,(x,y)\mapsto x}
58:. A regular map whose inverse is also regular is called
2559:
is a morphism. It corresponds to the ring homomorphism
4143:
induces an injection on the level of function fields:
1597:
of the same degree in the homogeneous coordinate ring
89:
An algebraic variety has naturally the structure of a
7574:
7441:
7290:
7263:
7224:
7059:
6977:
6919:
6810:
6780:
6646:
6521:
6412:
6167:
6046:
5963:
5922:
5883:
5832:
5773:
5728:
5629:
5527:
5449:
5393:
5337:
5297:
5260:
5224:
5208:
simultaneously (see Note at the end of the section.)
5054:
4872:
4665:
4583:
4500:
4417:
4398:. There is also a relative version of this fact; see
4367:
is a smooth complete curve, any rational function on
4270:
4248:. (More abstractly, this is the induced map from the
4152:
4088:
3937:
3829:
3710:
3639:
3497:
3417:
3306:
3255:
3121:
3043:
2957:
2882:
2851:, which is not an isomorphism and so the restriction
2738:
2706:
2565:
2473:
2420:
2351:
2285:
2252:
2118:
2045:
1966:
1930:
1788:
1690:
1639:
1603:
1552:
1353:
is an isomorphism of affine varieties if and only if
1310:
1268:
1241:
1207:
1107:
1024:
982:
929:
900:
833:
781:
702:
530:
498:
362:
324:
263:
216:
181:
148:
119:
5510:{\displaystyle \{(x:y:z)\in X\mid x\neq 0,z\neq 0\}}
5045:. Hence, going back to the homogeneous coordinates,
4956:{\displaystyle f|_{U}(x)=(g_{1}(x),\dots ,g_{m}(x))}
3479:
and ∞ = (1 : 0). There is an automorphism σ of
247:{\displaystyle \mathbb {A} ^{n}\to \mathbb {A} ^{m}}
7109:is algebraically closed, then each geometric fiber
6968:is positive, comparing the leading terms, one has:
4636:{\displaystyle f:U\to \mathbf {P} ^{m}-\{y_{0}=0\}}
4019:is a morphism between varieties, then the image of
1676:if and only if there are some homogeneous elements
7663:
7560:
7410:
7276:
7237:
7086:
7038:
6956:
6905:
6796:
6759:
6629:
6470:
6219:
6135:
5996:
5947:
5908:
5853:
5806:
5753:
5656:
5615:
5509:
5435:
5379:
5319:
5273:
5237:
5147:
4955:
4845:
4635:
4527:
4436:
4300:
4229:
4131:
4007:
3907:
3815:
3688:
3542:
3459:
3352:
3292:
3241:
3107:
2999:
2927:
2843:
2724:
2665:
2551:
2459:
2391:
2337:
2271:
2183:
2088:
2028:
1949:
1853:
1712:
1652:
1625:
1565:
1341:
1296:
1254:
1227:
1190:
1091:
1007:
965:
915:
886:
811:
764:
614:is regular as a function on some affine charts of
536:
516:
429:
337:
307:
246:
199:
163:
134:
6323:such that the restriction of the structure sheaf
4404:A morphism between algebraic varieties that is a
3689:{\displaystyle p:X\times Y\to X,\,(x,y)\mapsto x}
3353:{\displaystyle f={g \over x^{n}}={h \over y^{m}}}
765:{\displaystyle f^{\#}:k\to k,\,g\mapsto g\circ f}
6062:
5211:In fact, the above description is valid for any
4448:is bijective birational and the target space of
3908:{\displaystyle (f\otimes 1)(x,y)=f(p(x,y))=f(x)}
2345:is a morphism; it is bijective with the inverse
488:, and hence satisfies the defining equations of
6220:{\displaystyle X_{n}=\{x\in X\mid e(x)\geq n\}}
6029:be a morphism of algebraic varieties. For each
7201:It is not clear how to prove this, though. If
6471:{\displaystyle \chi (f^{*}F)=\deg(f)\chi (F).}
5616:{\displaystyle (x:y)=(xy:y^{2})=(xy:xz)=(y:z)}
4244:runs over all nonempty open affine subsets of
4062:, then there is a nonempty open affine subset
3614:is actually constant, we have to attribute to
3400:is determined by its coordinate ring and thus
887:{\displaystyle g\circ f=g(f_{1},\dots ,f_{m})}
50:. A morphism from an algebraic variety to the
5254:'s are in the homogeneous coordinate ring of
5218:, an open subvariety of a projective variety
1899:that are finitely generated algebras over an
8:
6957:{\displaystyle R^{q}f_{*}{\mathcal {O}}_{X}}
6214:
6181:
6127:
6065:
5861:and (b) for every irreducible closed subset
5504:
5450:
4630:
4611:
3454:
3448:
3102:
3090:
2178:
2136:
1364:is a closed subvariety of an affine variety
7771:
6365:is then also the rank of this free module.
6011:
5676:
1507:. Caution: the condition is for some pair (
1376:is the restriction of regular functions on
1357:is an isomorphism of the coordinate rings.
7820:
7808:
7783:
7568:since φ is an algebra homomorphism. Also,
674:where the morphisms are the regular maps.
7632:
7626:
7600:
7594:
7579:
7573:
7526:
7520:
7491:
7485:
7455:
7448:
7443:
7440:
7389:
7363:
7357:
7328:
7322:
7301:
7289:
7268:
7262:
7229:
7223:
7078:
7072:
7071:
7064:
7058:
6991:
6976:
6948:
6942:
6941:
6934:
6924:
6918:
6894:
6881:
6875:
6874:
6867:
6857:
6838:
6825:
6815:
6809:
6785:
6779:
6742:
6732:
6722:
6706:
6687:
6676:
6657:
6645:
6615:
6587:
6568:
6558:
6548:
6526:
6520:
6423:
6411:
6172:
6166:
6119:
6092:
6083:
6045:
5962:
5927:
5921:
5888:
5882:
5831:
5772:
5733:
5727:
5648:
5643:
5628:
5562:
5526:
5448:
5392:
5336:
5302:
5296:
5261:
5259:
5225:
5223:
5127:
5099:
5077:
5053:
4935:
4907:
4882:
4877:
4871:
4834:
4825:
4819:
4800:
4791:
4785:
4766:
4757:
4751:
4732:
4723:
4717:
4692:
4673:
4664:
4618:
4602:
4597:
4582:
4519:
4514:
4499:
4428:
4416:
4335:is a smooth complete curve (for example,
4269:
4264:.) Conversely, every inclusion of fields
4211:
4168:
4151:
4093:
4087:
3968:
3959:
3954:
3944:
3939:
3936:
3828:
3797:
3776:
3715:
3709:
3664:
3638:
3529:
3502:
3496:
3439:
3434:
3424:
3419:
3416:
3342:
3333:
3322:
3313:
3305:
3278:
3254:
3221:
3190:
3174:
3169:
3139:
3134:
3132:
3120:
3081:
3076:
3055:
3050:
3048:
3042:
2979:
2956:
2904:
2881:
2829:
2783:
2764:
2737:
2705:
2645:
2626:
2609:
2570:
2564:
2534:
2515:
2501:
2486:
2481:
2472:
2451:
2438:
2425:
2419:
2380:
2350:
2313:
2304:
2299:
2284:
2263:
2251:
2127:
2121:
2120:
2117:
2071:
2065:
2064:
2054:
2048:
2047:
2044:
2017:
2011:
2010:
1997:
1975:
1969:
1968:
1965:
1941:
1929:
1842:
1841:
1829:
1816:
1815:
1814:
1793:
1787:
1697:
1689:
1640:
1638:
1610:
1602:
1553:
1551:
1324:
1317:
1312:
1309:
1282:
1275:
1270:
1267:
1246:
1240:
1219:
1209:
1206:
1171:
1165:
1136:
1130:
1112:
1106:
1078:
1069:
1050:
1023:
987:
981:
928:
899:
875:
856:
832:
780:
746:
707:
701:
529:
497:
416:
407:
388:
361:
329:
323:
296:
277:
262:
238:
234:
233:
223:
219:
218:
215:
180:
155:
151:
150:
147:
126:
122:
121:
118:
4301:{\displaystyle k(Y)\hookrightarrow k(X)}
1463:if, in some open affine neighborhood of
78:maps are widely used as well; they are
7832:
7764:
7173:
7087:{\displaystyle f_{*}{\mathcal {O}}_{X}}
6085: an irreducible component of
5657:{\displaystyle f:X\to \mathbf {P} ^{1}}
4853:. Thus, by definition, the restriction
4528:{\displaystyle f:X\to \mathbf {P} ^{m}}
2235:variables and the regular functions on
7844:
6238:. For an algebraic approach where the
3543:{\displaystyle \sigma ^{\#}(f)=f(1/z)}
2680:Continuing the preceding example, let
1424:studied in differential geometry. The
1297:{\displaystyle {\phi ^{a}}^{\#}=\phi }
308:{\displaystyle f=(f_{1},\dots ,f_{m})}
6315:, there is some nonempty open subset
6288:. Then, by definition, the degree of
6242:plays a main role and the notion of "
4139:is injective. Thus, the dominant map
4046:of algebraic varieties is said to be
2928:{\displaystyle f(x,y)={1-y \over x}.}
2673:which is seen to be injective (since
1744:Comparison with a morphism of schemes
1546:) is the same as that of the closure
7:
6492:is a finite surjective morphism, if
5818:There exists a nonempty open subset
5807:{\displaystyle \dim Z\geq \dim W+r.}
5710:For every irreducible closed subset
5436:{\displaystyle (x:y:z)\mapsto (y:z)}
5380:{\displaystyle (x:y:z)\mapsto (x:y)}
4058:is a nonempty open affine subset of
3608:completeness of projective varieties
3000:{\displaystyle f(x,y)={x \over 1+y}}
2844:{\displaystyle k=k\hookrightarrow k}
2692:is the complement of the hyperplane
1662:Projective variety#Variety structure
1475:; i.e., there are regular functions
630:, if it is regular at all points of
4050:if it has dense image. For such an
3396:cannot be affine since if it were,
2222:Morphism of schemes § Examples
2122:
2066:
2049:
2012:
1970:
1843:
1817:
1228:{\displaystyle {\overline {y}}_{i}}
7895:Algebraic Geometry, A First Course
7580:
7456:
6688:
6584:
6523:
4094:
3716:
3503:
3451:
2571:
1942:
1318:
1283:
708:
25:
8031:The Arithmetic of Elliptic Curves
7797:Foundations of algebraic geometry
2460:{\displaystyle y^{2}=x^{3}+x^{2}}
681:between the coordinate rings: if
649:are affine varieties, then a map
62:, and the biregular maps are the
5997:{\displaystyle \dim Z=\dim W+r.}
5873:and every irreducible component
5718:and every irreducible component
5644:
4598:
4515:
3955:
3940:
3924:A morphism between varieties is
3435:
3420:
3170:
3135:
3077:
3051:
2482:
2300:
1577:and thus a rational function on
1008:{\displaystyle \phi ^{a}:X\to Y}
164:{\displaystyle \mathbb {A} ^{m}}
135:{\displaystyle \mathbb {A} ^{n}}
6259:Zariski's connectedness theorem
5274:{\displaystyle {\overline {X}}}
5238:{\displaystyle {\overline {X}}}
4487:Morphisms to a projective space
3696:is a morphism of varieties. If
3603:
3554:is regular at ∞ if and only if
2407:is an isomorphism of varieties.
2231:are exactly the polynomials in
1907:
1653:{\displaystyle {\overline {X}}}
1566:{\displaystyle {\overline {X}}}
1420:, and are algebraic analogs of
1342:{\displaystyle {f^{\#}}^{a}=f.}
894:is a polynomial in elements of
672:category of algebraic varieties
254:. Explicitly, it has the form:
7649:
7591:
7555:
7549:
7540:
7537:
7517:
7502:
7482:
7476:
7467:
7461:
7399:
7386:
7377:
7374:
7354:
7339:
7319:
7313:
7284:is a map to the affine space,
7190:affine variety#Structure sheaf
7126:Degree of a continuous mapping
7033:
7027:
7018:
7012:
7000:
6984:
6847:
6831:
6751:
6715:
6703:
6693:
6666:
6650:
6624:
6602:
6580:
6577:
6535:
6462:
6456:
6450:
6444:
6432:
6416:
6205:
6199:
6116:
6113:
6107:
6101:
6056:
6050:
5942:
5936:
5903:
5897:
5848:
5842:
5748:
5742:
5639:
5610:
5598:
5592:
5574:
5568:
5546:
5540:
5528:
5471:
5453:
5430:
5418:
5415:
5412:
5394:
5374:
5362:
5359:
5356:
5338:
5142:
5139:
5133:
5111:
5105:
5089:
5083:
5070:
5064:
5058:
5032:for some homogeneous elements
4950:
4947:
4941:
4919:
4913:
4900:
4894:
4888:
4878:
4840:
4778:
4772:
4704:
4698:
4666:
4593:
4554:-th homogeneous coordinate of
4510:
4437:{\displaystyle t\mapsto t^{p}}
4421:
4295:
4289:
4283:
4280:
4274:
4215:
4205:
4199:
4193:
4190:
4184:
4162:
4156:
4126:
4120:
4114:
4111:
4105:
4002:
3987:
3984:
3981:
3969:
3950:
3902:
3896:
3887:
3884:
3872:
3866:
3857:
3845:
3842:
3830:
3801:
3791:
3785:
3769:
3763:
3754:
3742:
3736:
3733:
3727:
3680:
3677:
3665:
3655:
3618:the value ∞ at some points of
3537:
3523:
3514:
3508:
3287:
3259:
3236:
3208:
3199:
3183:
3180:
3165:
3156:
3153:
3147:
3125:
3069:
3063:
2973:
2961:
2898:
2886:
2838:
2826:
2813:
2804:
2798:
2795:
2757:
2748:
2742:
2716:
2657:
2619:
2613:
2603:
2597:
2591:
2588:
2582:
2546:
2508:
2505:
2492:
2392:{\displaystyle g(x)=(x,x^{2})}
2386:
2367:
2361:
2355:
2329:
2326:
2314:
2295:
2169:
2163:
2154:
2148:
2081:
2075:
2023:
2006:
1985:
1979:
1848:
1838:
1822:
1805:
1768:, then each ring homomorphism
1707:
1694:
1620:
1607:
1589:for some homogeneous elements
1185:
1182:
1162:
1147:
1127:
1121:
1075:
1043:
1034:
1028:
999:
960:
954:
948:
945:
939:
910:
904:
881:
849:
806:
800:
791:
785:
750:
740:
734:
728:
725:
719:
508:
413:
381:
372:
366:
302:
270:
229:
200:{\displaystyle f\colon X\to Y}
191:
1:
6797:{\displaystyle L^{\otimes n}}
6236:Noether's normalization lemma
5854:{\displaystyle U\subset f(X)}
4483:(complex-analytic function).
4132:{\displaystyle f^{\#}:k\to k}
1524:
1385:
7835:, Ch. I, § 8. Theorems 2, 3.
7638:
7606:
7532:
7497:
7394:
7369:
7334:
7149:– The algebraic analogue of
6964:has positive codimension if
5623:) and so defines a morphism
5266:
5245:; the difference being that
5230:
4975:'s are regular functions on
4371:may be viewed as a morphism
3625:For any algebraic varieties
2100:corresponding to the points
1950:{\displaystyle \phi =f^{\#}}
1924:is a morphism, then writing
1887:are affine varieties; i.e.,
1702:
1664:.) Then a rational function
1645:
1615:
1558:
1396:In the particular case that
1214:
1177:
1142:
966:{\displaystyle \phi :k\to k}
819:are the coordinate rings of
602:and the restricted function
570:if there is a neighbourhood
460:define the same function on
82:that are defined locally by
7157:Resolution of singularities
6264:Degree of a finite morphism
4659:through the identification
4542:to a projective space. Let
4469:complex algebraic varieties
3471: : 1) with the points
3467:by identifying the points (
2700:is affine. The restriction
827:; it is well-defined since
8090:
7995:Basic Algebraic Geometry 1
7823:, Ch. I, Proposition 6.8..
7257:, then, a priori thinking
7123:
7117:) consists exactly of deg(
6256:
5166:and by continuity for all
4396:Hartogs' extension theorem
3487:is a rational function on
2939:is a rational function on
2239:are exactly the constants.
2219:
1901:algebraically closed field
26:
8008:10.1007/978-3-642-37956-7
7731:is a regular function on
7277:{\displaystyle \phi ^{a}}
7238:{\displaystyle \phi ^{a}}
6913:and since the support of
6244:universally catenary ring
6154:; i.e., for each integer
5948:{\displaystyle f^{-1}(U)}
5909:{\displaystyle f^{-1}(W)}
5754:{\displaystyle f^{-1}(W)}
5443:agree on the open subset
5204:'s that do not vanish at
4444:.) On the other hand, if
4308:is induced by a dominant
3029:is a regular function on
2227:The regular functions on
1426:ring of regular functions
1388:below for more examples.
492:. That is, a regular map
7750:integrally closed domain
5320:{\displaystyle y^{2}=xz}
5213:quasi-projective variety
3392:. (This also shows that
2725:{\displaystyle f:U\to X}
2688: − {1}. Since
1536:quasi-projective variety
517:{\displaystyle f:X\to Y}
207:is the restriction of a
86:instead of polynomials.
6804:of a line bundle, then
6514:Leray spectral sequence
6483:Riemann–Hurwitz formula
5672:The important fact is:
4343:is a rational map from
2951:can also be written as
2272:{\displaystyle y=x^{2}}
2198:of affine schemes over
1372:is the inclusion, then
452:(note: two polynomials
8074:Functions and mappings
7951:, old version v. 5.xx.
7811:, Ch. I, Theorem 4.4..
7665:
7562:
7412:
7278:
7239:
7088:
7040:
6958:
6907:
6798:
6761:
6692:
6631:
6472:
6221:
6137:
6121: containing
5998:
5949:
5910:
5855:
5808:
5755:
5658:
5617:
5511:
5437:
5381:
5321:
5275:
5239:
5149:
4957:
4847:
4637:
4529:
4467:A regular map between
4462:Zariski's main theorem
4438:
4363:. In particular, when
4347:to a projective space
4302:
4231:
4133:
4009:
3909:
3817:
3690:
3544:
3461:
3354:
3300:. Thus, we can write:
3294:
3249:. Similarly, it is in
3243:
3109:
3001:
2929:
2861:is not an isomorphism.
2845:
2726:
2667:
2553:
2461:
2393:
2339:
2273:
2209:For more details, see
2191:. This is immediate.)
2185:
2090:
2030:
1951:
1855:
1779:determines a morphism
1714:
1684:of the same degree in
1672:is regular at a point
1654:
1627:
1567:
1459:is regular at a point
1343:
1298:
1256:
1229:
1192:
1093:
1092:{\displaystyle k=k/J,}
1009:
967:
917:
888:
813:
766:
547:More generally, a map
538:
518:
431:
430:{\displaystyle k=k/I,}
339:
309:
248:
201:
175:), then a regular map
165:
136:
46:. It is also called a
7666:
7563:
7413:
7279:
7240:
7151:local diffeomorphisms
7089:
7041:
6959:
6908:
6799:
6762:
6672:
6632:
6473:
6222:
6138:
5999:
5950:
5911:
5856:
5809:
5756:
5659:
5618:
5512:
5438:
5382:
5322:
5276:
5240:
5150:
4958:
4848:
4646:is a morphism, where
4638:
4538:be a morphism from a
4530:
4439:
4303:
4232:
4134:
4010:
3910:
3818:
3691:
3562:) is regular at zero.
3545:
3462:
3355:
3295:
3244:
3242:{\displaystyle k=k=k}
3110:
3002:
2930:
2846:
2727:
2668:
2554:
2462:
2394:
2340:
2274:
2186:
2091:
2031:
1952:
1856:
1715:
1655:
1628:
1568:
1515:) not for all pairs (
1344:
1299:
1257:
1255:{\displaystyle y_{i}}
1230:
1193:
1094:
1010:
968:
918:
889:
814:
767:
679:algebra homomorphisms
539:
519:
432:
340:
338:{\displaystyle f_{i}}
310:
249:
202:
166:
137:
38:, a morphism between
8026:Silverman, Joseph H.
7990:Shafarevich, Igor R.
7799:, Proposition 6.5.7.
7572:
7439:
7288:
7261:
7222:
7162:contraction morphism
7057:
6975:
6917:
6808:
6778:
6644:
6519:
6508:a coherent sheaf on
6410:
6401:Euler characteristic
6399:, writing χ for the
6165:
6152:upper-semicontinuous
6044:
5961:
5920:
5881:
5830:
5771:
5726:
5668:Fibers of a morphism
5627:
5525:
5447:
5391:
5335:
5295:
5258:
5222:
5183:'s do not vanish at
5052:
5007:. Then we can write
4983:is projective, each
4870:
4663:
4581:
4498:
4481:holomorphic function
4415:
4268:
4150:
4086:
4027:of its closure (cf.
3935:
3827:
3708:
3637:
3495:
3415:
3304:
3253:
3119:
3041:
2955:
2880:
2868:be the affine curve
2736:
2704:
2563:
2471:
2418:
2414:be the affine curve
2403:is also a morphism,
2349:
2283:
2250:
2246:be the affine curve
2116:
2043:
1964:
1928:
1786:
1688:
1637:
1601:
1550:
1308:
1266:
1239:
1205:
1105:
1022:
980:
927:
898:
831:
779:
700:
578:and a neighbourhood
528:
496:
360:
322:
261:
214:
179:
146:
117:
91:locally ringed space
68:projective varieties
8064:Algebraic varieties
7867:Intersection Theory
7774:, p. 25, Def..
7253:is a polynomial in
6015: —
5680: —
4562:) is nonzero; say,
4460:is biregular. (cf.
3610:). That is, unless
3372:. But this implies
3368:are polynomials in
1874:morphism of schemes
1736:does not vanish at
1503:does not vanish at
1471:that is regular at
1438:Liouville's theorem
812:{\displaystyle k,k}
40:algebraic varieties
8069:Types of functions
7949:Algebraic geometry
7923:Algebraic Geometry
7847:, Example 18.3.9..
7661:
7558:
7408:
7274:
7235:
7137:Algebraic function
7084:
7036:
6954:
6903:
6794:
6774:is a tensor power
6770:In particular, if
6757:
6627:
6468:
6217:
6133:
6013:
5994:
5945:
5906:
5851:
5804:
5751:
5678:
5654:
5613:
5507:
5433:
5377:
5317:
5271:
5235:
5145:
4953:
4843:
4633:
4540:projective variety
4525:
4434:
4410:Frobenius morphism
4390:(in particular, a
4298:
4227:
4176:
4129:
4005:
3905:
3813:
3686:
3540:
3457:
3350:
3290:
3239:
3105:
2997:
2925:
2841:
2722:
2663:
2549:
2457:
2389:
2335:
2269:
2181:
2086:
2026:
1957:, we need to show
1947:
1851:
1710:
1650:
1623:
1563:
1447:A scalar function
1434:projective variety
1339:
1294:
1252:
1235:are the images of
1225:
1188:
1089:
1018:given by: writing
1005:
963:
913:
884:
809:
762:
565:regular at a point
534:
514:
427:
335:
305:
244:
197:
161:
132:
84:rational fractions
70:– the concepts of
36:algebraic geometry
8045:978-0-387-09494-6
8017:978-0-387-97716-4
7918:Hartshorne, Robin
7909:978-1-4757-2189-8
7881:978-0-387-98549-7
7707:and some nonzero
7641:
7609:
7535:
7500:
7397:
7372:
7337:
7094:is the degree of
6122:
6086:
5702: − dim
5269:
5233:
4355:is a regular map
4169:
4029:constructible set
4023:contains an open
3633:, the projection
3348:
3328:
3293:{\displaystyle k}
2995:
2920:
1713:{\displaystyle k}
1705:
1648:
1626:{\displaystyle k}
1618:
1561:
1469:rational function
1418:regular functions
1404:the regular maps
1392:Regular functions
1217:
1180:
1145:
923:. Conversely, if
916:{\displaystyle k}
537:{\displaystyle Y}
80:partial functions
54:is also called a
18:Dominant morphism
16:(Redirected from
8081:
8049:
8034:(2nd ed.).
8021:
8000:Springer Science
7985:
7941:
7913:
7885:
7872:Springer Science
7848:
7842:
7836:
7830:
7824:
7818:
7812:
7806:
7800:
7793:
7787:
7781:
7775:
7772:Shafarevich 2013
7769:
7753:
7746:
7740:
7677:
7671:
7670:
7668:
7667:
7662:
7642:
7637:
7636:
7627:
7610:
7605:
7604:
7595:
7587:
7586:
7567:
7565:
7564:
7559:
7536:
7531:
7530:
7521:
7501:
7496:
7495:
7486:
7460:
7459:
7454:
7453:
7452:
7433:
7427:
7417:
7415:
7414:
7409:
7398:
7390:
7373:
7368:
7367:
7358:
7338:
7333:
7332:
7323:
7306:
7305:
7283:
7281:
7280:
7275:
7273:
7272:
7244:
7242:
7241:
7236:
7234:
7233:
7216:
7210:
7199:
7193:
7178:
7093:
7091:
7090:
7085:
7083:
7082:
7077:
7076:
7069:
7068:
7045:
7043:
7042:
7037:
6996:
6995:
6963:
6961:
6960:
6955:
6953:
6952:
6947:
6946:
6939:
6938:
6929:
6928:
6912:
6910:
6909:
6904:
6902:
6901:
6886:
6885:
6880:
6879:
6872:
6871:
6862:
6861:
6843:
6842:
6830:
6829:
6820:
6819:
6803:
6801:
6800:
6795:
6793:
6792:
6766:
6764:
6763:
6758:
6747:
6746:
6737:
6736:
6727:
6726:
6711:
6710:
6691:
6686:
6662:
6661:
6636:
6634:
6633:
6628:
6620:
6619:
6598:
6597:
6573:
6572:
6563:
6562:
6553:
6552:
6531:
6530:
6512:, then from the
6477:
6475:
6474:
6469:
6428:
6427:
6361:. The degree of
6342:
6313:generic freeness
6240:generic freeness
6226:
6224:
6223:
6218:
6177:
6176:
6142:
6140:
6139:
6134:
6123:
6120:
6100:
6099:
6087:
6084:
6016:
6003:
6001:
6000:
5995:
5954:
5952:
5951:
5946:
5935:
5934:
5915:
5913:
5912:
5907:
5896:
5895:
5860:
5858:
5857:
5852:
5813:
5811:
5810:
5805:
5760:
5758:
5757:
5752:
5741:
5740:
5681:
5663:
5661:
5660:
5655:
5653:
5652:
5647:
5622:
5620:
5619:
5614:
5567:
5566:
5516:
5514:
5513:
5508:
5442:
5440:
5439:
5434:
5386:
5384:
5383:
5378:
5331:. Then two maps
5326:
5324:
5323:
5318:
5307:
5306:
5280:
5278:
5277:
5272:
5270:
5262:
5244:
5242:
5241:
5236:
5234:
5226:
5154:
5152:
5151:
5146:
5132:
5131:
5104:
5103:
5082:
5081:
4962:
4960:
4959:
4954:
4940:
4939:
4912:
4911:
4887:
4886:
4881:
4852:
4850:
4849:
4844:
4839:
4838:
4829:
4824:
4823:
4805:
4804:
4795:
4790:
4789:
4771:
4770:
4761:
4756:
4755:
4737:
4736:
4727:
4722:
4721:
4697:
4696:
4678:
4677:
4642:
4640:
4639:
4634:
4623:
4622:
4607:
4606:
4601:
4534:
4532:
4531:
4526:
4524:
4523:
4518:
4443:
4441:
4440:
4435:
4433:
4432:
4307:
4305:
4304:
4299:
4236:
4234:
4233:
4228:
4177:
4138:
4136:
4135:
4130:
4098:
4097:
4014:
4012:
4011:
4006:
3964:
3963:
3958:
3949:
3948:
3943:
3914:
3912:
3911:
3906:
3822:
3820:
3819:
3814:
3781:
3780:
3720:
3719:
3695:
3693:
3692:
3687:
3586:, the functions
3549:
3547:
3546:
3541:
3533:
3507:
3506:
3466:
3464:
3463:
3458:
3444:
3443:
3438:
3429:
3428:
3423:
3376:is divisible by
3359:
3357:
3356:
3351:
3349:
3347:
3346:
3334:
3329:
3327:
3326:
3314:
3299:
3297:
3296:
3291:
3286:
3285:
3248:
3246:
3245:
3240:
3229:
3228:
3198:
3197:
3179:
3178:
3173:
3146:
3145:
3144:
3143:
3138:
3114:
3112:
3111:
3106:
3086:
3085:
3080:
3062:
3061:
3060:
3059:
3054:
3020:
3006:
3004:
3003:
2998:
2996:
2994:
2980:
2934:
2932:
2931:
2926:
2921:
2916:
2905:
2850:
2848:
2847:
2842:
2837:
2836:
2788:
2787:
2769:
2768:
2731:
2729:
2728:
2723:
2672:
2670:
2669:
2664:
2650:
2649:
2631:
2630:
2575:
2574:
2558:
2556:
2555:
2550:
2539:
2538:
2520:
2519:
2491:
2490:
2485:
2466:
2464:
2463:
2458:
2456:
2455:
2443:
2442:
2430:
2429:
2398:
2396:
2395:
2390:
2385:
2384:
2344:
2342:
2341:
2336:
2309:
2308:
2303:
2278:
2276:
2275:
2270:
2268:
2267:
2196:full subcategory
2190:
2188:
2187:
2182:
2132:
2131:
2126:
2125:
2095:
2093:
2092:
2087:
2085:
2084:
2070:
2069:
2059:
2058:
2053:
2052:
2035:
2033:
2032:
2027:
2022:
2021:
2016:
2015:
2005:
2004:
1989:
1988:
1974:
1973:
1956:
1954:
1953:
1948:
1946:
1945:
1923:
1897:integral domains
1860:
1858:
1857:
1852:
1847:
1846:
1837:
1836:
1821:
1820:
1798:
1797:
1778:
1719:
1717:
1716:
1711:
1706:
1698:
1659:
1657:
1656:
1651:
1649:
1641:
1632:
1630:
1629:
1624:
1619:
1611:
1572:
1570:
1569:
1564:
1562:
1554:
1442:complex analysis
1422:smooth functions
1360:For example, if
1348:
1346:
1345:
1340:
1329:
1328:
1323:
1322:
1321:
1303:
1301:
1300:
1295:
1287:
1286:
1281:
1280:
1279:
1261:
1259:
1258:
1253:
1251:
1250:
1234:
1232:
1231:
1226:
1224:
1223:
1218:
1210:
1197:
1195:
1194:
1189:
1181:
1176:
1175:
1166:
1146:
1141:
1140:
1131:
1117:
1116:
1098:
1096:
1095:
1090:
1082:
1074:
1073:
1055:
1054:
1014:
1012:
1011:
1006:
992:
991:
972:
970:
969:
964:
922:
920:
919:
914:
893:
891:
890:
885:
880:
879:
861:
860:
818:
816:
815:
810:
771:
769:
768:
763:
712:
711:
543:
541:
540:
535:
523:
521:
520:
515:
436:
434:
433:
428:
420:
412:
411:
393:
392:
344:
342:
341:
336:
334:
333:
314:
312:
311:
306:
301:
300:
282:
281:
253:
251:
250:
245:
243:
242:
237:
228:
227:
222:
206:
204:
203:
198:
173:affine varieties
170:
168:
167:
162:
160:
159:
154:
141:
139:
138:
133:
131:
130:
125:
56:regular function
21:
8089:
8088:
8084:
8083:
8082:
8080:
8079:
8078:
8054:
8053:
8052:
8046:
8036:Springer Verlag
8024:
8018:
7988:
7982:
7964:Springer-Verlag
7954:
7938:
7928:Springer-Verlag
7916:
7910:
7900:Springer Verlag
7888:
7882:
7862:Fulton, William
7860:
7856:
7851:
7843:
7839:
7831:
7827:
7821:Hartshorne 1997
7819:
7815:
7809:Hartshorne 1997
7807:
7803:
7794:
7790:
7784:Hartshorne 1997
7782:
7778:
7770:
7766:
7762:
7757:
7756:
7747:
7743:
7678:
7674:
7628:
7596:
7575:
7570:
7569:
7522:
7487:
7444:
7442:
7437:
7436:
7434:
7430:
7359:
7324:
7297:
7286:
7285:
7264:
7259:
7258:
7225:
7220:
7219:
7217:
7213:
7200:
7196:
7179:
7175:
7170:
7147:Étale morphisms
7142:Smooth morphism
7133:
7128:
7070:
7060:
7055:
7054:
6987:
6973:
6972:
6940:
6930:
6920:
6915:
6914:
6890:
6873:
6863:
6853:
6834:
6821:
6811:
6806:
6805:
6781:
6776:
6775:
6738:
6728:
6718:
6702:
6653:
6642:
6641:
6611:
6583:
6564:
6554:
6544:
6522:
6517:
6516:
6488:In general, if
6419:
6408:
6407:
6388:, then for any
6358:
6352:
6333:
6331:
6266:
6261:
6232:
6168:
6163:
6162:
6088:
6042:
6041:
6014:
6008:
5959:
5958:
5923:
5918:
5917:
5884:
5879:
5878:
5828:
5827:
5769:
5768:
5729:
5724:
5723:
5679:
5670:
5642:
5625:
5624:
5558:
5523:
5522:
5445:
5444:
5389:
5388:
5333:
5332:
5298:
5293:
5292:
5256:
5255:
5253:
5220:
5219:
5203:
5182:
5174:as long as the
5123:
5095:
5073:
5050:
5049:
5040:
5031:
5024:
5015:
5006:
4991:
4974:
4931:
4903:
4876:
4868:
4867:
4862:
4830:
4815:
4796:
4781:
4762:
4747:
4728:
4713:
4688:
4669:
4661:
4660:
4654:
4614:
4596:
4579:
4578:
4513:
4496:
4495:
4489:
4473:holomorphic map
4424:
4413:
4412:
4266:
4265:
4148:
4147:
4089:
4084:
4083:
3953:
3938:
3933:
3932:
3922:
3825:
3824:
3772:
3711:
3706:
3705:
3635:
3634:
3596:projective line
3581:algebraic curve
3498:
3493:
3492:
3433:
3418:
3413:
3412:
3338:
3318:
3302:
3301:
3274:
3251:
3250:
3217:
3186:
3168:
3133:
3128:
3117:
3116:
3075:
3049:
3044:
3039:
3038:
3011:
2984:
2953:
2952:
2906:
2878:
2877:
2860:
2825:
2779:
2760:
2734:
2733:
2702:
2701:
2677:is surjective).
2641:
2622:
2566:
2561:
2560:
2530:
2511:
2480:
2469:
2468:
2447:
2434:
2421:
2416:
2415:
2376:
2347:
2346:
2298:
2281:
2280:
2259:
2248:
2247:
2224:
2218:
2119:
2114:
2113:
2063:
2046:
2041:
2040:
2009:
1993:
1967:
1962:
1961:
1937:
1926:
1925:
1911:
1825:
1789:
1784:
1783:
1769:
1746:
1686:
1685:
1635:
1634:
1599:
1598:
1581:is of the form
1548:
1547:
1430:coordinate ring
1394:
1349:In particular,
1313:
1311:
1306:
1305:
1271:
1269:
1264:
1263:
1242:
1237:
1236:
1208:
1203:
1202:
1167:
1132:
1108:
1103:
1102:
1065:
1046:
1020:
1019:
983:
978:
977:
925:
924:
896:
895:
871:
852:
829:
828:
777:
776:
703:
698:
697:
526:
525:
494:
493:
464:if and only if
403:
384:
358:
357:
347:coordinate ring
325:
320:
319:
292:
273:
259:
258:
232:
217:
212:
211:
177:
176:
149:
144:
143:
120:
115:
114:
99:
32:
29:Biregular graph
23:
22:
15:
12:
11:
5:
8087:
8085:
8077:
8076:
8071:
8066:
8056:
8055:
8051:
8050:
8044:
8022:
8016:
7986:
7980:
7972:10.1007/b62130
7956:Mumford, David
7952:
7942:
7936:
7914:
7908:
7886:
7880:
7857:
7855:
7852:
7850:
7849:
7837:
7825:
7813:
7801:
7788:
7786:, Ch. I, § 3..
7776:
7763:
7761:
7758:
7755:
7754:
7741:
7672:
7660:
7657:
7654:
7651:
7648:
7645:
7640:
7635:
7631:
7625:
7622:
7619:
7616:
7613:
7608:
7603:
7599:
7593:
7590:
7585:
7582:
7578:
7557:
7554:
7551:
7548:
7545:
7542:
7539:
7534:
7529:
7525:
7519:
7516:
7513:
7510:
7507:
7504:
7499:
7494:
7490:
7484:
7481:
7478:
7475:
7472:
7469:
7466:
7463:
7458:
7451:
7447:
7428:
7407:
7404:
7401:
7396:
7393:
7388:
7385:
7382:
7379:
7376:
7371:
7366:
7362:
7356:
7353:
7350:
7347:
7344:
7341:
7336:
7331:
7327:
7321:
7318:
7315:
7312:
7309:
7304:
7300:
7296:
7293:
7271:
7267:
7232:
7228:
7211:
7194:
7172:
7171:
7169:
7166:
7165:
7164:
7159:
7154:
7144:
7139:
7132:
7129:
7081:
7075:
7067:
7063:
7047:
7046:
7035:
7032:
7029:
7026:
7023:
7020:
7017:
7014:
7011:
7008:
7005:
7002:
6999:
6994:
6990:
6986:
6983:
6980:
6951:
6945:
6937:
6933:
6927:
6923:
6900:
6897:
6893:
6889:
6884:
6878:
6870:
6866:
6860:
6856:
6852:
6849:
6846:
6841:
6837:
6833:
6828:
6824:
6818:
6814:
6791:
6788:
6784:
6768:
6767:
6756:
6753:
6750:
6745:
6741:
6735:
6731:
6725:
6721:
6717:
6714:
6709:
6705:
6701:
6698:
6695:
6690:
6685:
6682:
6679:
6675:
6671:
6668:
6665:
6660:
6656:
6652:
6649:
6626:
6623:
6618:
6614:
6610:
6607:
6604:
6601:
6596:
6593:
6590:
6586:
6582:
6579:
6576:
6571:
6567:
6561:
6557:
6551:
6547:
6543:
6540:
6537:
6534:
6529:
6525:
6479:
6478:
6467:
6464:
6461:
6458:
6455:
6452:
6449:
6446:
6443:
6440:
6437:
6434:
6431:
6426:
6422:
6418:
6415:
6390:coherent sheaf
6354:
6348:
6327:
6265:
6262:
6228:
6227:
6216:
6213:
6210:
6207:
6204:
6201:
6198:
6195:
6192:
6189:
6186:
6183:
6180:
6175:
6171:
6144:
6143:
6132:
6129:
6126:
6118:
6115:
6112:
6109:
6106:
6103:
6098:
6095:
6091:
6082:
6079:
6076:
6073:
6070:
6067:
6064:
6061:
6058:
6055:
6052:
6049:
6009:
6007:
6006:
6005:
6004:
5993:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5969:
5966:
5944:
5941:
5938:
5933:
5930:
5926:
5905:
5902:
5899:
5894:
5891:
5887:
5850:
5847:
5844:
5841:
5838:
5835:
5826:such that (a)
5816:
5815:
5814:
5803:
5800:
5797:
5794:
5791:
5788:
5785:
5782:
5779:
5776:
5750:
5747:
5744:
5739:
5736:
5732:
5674:
5669:
5666:
5651:
5646:
5641:
5638:
5635:
5632:
5612:
5609:
5606:
5603:
5600:
5597:
5594:
5591:
5588:
5585:
5582:
5579:
5576:
5573:
5570:
5565:
5561:
5557:
5554:
5551:
5548:
5545:
5542:
5539:
5536:
5533:
5530:
5506:
5503:
5500:
5497:
5494:
5491:
5488:
5485:
5482:
5479:
5476:
5473:
5470:
5467:
5464:
5461:
5458:
5455:
5452:
5432:
5429:
5426:
5423:
5420:
5417:
5414:
5411:
5408:
5405:
5402:
5399:
5396:
5376:
5373:
5370:
5367:
5364:
5361:
5358:
5355:
5352:
5349:
5346:
5343:
5340:
5316:
5313:
5310:
5305:
5301:
5268:
5265:
5249:
5232:
5229:
5199:
5178:
5156:
5155:
5144:
5141:
5138:
5135:
5130:
5126:
5122:
5119:
5116:
5113:
5110:
5107:
5102:
5098:
5094:
5091:
5088:
5085:
5080:
5076:
5072:
5069:
5066:
5063:
5060:
5057:
5036:
5029:
5020:
5011:
5004:
4987:
4970:
4964:
4963:
4952:
4949:
4946:
4943:
4938:
4934:
4930:
4927:
4924:
4921:
4918:
4915:
4910:
4906:
4902:
4899:
4896:
4893:
4890:
4885:
4880:
4875:
4858:
4842:
4837:
4833:
4828:
4822:
4818:
4814:
4811:
4808:
4803:
4799:
4794:
4788:
4784:
4780:
4777:
4774:
4769:
4765:
4760:
4754:
4750:
4746:
4743:
4740:
4735:
4731:
4726:
4720:
4716:
4712:
4709:
4706:
4703:
4700:
4695:
4691:
4687:
4684:
4681:
4676:
4672:
4668:
4650:
4644:
4643:
4632:
4629:
4626:
4621:
4617:
4613:
4610:
4605:
4600:
4595:
4592:
4589:
4586:
4546:be a point of
4536:
4535:
4522:
4517:
4512:
4509:
4506:
4503:
4488:
4485:
4454:normal variety
4431:
4427:
4423:
4420:
4392:smooth variety
4388:normal variety
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4276:
4273:
4238:
4237:
4226:
4223:
4220:
4217:
4214:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4180:
4175:
4172:
4167:
4164:
4161:
4158:
4155:
4128:
4125:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4096:
4092:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
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3980:
3977:
3974:
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3957:
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3942:
3921:
3918:
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3916:
3904:
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3898:
3895:
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3809:
3806:
3803:
3800:
3796:
3793:
3790:
3787:
3784:
3779:
3775:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3726:
3723:
3718:
3714:
3685:
3682:
3679:
3676:
3673:
3670:
3667:
3663:
3660:
3657:
3654:
3651:
3648:
3645:
3642:
3623:
3567:function field
3563:
3539:
3536:
3532:
3528:
3525:
3522:
3519:
3516:
3513:
3510:
3505:
3501:
3456:
3453:
3450:
3447:
3442:
3437:
3432:
3427:
3422:
3409:
3345:
3341:
3337:
3332:
3325:
3321:
3317:
3312:
3309:
3289:
3284:
3281:
3277:
3273:
3270:
3267:
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3261:
3258:
3238:
3235:
3232:
3227:
3224:
3220:
3216:
3213:
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3201:
3196:
3193:
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3182:
3177:
3172:
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3137:
3131:
3127:
3124:
3104:
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3095:
3092:
3089:
3084:
3079:
3074:
3071:
3068:
3065:
3058:
3053:
3047:
3037:is regular on
3008:
2993:
2990:
2987:
2983:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2924:
2919:
2915:
2912:
2909:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2862:
2856:
2840:
2835:
2832:
2828:
2824:
2821:
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2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2786:
2782:
2778:
2775:
2772:
2767:
2763:
2759:
2756:
2753:
2750:
2747:
2744:
2741:
2721:
2718:
2715:
2712:
2709:
2678:
2662:
2659:
2656:
2653:
2648:
2644:
2640:
2637:
2634:
2629:
2625:
2621:
2618:
2615:
2612:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2573:
2569:
2548:
2545:
2542:
2537:
2533:
2529:
2526:
2523:
2518:
2514:
2510:
2507:
2504:
2500:
2497:
2494:
2489:
2484:
2479:
2476:
2454:
2450:
2446:
2441:
2437:
2433:
2428:
2424:
2408:
2388:
2383:
2379:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2312:
2307:
2302:
2297:
2294:
2291:
2288:
2266:
2262:
2258:
2255:
2240:
2217:
2214:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2130:
2124:
2098:maximal ideals
2083:
2080:
2077:
2074:
2068:
2062:
2057:
2051:
2037:
2036:
2025:
2020:
2014:
2008:
2003:
2000:
1996:
1992:
1987:
1984:
1981:
1978:
1972:
1944:
1940:
1936:
1933:
1864:by taking the
1862:
1861:
1850:
1845:
1840:
1835:
1832:
1828:
1824:
1819:
1813:
1810:
1807:
1804:
1801:
1796:
1792:
1766:affine schemes
1745:
1742:
1709:
1704:
1701:
1696:
1693:
1647:
1644:
1622:
1617:
1614:
1609:
1606:
1560:
1557:
1393:
1390:
1338:
1335:
1332:
1327:
1320:
1316:
1293:
1290:
1285:
1278:
1274:
1249:
1245:
1222:
1216:
1213:
1199:
1198:
1187:
1184:
1179:
1174:
1170:
1164:
1161:
1158:
1155:
1152:
1149:
1144:
1139:
1135:
1129:
1126:
1123:
1120:
1115:
1111:
1088:
1085:
1081:
1077:
1072:
1068:
1064:
1061:
1058:
1053:
1049:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1016:
1015:
1004:
1001:
998:
995:
990:
986:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
912:
909:
906:
903:
883:
878:
874:
870:
867:
864:
859:
855:
851:
848:
845:
842:
839:
836:
808:
805:
802:
799:
796:
793:
790:
787:
784:
773:
772:
761:
758:
755:
752:
749:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
710:
706:
668:
667:
533:
513:
510:
507:
504:
501:
438:
437:
426:
423:
419:
415:
410:
406:
402:
399:
396:
391:
387:
383:
380:
377:
374:
371:
368:
365:
332:
328:
316:
315:
304:
299:
295:
291:
288:
285:
280:
276:
272:
269:
266:
241:
236:
231:
226:
221:
209:polynomial map
196:
193:
190:
187:
184:
158:
153:
129:
124:
98:
95:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8086:
8075:
8072:
8070:
8067:
8065:
8062:
8061:
8059:
8047:
8041:
8037:
8033:
8032:
8027:
8023:
8019:
8013:
8009:
8005:
8001:
7997:
7996:
7991:
7987:
7983:
7977:
7973:
7969:
7965:
7961:
7957:
7953:
7950:
7946:
7943:
7939:
7937:0-387-90244-9
7933:
7929:
7925:
7924:
7919:
7915:
7911:
7905:
7901:
7897:
7896:
7891:
7887:
7883:
7877:
7873:
7869:
7868:
7863:
7859:
7858:
7853:
7846:
7841:
7838:
7834:
7829:
7826:
7822:
7817:
7814:
7810:
7805:
7802:
7798:
7792:
7789:
7785:
7780:
7777:
7773:
7768:
7765:
7759:
7751:
7745:
7742:
7738:
7734:
7730:
7726:
7722:
7718:
7714:
7710:
7706:
7702:
7698:
7694:
7690:
7686:
7682:
7676:
7673:
7658:
7655:
7652:
7646:
7643:
7633:
7629:
7623:
7620:
7617:
7614:
7611:
7601:
7597:
7588:
7583:
7576:
7552:
7546:
7543:
7527:
7523:
7514:
7511:
7508:
7505:
7492:
7488:
7479:
7473:
7470:
7464:
7449:
7445:
7432:
7429:
7425:
7421:
7405:
7402:
7391:
7383:
7380:
7364:
7360:
7351:
7348:
7345:
7342:
7329:
7325:
7316:
7310:
7307:
7302:
7298:
7294:
7291:
7269:
7265:
7256:
7252:
7248:
7230:
7226:
7218:The image of
7215:
7212:
7208:
7204:
7198:
7195:
7191:
7187:
7183:
7177:
7174:
7167:
7163:
7160:
7158:
7155:
7152:
7148:
7145:
7143:
7140:
7138:
7135:
7134:
7130:
7127:
7122:
7120:
7116:
7112:
7108:
7105:is étale and
7104:
7099:
7097:
7079:
7065:
7061:
7052:
7030:
7024:
7021:
7015:
7009:
7006:
7003:
6997:
6992:
6988:
6981:
6978:
6971:
6970:
6969:
6967:
6949:
6935:
6931:
6925:
6921:
6898:
6895:
6891:
6887:
6882:
6868:
6864:
6858:
6854:
6850:
6844:
6839:
6835:
6826:
6822:
6816:
6812:
6789:
6786:
6782:
6773:
6754:
6748:
6743:
6739:
6733:
6729:
6723:
6719:
6712:
6707:
6699:
6696:
6683:
6680:
6677:
6673:
6669:
6663:
6658:
6654:
6647:
6640:
6639:
6638:
6621:
6616:
6612:
6608:
6605:
6599:
6594:
6591:
6588:
6574:
6569:
6565:
6559:
6555:
6549:
6545:
6541:
6538:
6532:
6527:
6515:
6511:
6507:
6503:
6499:
6495:
6491:
6486:
6484:
6465:
6459:
6453:
6447:
6441:
6438:
6435:
6429:
6424:
6420:
6413:
6406:
6405:
6404:
6402:
6398:
6394:
6391:
6387:
6383:
6379:
6375:
6371:
6366:
6364:
6360:
6357:
6351:
6347:
6340:
6336:
6330:
6326:
6322:
6318:
6314:
6310:
6306:
6303:
6299:
6295:
6291:
6287:
6283:
6279:
6275:
6271:
6263:
6260:
6255:
6253:
6249:
6245:
6241:
6237:
6231:
6211:
6208:
6202:
6196:
6193:
6190:
6187:
6184:
6178:
6173:
6169:
6161:
6160:
6159:
6157:
6153:
6149:
6130:
6124:
6110:
6104:
6096:
6093:
6089:
6080:
6077:
6074:
6071:
6068:
6059:
6053:
6047:
6040:
6039:
6038:
6036:
6032:
6028:
6024:
6020:
5991:
5988:
5985:
5982:
5979:
5976:
5973:
5970:
5967:
5964:
5957:
5956:
5939:
5931:
5928:
5924:
5916:intersecting
5900:
5892:
5889:
5885:
5876:
5872:
5869:intersecting
5868:
5864:
5845:
5839:
5836:
5833:
5825:
5821:
5817:
5801:
5798:
5795:
5792:
5789:
5786:
5783:
5780:
5777:
5774:
5767:
5766:
5764:
5745:
5737:
5734:
5730:
5721:
5717:
5713:
5709:
5708:
5707:
5705:
5701:
5697:
5693:
5689:
5685:
5673:
5667:
5665:
5649:
5636:
5633:
5630:
5607:
5604:
5601:
5595:
5589:
5586:
5583:
5580:
5577:
5571:
5563:
5559:
5555:
5552:
5549:
5543:
5537:
5534:
5531:
5520:
5501:
5498:
5495:
5492:
5489:
5486:
5483:
5480:
5477:
5474:
5468:
5465:
5462:
5459:
5456:
5427:
5424:
5421:
5409:
5406:
5403:
5400:
5397:
5371:
5368:
5365:
5353:
5350:
5347:
5344:
5341:
5330:
5314:
5311:
5308:
5303:
5299:
5291:be the conic
5290:
5286:
5282:
5263:
5252:
5248:
5227:
5217:
5214:
5209:
5207:
5202:
5198:
5194:
5190:
5186:
5181:
5177:
5173:
5169:
5165:
5161:
5136:
5128:
5124:
5120:
5117:
5114:
5108:
5100:
5096:
5092:
5086:
5078:
5074:
5067:
5061:
5055:
5048:
5047:
5046:
5044:
5039:
5035:
5028:
5023:
5019:
5014:
5010:
5003:
4999:
4995:
4990:
4986:
4982:
4978:
4973:
4969:
4944:
4936:
4932:
4928:
4925:
4922:
4916:
4908:
4904:
4897:
4891:
4883:
4873:
4866:
4865:
4864:
4861:
4856:
4835:
4831:
4826:
4820:
4816:
4812:
4809:
4806:
4801:
4797:
4792:
4786:
4782:
4775:
4767:
4763:
4758:
4752:
4748:
4744:
4741:
4738:
4733:
4729:
4724:
4718:
4714:
4710:
4707:
4701:
4693:
4689:
4685:
4682:
4679:
4674:
4670:
4658:
4653:
4649:
4627:
4624:
4619:
4615:
4608:
4603:
4590:
4587:
4584:
4577:
4576:
4575:
4573:
4569:
4565:
4561:
4557:
4553:
4549:
4545:
4541:
4520:
4507:
4504:
4501:
4494:
4493:
4492:
4486:
4484:
4482:
4478:
4474:
4470:
4465:
4463:
4459:
4455:
4451:
4447:
4429:
4425:
4418:
4411:
4407:
4406:homeomorphism
4402:
4400:
4397:
4393:
4389:
4384:
4382:
4378:
4374:
4370:
4366:
4362:
4358:
4354:
4350:
4346:
4342:
4338:
4334:
4329:
4327:
4323:
4319:
4315:
4311:
4292:
4286:
4277:
4271:
4263:
4259:
4255:
4254:generic point
4251:
4250:residue field
4247:
4243:
4224:
4221:
4218:
4212:
4208:
4202:
4196:
4187:
4181:
4178:
4173:
4170:
4165:
4159:
4153:
4146:
4145:
4144:
4142:
4123:
4117:
4108:
4102:
4099:
4090:
4081:
4077:
4073:
4069:
4065:
4061:
4057:
4053:
4049:
4045:
4041:
4037:
4032:
4030:
4026:
4022:
4018:
3999:
3996:
3993:
3990:
3978:
3975:
3972:
3965:
3960:
3945:
3929:
3927:
3919:
3899:
3893:
3890:
3881:
3878:
3875:
3869:
3863:
3860:
3854:
3851:
3848:
3839:
3836:
3833:
3810:
3807:
3804:
3798:
3794:
3788:
3782:
3777:
3773:
3766:
3760:
3757:
3751:
3748:
3745:
3739:
3730:
3724:
3721:
3712:
3703:
3699:
3683:
3674:
3671:
3668:
3661:
3658:
3652:
3649:
3646:
3643:
3640:
3632:
3628:
3624:
3621:
3617:
3613:
3609:
3605:
3601:
3597:
3593:
3589:
3585:
3582:
3579:
3575:
3571:
3568:
3564:
3561:
3557:
3553:
3534:
3530:
3526:
3520:
3517:
3511:
3499:
3490:
3486:
3482:
3478:
3474:
3470:
3445:
3440:
3430:
3425:
3410:
3407:
3403:
3399:
3395:
3391:
3387:
3383:
3379:
3375:
3371:
3367:
3363:
3343:
3339:
3335:
3330:
3323:
3319:
3315:
3310:
3307:
3282:
3279:
3275:
3271:
3268:
3265:
3262:
3256:
3233:
3230:
3225:
3222:
3218:
3214:
3211:
3205:
3202:
3194:
3191:
3187:
3175:
3162:
3159:
3150:
3140:
3129:
3122:
3115:and so is in
3099:
3096:
3093:
3087:
3082:
3072:
3066:
3056:
3045:
3036:
3032:
3028:
3024:
3018:
3014:
3009:
2991:
2988:
2985:
2981:
2976:
2970:
2967:
2964:
2958:
2950:
2946:
2942:
2938:
2922:
2917:
2913:
2910:
2907:
2901:
2895:
2892:
2889:
2883:
2875:
2871:
2867:
2863:
2859:
2854:
2833:
2830:
2822:
2819:
2816:
2810:
2807:
2801:
2792:
2789:
2784:
2780:
2776:
2773:
2770:
2765:
2761:
2754:
2751:
2745:
2739:
2719:
2713:
2710:
2707:
2699:
2695:
2691:
2687:
2683:
2679:
2676:
2660:
2654:
2651:
2646:
2642:
2638:
2635:
2632:
2627:
2623:
2616:
2610:
2606:
2600:
2594:
2585:
2579:
2576:
2567:
2543:
2540:
2535:
2531:
2527:
2524:
2521:
2516:
2512:
2502:
2498:
2495:
2487:
2477:
2474:
2452:
2448:
2444:
2439:
2435:
2431:
2426:
2422:
2413:
2409:
2406:
2402:
2381:
2377:
2373:
2370:
2364:
2358:
2352:
2332:
2323:
2320:
2317:
2310:
2305:
2292:
2289:
2286:
2264:
2260:
2256:
2253:
2245:
2241:
2238:
2234:
2230:
2226:
2225:
2223:
2215:
2213:
2211:
2207:
2205:
2201:
2197:
2192:
2175:
2172:
2166:
2160:
2157:
2151:
2145:
2142:
2139:
2133:
2128:
2111:
2107:
2103:
2099:
2078:
2072:
2060:
2055:
2018:
2001:
1998:
1994:
1990:
1982:
1976:
1960:
1959:
1958:
1938:
1934:
1931:
1922:
1918:
1914:
1910:. (Proof: If
1909:
1905:
1902:
1898:
1894:
1890:
1886:
1882:
1877:
1875:
1871:
1867:
1833:
1830:
1826:
1811:
1808:
1802:
1799:
1794:
1790:
1782:
1781:
1780:
1777:
1773:
1767:
1763:
1759:
1755:
1751:
1743:
1741:
1739:
1735:
1731:
1727:
1723:
1699:
1691:
1683:
1679:
1675:
1671:
1667:
1663:
1642:
1612:
1604:
1596:
1592:
1588:
1584:
1580:
1576:
1555:
1545:
1541:
1537:
1533:
1528:
1526:
1522:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1490:
1486:
1482:
1478:
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1445:
1443:
1439:
1435:
1431:
1428:(that is the
1427:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1391:
1389:
1387:
1383:
1379:
1375:
1371:
1367:
1363:
1358:
1356:
1352:
1336:
1333:
1330:
1325:
1314:
1291:
1288:
1276:
1272:
1247:
1243:
1220:
1211:
1172:
1168:
1159:
1156:
1153:
1150:
1137:
1133:
1124:
1118:
1113:
1109:
1101:
1100:
1099:
1086:
1083:
1079:
1070:
1066:
1062:
1059:
1056:
1051:
1047:
1040:
1037:
1031:
1025:
1002:
996:
993:
988:
984:
976:
975:
974:
957:
951:
942:
936:
933:
930:
907:
901:
876:
872:
868:
865:
862:
857:
853:
846:
843:
840:
837:
834:
826:
822:
803:
797:
794:
788:
782:
759:
756:
753:
747:
743:
737:
731:
722:
716:
713:
704:
696:
695:
694:
692:
688:
684:
680:
675:
673:
665:
660:
656:
652:
648:
644:
640:
637:
636:
635:
633:
629:
625:
621:
617:
613:
609:
605:
601:
597:
593:
589:
585:
581:
577:
573:
569:
566:
562:
558:
554:
550:
545:
531:
511:
505:
502:
499:
491:
487:
483:
479:
476:). The image
475:
471:
468: −
467:
463:
459:
455:
451:
447:
443:
424:
421:
417:
408:
404:
400:
397:
394:
389:
385:
378:
375:
369:
363:
356:
355:
354:
352:
348:
345:s are in the
330:
326:
297:
293:
289:
286:
283:
278:
274:
267:
264:
257:
256:
255:
239:
224:
210:
194:
188:
185:
182:
174:
171:(so they are
156:
127:
112:
108:
104:
96:
94:
92:
87:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
30:
19:
8030:
7994:
7959:
7921:
7894:
7866:
7840:
7833:Mumford 1999
7828:
7816:
7804:
7791:
7779:
7767:
7744:
7736:
7732:
7728:
7724:
7720:
7716:
7712:
7708:
7704:
7700:
7696:
7692:
7688:
7684:
7680:
7675:
7431:
7423:
7419:
7254:
7250:
7246:
7214:
7206:
7202:
7197:
7185:
7181:
7176:
7118:
7114:
7110:
7106:
7102:
7100:
7095:
7051:generic rank
7048:
6965:
6771:
6769:
6637:, one gets:
6509:
6505:
6497:
6493:
6489:
6487:
6480:
6396:
6392:
6381:
6377:
6369:
6367:
6362:
6355:
6349:
6345:
6338:
6334:
6328:
6324:
6320:
6316:
6308:
6304:
6301:
6297:
6293:
6289:
6285:
6277:
6273:
6269:
6267:
6247:
6233:
6229:
6155:
6147:
6145:
6034:
6030:
6026:
6022:
6018:
6010:
5874:
5870:
5866:
5862:
5823:
5819:
5762:
5719:
5715:
5711:
5703:
5699:
5695:
5691:
5687:
5683:
5675:
5671:
5518:
5328:
5288:
5284:
5283:
5250:
5246:
5215:
5210:
5205:
5200:
5196:
5192:
5188:
5184:
5179:
5175:
5171:
5167:
5163:
5159:
5157:
5042:
5037:
5033:
5026:
5021:
5017:
5012:
5008:
5001:
4997:
4993:
4988:
4984:
4980:
4976:
4971:
4967:
4965:
4863:is given by
4859:
4854:
4656:
4651:
4647:
4645:
4571:
4567:
4563:
4559:
4555:
4551:
4550:. Then some
4547:
4543:
4537:
4490:
4466:
4457:
4449:
4445:
4403:
4385:
4380:
4376:
4372:
4368:
4364:
4360:
4356:
4352:
4348:
4344:
4340:
4336:
4332:
4330:
4325:
4321:
4317:
4313:
4310:rational map
4261:
4257:
4245:
4242:direct limit
4239:
4140:
4079:
4075:
4071:
4067:
4063:
4059:
4055:
4051:
4047:
4043:
4039:
4035:
4033:
4025:dense subset
4020:
4016:
3930:
3923:
3701:
3697:
3630:
3626:
3619:
3615:
3611:
3599:
3591:
3587:
3583:
3573:
3569:
3559:
3555:
3551:
3488:
3484:
3480:
3476:
3472:
3468:
3405:
3401:
3397:
3393:
3389:
3385:
3381:
3377:
3373:
3369:
3365:
3361:
3034:
3030:
3026:
3022:
3016:
3012:
2948:
2944:
2940:
2936:
2876:= 1 and let
2873:
2869:
2865:
2857:
2852:
2697:
2693:
2689:
2685:
2681:
2674:
2411:
2404:
2400:
2243:
2236:
2232:
2228:
2208:
2203:
2199:
2193:
2109:
2105:
2101:
2038:
1920:
1916:
1912:
1903:
1892:
1888:
1884:
1880:
1878:
1876:in general.
1870:prime ideals
1863:
1775:
1771:
1761:
1757:
1753:
1749:
1747:
1737:
1733:
1729:
1725:
1721:
1681:
1677:
1673:
1669:
1665:
1594:
1590:
1586:
1582:
1578:
1574:
1543:
1539:
1531:
1529:
1520:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1488:
1484:
1480:
1476:
1472:
1464:
1460:
1456:
1452:
1448:
1446:
1417:
1413:
1409:
1405:
1401:
1397:
1395:
1381:
1377:
1373:
1369:
1365:
1361:
1359:
1354:
1350:
1200:
1017:
824:
820:
774:
690:
686:
682:
676:
669:
664:ringed space
658:
654:
650:
646:
642:
638:
631:
627:
623:
619:
615:
611:
607:
603:
599:
595:
591:
590:) such that
587:
583:
579:
575:
571:
567:
564:
559:between two
556:
552:
548:
546:
489:
485:
481:
477:
473:
469:
465:
461:
457:
453:
449:
441:
439:
350:
317:
111:subvarieties
106:
102:
100:
88:
64:isomorphisms
59:
55:
47:
33:
7945:James Milne
7890:Harris, Joe
7845:Fulton 1998
7727:; that is,
7679:Proof: Let
7049:(since the
6343:is free as
6230:is closed.
5761:dominating
4260:to that of
4034:A morphism
3604:#Properties
3578:irreducible
3565:Taking the
1908:#Definition
1416:are called
1304:as well as
109:are closed
52:affine line
48:regular map
44:polynomials
8058:Categories
7981:354063293X
7854:References
7699:with some
7124:See also:
7121:) points.
6257:See also:
6158:, the set
4574:such that
4240:where the
4070:such that
3926:continuous
3920:Properties
2220:See also:
1866:pre-images
1720:such that
1487:such that
1467:, it is a
626:is called
484:) lies in
318:where the
97:Definition
76:birational
7760:Citations
7644:∘
7639:¯
7621:…
7612:∘
7607:¯
7581:#
7547:ϕ
7533:¯
7515:ϕ
7509:…
7498:¯
7480:ϕ
7457:#
7446:ϕ
7395:¯
7384:ϕ
7370:¯
7352:ϕ
7346:…
7335:¯
7317:ϕ
7299:ϕ
7295:∘
7266:ϕ
7249:since if
7227:ϕ
7066:∗
7025:
7010:
6993:∗
6982:
6936:∗
6896:⊗
6888:⊗
6869:∗
6840:∗
6827:∗
6787:⊗
6744:∗
6734:∗
6713:χ
6697:−
6689:∞
6674:∑
6659:∗
6648:χ
6617:∗
6600:
6581:⇒
6570:∗
6560:∗
6533:
6454:χ
6442:
6425:∗
6414:χ
6209:≥
6194:∣
6188:∈
6094:−
6078:∣
6072:
6037:, define
6012:Corollary
5980:
5968:
5929:−
5890:−
5837:⊂
5790:
5784:≥
5778:
5735:−
5640:→
5499:≠
5487:≠
5481:∣
5475:∈
5416:↦
5360:↦
5267:¯
5231:¯
5118:⋯
4926:…
4810:…
4776:∼
4742:⋯
4683:⋯
4609:−
4594:→
4511:→
4477:removable
4422:↦
4339:) and if
4284:↪
4222:∘
4216:↦
4194:↪
4179:
4174:→
4115:→
4095:#
4082:and then
3985:↦
3951:→
3837:⊗
3808:⊗
3802:↦
3774:⊗
3749:×
3737:→
3717:#
3681:↦
3656:→
3650:×
3504:#
3500:σ
3452:∞
3446:∪
3280:−
3223:−
3192:−
3088:−
2911:−
2831:−
2820:−
2799:↪
2790:−
2771:−
2717:→
2652:−
2633:−
2614:↦
2592:→
2572:#
2541:−
2522:−
2506:↦
2493:→
2330:↦
2296:→
2158:∣
2143:∈
2112:); i.e.,
1999:−
1995:ϕ
1943:#
1932:ϕ
1831:−
1827:ϕ
1823:↦
1806:→
1791:ϕ
1770:ϕ :
1703:¯
1646:¯
1616:¯
1559:¯
1386:#Examples
1319:#
1292:ϕ
1284:#
1273:ϕ
1262:'s. Note
1215:¯
1178:¯
1160:ϕ
1154:…
1143:¯
1125:ϕ
1110:ϕ
1060:…
1000:→
985:ϕ
949:→
931:ϕ
866:…
838:∘
757:∘
751:↦
729:→
709:#
561:varieties
509:→
448:defining
398:…
287:…
230:→
192:→
186::
60:biregular
8028:(2009).
7992:(2013).
7958:(1999).
7920:(1997).
7892:(1992).
7864:(1998).
7245:lies in
7131:See also
6502:complete
6386:complete
5158:for all
4979:. Since
4048:dominant
3576:) of an
3411:Suppose
3388:is just
3019:− (0, 0)
2399:. Since
2216:Examples
2096:are the
1915: :
1879:Now, if
1525:Examples
72:rational
7795:Vakil,
7715:, then
7435:Proof:
6376:and if
6359:-module
6300:) over
5706:. Then
5677:Theorem
5521:(since
4456:, then
4351:, then
4252:of the
3602:. (cf.
3594:to the
3491:, then
3380:and so
3033:, then
3021:. Then
2467:. Then
2279:. Then
1760:= Spec
1752:= Spec
1523:); see
1400:equals
628:regular
622:. Then
444:is the
8042:
8014:
7978:
7934:
7906:
7878:
7719:is in
7422:is in
7418:since
6353:|
6311:). By
6282:finite
5698:= dim
5041:'s in
4966:where
3823:where
3360:where
2039:where
1384:. See
1201:where
775:where
472:is in
440:where
7687:. If
7168:Notes
6481:(The
6374:étale
6280:be a
6146:Then
4471:is a
4452:is a
4386:On a
4312:from
4054:, if
3598:over
2935:Then
2696:= 1,
1660:(cf.
1534:is a
1483:near
639:Note:
446:ideal
8040:ISBN
8012:ISBN
7976:ISBN
7932:ISBN
7904:ISBN
7876:ISBN
6504:and
6500:are
6384:are
6268:Let
6252:flat
6017:Let
5682:Let
5387:and
5285:Note
4491:Let
4078:) ⊂
3700:and
3550:and
3010:Let
2864:Let
2410:Let
2242:Let
2104:and
1895:are
1764:are
1756:and
1732:and
1499:and
1368:and
823:and
645:and
618:and
598:) ⊂
456:and
142:and
105:and
74:and
8004:doi
7968:doi
7711:in
7703:in
7101:If
7098:.)
7053:of
7022:deg
7007:deg
6979:deg
6439:deg
6403:,
6395:on
6372:is
6368:If
6332:to
6319:in
6250:is
6150:is
6069:dim
6063:max
6033:in
5977:dim
5965:dim
5877:of
5865:of
5822:in
5787:dim
5775:dim
5722:of
5714:of
5517:of
5327:in
5191:of
5170:in
5162:in
4996:of
4570:of
4464:.)
4331:If
4316:to
4256:of
4171:lim
4066:of
4031:).
3558:(1/
3475:on
1868:of
1748:If
1668:on
1633:of
1573:of
1530:If
1444:).
1440:in
1380:to
582:of
574:of
563:is
349:of
113:of
101:If
34:In
8060::
8038:.
8010:.
8002:.
7998:.
7974:.
7966:.
7947:,
7930:.
7926:.
7902:.
7898:.
7874:.
7870:.
7739:).
7723:=
7691:=
7205:,
7184:=
6496:,
6380:,
6276:→
6272::
6025:→
6021::
5955:,
5765:,
5690:→
5686::
5664:.
5281:.
5016:=
4401:.
4383:.
4375:→
4359:→
4328:.
3629:,
3408:.)
3404:=
3364:,
3015:=
2947:,
2872:+
2684:=
2212:.
2206:.
1919:→
1891:,
1883:,
1774:→
1724:=
1680:,
1593:,
1527:.
1519:,
1511:,
1491:=
1479:,
634:.
544:.
353::
8048:.
8020:.
8006::
7984:.
7970::
7940:.
7912:.
7884:.
7737:h
7735:(
7733:D
7729:f
7725:k
7721:A
7717:f
7713:A
7709:h
7705:A
7701:g
7697:h
7695:/
7693:g
7689:f
7685:x
7681:A
7659:.
7656:f
7653:=
7650:)
7647:f
7634:m
7630:y
7624:,
7618:,
7615:f
7602:1
7598:y
7592:(
7589:=
7584:a
7577:f
7556:)
7553:g
7550:(
7544:=
7541:)
7538:)
7528:m
7524:y
7518:(
7512:,
7506:,
7503:)
7493:1
7489:y
7483:(
7477:(
7474:g
7471:=
7468:)
7465:g
7462:(
7450:a
7426:.
7424:J
7420:g
7406:0
7403:=
7400:)
7392:g
7387:(
7381:=
7378:)
7375:)
7365:m
7361:y
7355:(
7349:,
7343:,
7340:)
7330:1
7326:y
7320:(
7314:(
7311:g
7308:=
7303:a
7292:g
7270:a
7255:J
7251:g
7247:Y
7231:a
7207:Y
7203:X
7192:.
7186:A
7182:Y
7153:.
7119:f
7115:y
7113:(
7111:f
7107:k
7103:f
7096:f
7080:X
7074:O
7062:f
7034:)
7031:L
7028:(
7019:)
7016:f
7013:(
7004:=
7001:)
6998:L
6989:f
6985:(
6966:q
6950:X
6944:O
6932:f
6926:q
6922:R
6899:n
6892:L
6883:X
6877:O
6865:f
6859:q
6855:R
6851:=
6848:)
6845:F
6836:f
6832:(
6823:f
6817:q
6813:R
6790:n
6783:L
6772:F
6755:.
6752:)
6749:F
6740:f
6730:f
6724:q
6720:R
6716:(
6708:q
6704:)
6700:1
6694:(
6684:0
6681:=
6678:q
6670:=
6667:)
6664:F
6655:f
6651:(
6625:)
6622:F
6613:f
6609:,
6606:X
6603:(
6595:q
6592:+
6589:p
6585:H
6578:)
6575:F
6566:f
6556:f
6550:q
6546:R
6542:,
6539:Y
6536:(
6528:p
6524:H
6510:Y
6506:F
6498:Y
6494:X
6490:f
6466:.
6463:)
6460:F
6457:(
6451:)
6448:f
6445:(
6436:=
6433:)
6430:F
6421:f
6417:(
6397:Y
6393:F
6382:Y
6378:X
6370:f
6363:f
6356:U
6350:Y
6346:O
6341:)
6339:U
6337:(
6335:f
6329:X
6325:O
6321:Y
6317:U
6309:Y
6307:(
6305:k
6302:f
6298:X
6296:(
6294:k
6290:f
6286:k
6278:Y
6274:X
6270:f
6248:f
6215:}
6212:n
6206:)
6203:x
6200:(
6197:e
6191:X
6185:x
6182:{
6179:=
6174:n
6170:X
6156:n
6148:e
6131:.
6128:}
6125:x
6117:)
6114:)
6111:x
6108:(
6105:f
6102:(
6097:1
6090:f
6081:Z
6075:Z
6066:{
6060:=
6057:)
6054:x
6051:(
6048:e
6035:X
6031:x
6027:Y
6023:X
6019:f
5992:.
5989:r
5986:+
5983:W
5974:=
5971:Z
5943:)
5940:U
5937:(
5932:1
5925:f
5904:)
5901:W
5898:(
5893:1
5886:f
5875:Z
5871:U
5867:Y
5863:W
5849:)
5846:X
5843:(
5840:f
5834:U
5824:Y
5820:U
5802:.
5799:r
5796:+
5793:W
5781:Z
5763:W
5749:)
5746:W
5743:(
5738:1
5731:f
5720:Z
5716:Y
5712:W
5704:Y
5700:X
5696:r
5692:Y
5688:X
5684:f
5650:1
5645:P
5637:X
5634::
5631:f
5611:)
5608:z
5605::
5602:y
5599:(
5596:=
5593:)
5590:z
5587:x
5584::
5581:y
5578:x
5575:(
5572:=
5569:)
5564:2
5560:y
5556::
5553:y
5550:x
5547:(
5544:=
5541:)
5538:y
5535::
5532:x
5529:(
5519:X
5505:}
5502:0
5496:z
5493:,
5490:0
5484:x
5478:X
5472:)
5469:z
5466::
5463:y
5460::
5457:x
5454:(
5451:{
5431:)
5428:z
5425::
5422:y
5419:(
5413:)
5410:z
5407::
5404:y
5401::
5398:x
5395:(
5375:)
5372:y
5369::
5366:x
5363:(
5357:)
5354:z
5351::
5348:y
5345::
5342:x
5339:(
5329:P
5315:z
5312:x
5309:=
5304:2
5300:y
5289:X
5264:X
5251:i
5247:f
5228:X
5216:X
5206:x
5201:i
5197:f
5193:X
5189:x
5185:x
5180:i
5176:f
5172:X
5168:x
5164:U
5160:x
5143:)
5140:)
5137:x
5134:(
5129:m
5125:f
5121::
5115::
5112:)
5109:x
5106:(
5101:1
5097:f
5093::
5090:)
5087:x
5084:(
5079:0
5075:f
5071:(
5068:=
5065:)
5062:x
5059:(
5056:f
5043:k
5038:i
5034:f
5030:0
5027:f
5025:/
5022:i
5018:f
5013:i
5009:g
5005:0
5002:f
4998:X
4994:k
4989:i
4985:g
4981:X
4977:U
4972:i
4968:g
4951:)
4948:)
4945:x
4942:(
4937:m
4933:g
4929:,
4923:,
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3778:k
3770:]
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3758:=
3755:]
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3734:]
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3662:,
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3644::
3641:p
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3620:V
3616:F
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3455:}
3449:{
3441:1
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3431:=
3426:1
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3406:A
3402:X
3398:X
3394:X
3390:k
3386:X
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3374:g
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3340:y
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3324:n
3320:x
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3288:]
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2478::
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2365:=
2362:)
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2324:y
2321:,
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2311:,
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2257:=
2254:y
2244:X
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2134:=
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2106:f
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2019:x
2013:m
2007:(
2002:1
1991:=
1986:)
1983:x
1980:(
1977:f
1971:m
1939:f
1935:=
1921:Y
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1893:B
1889:A
1885:Y
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1849:)
1844:p
1839:(
1834:1
1818:p
1812:,
1809:Y
1803:X
1800::
1795:a
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1772:B
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1754:A
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1730:h
1728:/
1726:g
1722:f
1708:]
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1621:]
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1608:[
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1579:X
1575:X
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1542:(
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1521:h
1517:g
1513:h
1509:g
1505:x
1501:h
1497:h
1495:/
1493:g
1489:f
1485:x
1481:h
1477:g
1473:x
1465:x
1461:x
1457:A
1455:→
1453:X
1451::
1449:f
1414:A
1412:→
1410:X
1408::
1406:f
1402:A
1398:Y
1382:X
1378:Y
1374:f
1370:f
1366:Y
1362:X
1355:f
1351:f
1337:.
1334:f
1331:=
1326:a
1315:f
1289:=
1277:a
1248:i
1244:y
1221:i
1212:y
1186:)
1183:)
1173:m
1169:y
1163:(
1157:,
1151:,
1148:)
1138:1
1134:y
1128:(
1122:(
1119:=
1114:a
1087:,
1084:J
1080:/
1076:]
1071:m
1067:y
1063:,
1057:,
1052:1
1048:y
1044:[
1041:k
1038:=
1035:]
1032:Y
1029:[
1026:k
1003:Y
997:X
994::
989:a
961:]
958:X
955:[
952:k
946:]
943:Y
940:[
937:k
934::
911:]
908:X
905:[
902:k
882:)
877:m
873:f
869:,
863:,
858:1
854:f
850:(
847:g
844:=
841:f
835:g
825:Y
821:X
807:]
804:Y
801:[
798:k
795:,
792:]
789:X
786:[
783:k
760:f
754:g
748:g
744:,
741:]
738:X
735:[
732:k
726:]
723:Y
720:[
717:k
714::
705:f
691:Y
689:→
687:X
685::
683:f
659:Y
657:→
655:X
653::
651:f
647:Y
643:X
632:X
624:f
620:V
616:U
612:V
610:→
608:U
606::
604:f
600:V
596:U
594:(
592:f
588:x
586:(
584:f
580:V
576:x
572:U
568:x
557:Y
555:→
553:X
551::
549:f
532:Y
512:Y
506:X
503::
500:f
490:Y
486:Y
482:X
480:(
478:f
474:I
470:g
466:f
462:X
458:g
454:f
450:X
442:I
425:,
422:I
418:/
414:]
409:n
405:x
401:,
395:,
390:1
386:x
382:[
379:k
376:=
373:]
370:X
367:[
364:k
351:X
331:i
327:f
303:)
298:m
294:f
290:,
284:,
279:1
275:f
271:(
268:=
265:f
240:m
235:A
225:n
220:A
195:Y
189:X
183:f
157:m
152:A
128:n
123:A
107:Y
103:X
31:.
20:)
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