4455:
2529:
algebraic geometry changed abruptly. ... The style of thinking that was fully developed in algebraic geometry at that time was too far removed from the set-theoretical and axiomatic spirit, which then determined the development of mathematics. ... Around the middle of the present century algebraic geometry had undergone to a large extent such a reshaping process. As a result, it can again lay claim to the position it once occupied in mathematics.
14557:
9567:
10317:
objects form "nice" families. Once a good concept of "nice families" is established, the existence of a coarse moduli space should be nearly automatic. The coarse moduli space is not the fundamental object any longer, rather it is only a convenient way to keep track of certain information that is only latent in the moduli functor or moduli stack.
9304:, define immersions as the composite of an open immersion followed by a closed immersion. These immersions are immersions in the sense above, but the converse is false. Furthermore, under this definition, the composite of two immersions is not necessarily an immersion. However, the two definitions are equivalent when
9896:(see below for the definition). Any affine scheme is separated, therefore any scheme is locally separated. However, the affine pieces may glue together pathologically to yield a non-separated scheme. The following is a (non-exhaustive) list of local properties of rings, which are applied to schemes. Let
10316:
While much of the early work on moduli, especially since , put the emphasis on the construction of fine or coarse moduli spaces, recently the emphasis shifted towards the study of the families of varieties, that is towards moduli functors and moduli stacks. The main task is to understand what kind of
2528:
Algebraic geometry occupied a central place in the mathematics of the last century. The deepest results of Abel, Riemann, Weierstrass, many of the most important papers of Klein and
Poincare belong to this domain. At the end of the last and the beginning of the present century the attitude towards
15987:
On
Grothendieck's own view there should be almost no history of schemes, but only a history of the resistance to them: ... There is no serious historical question of how Grothendieck found his definition of schemes. It was in the air. Serre has well said that no one invented schemes (conversation
5957:
15546:
4143:. Note that all complete smooth curves are projective in the sense that they admit embeddings into projective space, but for the degree to be well-defined a choice of such an embedding has to be explicitly specified. The arithmetic of a complete smooth curve over a
1843:
2766:
12935:
3149:
13856:
8054:
19431:
10857:
15250:
12568:
3947:
or even some constructive process. In algebraic geometry one distinguishes between discrete and continuous invariants. For continuous classifying invariants one additionally attempts to provide some geometric structure which leads to
15862:
15080:
5072:, by definition the maximal length of a chain of irreducible closed subschemes, is a global property. It can be seen locally if a scheme is irreducible. It depends only on the topology, not on the structure sheaf. See also
15953:
8146:
10966:
19006:
19146:
9884:. The fact that localizations of a Noetherian ring are still noetherian then means that the property of a scheme of being locally Noetherian is local in the above sense (whence the name). Another example: if a ring is
2314:
9294:
559:
432:
16793:
5604:
12105:
17807:
5322:
11171:
11047:
4906:
13986:
696:
8523:
5201:
3852:, if all chains between two irreducible closed subschemes have the same length. Examples include virtually everything, e.g. varieties over a field, and it is hard to construct examples that are not catenary.
18376:
14067:
6600:
5832:
2805:
15439:
18860:
3306:
12370:
6539:
18802:
15625:
11443:
17484:
16361:
3603:
2961:
7724:
2111:
14676:
13573:
11772:
6452:
3451:
4417:
8751:
6704:
5809:
13641:
5246:
8852:
8365:
8294:
4675:
16911:
6186:
3761:
18209:
16986:
4994:
4432:. The reason the definition of closed subschemes relies on such a construction is that, unlike open subsets, a closed subset of a scheme does not have a unique structure as a subscheme.
3252:
3163:
between schemes is a morphism that becomes an isomorphism after restricted to some open dense subset. One of the most common examples of a birational map is the map induced by a blowup.
17359:
16251:
16204:
15136:
12464:
10690:
10582:
16078:
14425:
14193:
10258:
9374:
9340:
8397:
8326:
2711:
2647:
19076:
11278:
10640:
10111:
9626:
7175:
7003:
6819:
2226:
1721:
1670:
1597:
1228:
18541:
8914:
7929:
1087:
942:
18498:
18291:
14299:
8984:
7444:
7400:
1363:
1279:
1192:
1130:
1035:
992:
315:
19338:
19207:
18147:
14601:
14233:
12779:
8618:
4340:
2389:
857:
15669:
15645:
11708:
18744:
16166:
13096:
12681:
12266:
12208:
10455:
4290:
3832:
1753:
1394:
896:
473:
350:
17287:
14457:
14141:
13709:
9938:
as a base, though, or even more general bases. Connected, irreducible, reduced, integral, normal, regular, Cohen-Macaulay, locally noetherian, dimension, catenary, Gorenstein.
6365:
6233:
18644:
16284:
12045:
9443:
6058:
4057:
2160:
19301:
19170:
19030:
17314:
17128:
14504:
11819:
10136:
10067:
7755:
7200:
7131:
7028:
6959:
6844:
6775:
5811:, which coincides with the usual notion of étale map in differential geometry. Étale morphisms form a very important class of morphisms; they are used to build the so-called
3494:
2720:
1547:
1511:
1468:
1433:
18095:
15368:
14700:
13917:
12705:
12650:
12290:
12232:
17875:
16835:
16698:
15285:
14817:
12843:
6315:
3943:
is a guiding principle in all of mathematics where one tries to describe all objects satisfying certain properties up to given equivalences by more accessible data such as
18117:
18037:
17645:
13458:
8585:
8563:
6646:
4729:
2866:
1948:
1926:
1904:
1323:
17569:
17421:
15415:
14916:
14868:
10495:
3067:
2429:
16656:
16533:
which are themselves semisimple groups with additional properties. Since all simple groups are reductive, a split simple group is a simple group that is split-reductive.
14259:
3203:
13738:
7625:
4185:
attaches invariants to a singularity which classify them. (b) For curves and surfaces resolution is known in any characteristic which also yields a classification. See
18245:
15700:
13023:
12177:
10405:
8742:
8449:
7958:
6098:
4805:
4625:
3909:
18328:
17941:
17908:
5504:
4117:
19343:
15288:
14780:
14754:
13286:
13053:
10739:
8255:
7292:
780:
17511:
15156:
12397:
11960:
10654:> 0. One thus says that "a curve is projectively normal if the linear system that embeds it is complete." The term "linearly normal" is synonymous with 1-normal.
9918:
9809:
6259:
5031:
4949:
3637:
3539:
3353:
238:
14936:
5058:
3326:
15165:
12495:
8174:
4083:
4020:
8675:
19277:
19257:
19227:
18924:
18904:
18057:
18004:
17984:
17961:
17827:
17723:
17703:
17665:
17615:
17595:
17531:
17441:
17379:
17251:
17231:
17208:
17188:
17168:
17148:
17101:
17081:
17061:
17041:
16471:
16451:
16427:
16128:
14477:
14373:
14353:
14319:
14161:
14095:
13478:
13428:
13408:
12971:(i.e. such that fiber products with it are closed maps), and of finite type. Projective morphisms are proper; but the converse is not in general true. See also
12799:
12745:
12725:
12612:
12588:
11992:
11930:
11906:
11874:
11843:
11792:
11728:
11675:
11648:
11628:
11608:
11588:
11568:
11545:
11521:
11487:
10176:
10156:
10042:
7586:
7544:
7240:
7220:
7106:
7068:
7048:
6934:
6884:
6864:
6750:
6666:
5741:
5721:
3994:
2131:
9392:
scheme structure is meant, that is, the scheme structure corresponding to the unique radical ideal consisting of all functions vanishing on that closed subset.
4207:
is an approach to birational classification of complete smooth varieties in higher dimension (at least 2). While the original goal is about smooth varieties,
15729:
6890:. Finite morphisms are quasi-finite, but not all morphisms having finite fibers are quasi-finite, and morphisms of finite type are usually not quasi-finite.
4297:
4190:
14961:
9308:
is quasi-compact. Note that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not:
18248:
15882:
8091:
16930:
15988:
1995). The question is, what made
Grothendieck believe he should use this definition to simplify an 80 page paper by Serre into some 1000 pages of
12981:, namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism.
10889:
18929:
20391:
19081:
2235:
47:
13480:. Any quasi-split reductive group is a split-reductive reductive group, but there are quasi-split reductive groups that are not split-reductive.
11069:
for simple normal crossing. Refers to several closely related notions such as nc divisor, nc singularity, snc divisor, and snc singularity. See
8590:
3. Some authors call a normal variety
Gorenstein if the canonical divisor is Cartier; note this usage is inconsistent with meaning 1.
20401:
9228:
482:
355:
18069:
play an important role. There is a close connection between linear Lie groups, their associated Lie algebras and linear algebraic groups over
16707:
5513:
12050:
8599:
19905:
17735:
15990:
12591:
5640:
5255:
20300:
20258:
20216:
20174:
20132:
20090:
20048:
20006:
7794:
11100:
10979:
19957:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics , vol. 2, Berlin, New York:
6607:
5649:
4822:
16514:
is another name for a smooth normal crossing divisor, i.e. a divisor that has only smooth normal crossing singularities. They appear in
9541:.) Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called its
5952:{\displaystyle 0\to {\mathcal {O}}_{\mathbf {P} ^{n}}\to {\mathcal {O}}_{\mathbf {P} ^{n}}(1)^{\oplus (n+1)}\to T\mathbf {P} ^{n}\to 0,}
3175:
is a birational transformation that replaces a closed subscheme with an effective
Cartier divisor. Precisely, given a noetherian scheme
619:
20406:
20372:
15541:{\displaystyle \operatorname {ch} (\pi _{!}E)\cdot \operatorname {td} (S)=\pi _{*}(\operatorname {ch} (E)\cdot \operatorname {td} (X))}
8454:
5111:
18337:
15382:
12993:
be a property of a scheme that is stable under base change (finite-type, proper, smooth, étale, etc.). Then a representable morphism
4148:
2771:
17:
16553:. There are many different characterisations of smoothness. The following are equivalent definitions of smoothness of the morphism
20396:
18811:
13922:
11995:
9388:
is not a radical ideal. When specifying a closed subset of a scheme without mentioning the scheme structure, usually the so-called
39:
3257:
2565:
Peña, Javier López; Lorscheid, Oliver (2009-08-31). "Mapping F_1-land:An overview of geometries over the field with one element".
18564:
13991:
6544:
4182:
3970:
20411:
16526:
16478:
15972:
13652:
13121:
9828:. It is usually the case that it is enough to check one cover, not all possible ones. One also says that a certain property is
4174:
3959:
12295:
20347:
19976:
18753:
15551:
11392:
9457:. For locally Noetherian schemes, to be integral is equivalent to being a connected scheme that is covered by the spectra of
5394:
is a scheme admitting an ample family of invertible sheaves. A scheme admitting an ample invertible sheaf is a basic example.
4163:
53:
For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme
9465:
of two integral schemes is not integral. However, for irreducible schemes, it is a local property.) For example, the scheme
2601:
A separated scheme of finite type over a field. For example, an algebraic variety is a reduced irreducible algebraic scheme.
19929:
The notion G-unramified is what is called "unramified" in EGA, but we follow
Raynaud's definition of "unramified", so that
17450:
16309:
15968:
4569:
3663:
3548:
2895:
4215:
10211:, the converse is false. For example, most schemes in finite-dimensional algebraic geometry are locally Noetherian, but
7668:
5328:
4167:
2067:
587:
20436:
20378:
16496:
16380:
15082:. Nowadays, the formula is viewed as a consequence of the more general formula (which is valid even if π is not tame):
14638:
13534:
11737:
8085:
3710:
3034:
6406:
4147:(in particular number and structure of its rational points) is governed by the classification of the associated curve
3401:
1607:
16515:
7810:
5069:
4921:
4353:
6674:
6484:
5750:
2482:
if the preimage of any open affine subset is again affine. In more fancy terms, affine morphisms are defined by the
13608:
5213:
18870:
A morphism has some property universally if all base changes of the morphism have this property. Examples include
16104:
14819:
is an algebraic space. If "algebraic space" is replaced by "scheme", then it is said to be strongly representable.
8808:
8331:
8260:
6622:
notions, i.e. conditions on morphisms rather than conditions on schemes themselves. The category of schemes has a
4630:
16876:
10358:
6111:
4186:
4178:
3720:
3030:
35:
18173:
16936:
4958:
3216:
20296:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie"
20254:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie"
19950:
18560:
17319:
16221:
16171:
16108:
15085:
13720:
12429:
11222:(and similarly for closed). For example, finitely presented flat morphisms are open and proper maps are closed.
10660:
10552:
9296:
is surjective. A composition of immersions is again an immersion. Some authors, such as
Hartshorne in his book
4510:
4132:
20170:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Première partie"
16045:
14382:
14166:
10214:
9345:
9311:
8370:
8299:
5703:
if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties
4197:
2684:
2620:
20339:
20212:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie"
19035:
11240:
10595:
10072:
9599:
7136:
6964:
6780:
2999:
2191:
1679:
1628:
1555:
1201:
18503:
8877:
7867:
3974:
3328:
is universal with respect to (1). Concretely, it is constructed as the relative Proj of the Rees algebra of
2662:
1049:
904:
19809:
19519:
18463:
18261:
14264:
12112:
10208:
9217:
is a closed immersion if, and only if, it induces a homeomorphism from the underlying topological space of
8949:
7409:
7365:
1850:
1838:{\displaystyle \mathbf {P} (V)=\operatorname {Proj} (k)=\operatorname {Proj} (\operatorname {Sym} (V^{*}))}
1328:
1244:
1157:
1095:
1000:
957:
280:
19306:
19175:
15967:
A heuristic term, roughly equivalent to "killing automorphisms". For example, one might say "we introduce
14574:
14201:
12750:
4306:
2330:
825:
20441:
18556:
17668:
16530:
16511:
15654:
15630:
14943:
14828:
11680:
9983:
9967:
9413:
7547:
4231:
4140:
18718:
16136:
13062:
12655:
12240:
12182:
10414:
4264:
4152:
3776:
2761:{\displaystyle \operatorname {Spec} \mathbb {C} \times _{\mathbb {R} }\operatorname {Spec} \mathbb {C} }
1368:
870:
447:
324:
19441:
17260:
14430:
14104:
13657:
9947:
8057:
7403:
6320:
6278:
6199:
5352:
3966:
3944:
3026:
18620:
16260:
16131:
12930:{\displaystyle \mathbf {P} (E)=\mathbf {Proj} (\operatorname {Sym} _{{\mathcal {O}}_{X}}(E^{\vee })).}
12017:
9419:
6028:
4033:
2136:
20334:
20086:"Eléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Première partie"
19550:
19437:
19282:
19151:
19011:
18552:
18458:
17292:
17106:
15872:
14482:
11797:
11293:
10346:
10116:
10047:
9995:
9699:
8944:
8603:
7729:
7180:
7111:
7008:
6939:
6824:
6755:
5475:
5340:
3467:
3011:
2844:
1520:
1484:
1441:
1406:
1140:
1040:
952:
275:
20373:
https://web.archive.org/web/20121108104319/http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf
20128:"Éléments de géométrie algébrique: III. Étude cohomologique des faisceaux cohérents, Seconde partie"
19873:
Harada, Megumi; Krepski, Derek (2013-02-02). "Global quotients among toric
Deligne-Mumford stacks".
18072:
17254:
15326:
14681:
12686:
12631:
12271:
12213:
4128:
3144:{\displaystyle \displaystyle \limsup _{l\to \infty }\operatorname {dim} \Gamma (X,L^{l})/l^{n}>0}
20044:"Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes"
18450:
17832:
16802:
16665:
15703:
15312:
15255:
14789:
13303:
10543:
8987:
6287:
1044:
948:
794:
18100:
18016:
17624:
13851:{\displaystyle \mathbf {P} ^{1}\to \mathbf {P} ^{d},\,(s:t)\mapsto (s^{d}:s^{d-1}t:\cdots :t^{d})}
13437:
9514:
when (as a topological space) it is not the union of two closed subsets except if one is equal to
8568:
8546:
6629:
5653:
picks up where the EGA left off. Today it is one of the standard references in algebraic geometry.
2849:
1931:
1909:
1887:
1306:
19445:
18704:
are isomorphism classes of curves with extra structure, say, then a universal object is called a
17536:
17388:
15956:
15388:
14889:
14841:
12727:
is affine or more generally if it is quasi-compact, separated and admits an ample sheaf, e.g. if
10460:
9644:
8049:{\displaystyle \dim \Gamma (X,\Omega _{X}^{n})=\dim \operatorname {H} ^{n}(X,{\mathcal {O}}_{X})}
2394:
43:
16632:
14238:
10879:
3182:
1471:
20446:
20287:
20245:
20203:
20161:
20119:
20077:
20035:
19993:
19434:
19426:{\displaystyle {\mathcal {O}}_{X,f(y)}/{\mathfrak {m}}\to {\mathcal {O}}_{Y,y}/{\mathfrak {n}}}
18136:
18063:
16522:
16100:
14333:
13885:
13879:
13199:
13124:
13110:
10852:{\displaystyle (I/I^{2})^{*}={\mathcal {H}}om_{{\mathcal {O}}_{Y}}(I/I^{2},{\mathcal {O}}_{Y})}
9871:
9742:
9477:
8867:
7935:
is the degree of the curve and δ is the number of nodes (which is zero if the curve is smooth).
7591:
7551:
5077:
3940:
3849:
2560:
598:
18218:
15678:
12996:
12150:
12115:, to all schemes of the fiber product operation a significant (if technically anodyne) result.
10378:
8688:
8422:
7466:
is locally of finite presentation if, and only if, it is locally of finite type. The morphism
6071:
4778:
4598:
3878:
19732:
Smith, Karen E.; Zhang, Wenliang (2014-09-03). "Frobenius
Splitting in Commutative Algebra".
18875:
18438:
18300:
17913:
17880:
14715:
13732:
13290:
11933:
10338:
9656:
9542:
7814:
7515:
7296:
5489:
4703:
4514:
4208:
4204:
4096:
3375:
15245:{\displaystyle R=\sum _{P\in X}\operatorname {length} _{{\mathcal {O}}_{P}}(\Omega _{X/Y})P}
14759:
14733:
13265:
13032:
12563:{\displaystyle \mathbb {P} _{X}^{n}:=\mathbb {P} ^{n}\times _{\mathrm {Spec} \mathbb {Z} }X}
8226:
7271:
1198:
under the natural group homomorphism from the group of
Cartier divisors to the Picard group
751:
20357:
20321:
20279:
20237:
20195:
20153:
20111:
20069:
20027:
19986:
18871:
18615:
18590:
is a normal variety with the action of a torus such that the torus has an open dense orbit.
18143:
17489:
16495:
A sheaf with a set of global sections that span the stalk of the sheaf at every point. See
16015:
16011:
15726:, then the left-hand side reduces to the Euler characteristic while the right-hand side is
15316:
15141:
12375:
11938:
11490:
11370:
10369:
9963:
9903:
9794:
9510:
8621:
is the free abelian group generated by isomorphism classes of varieties with the relation:
7802:
6238:
5644:
5643:
was an incomplete attempt to lay a foundation of algebraic geometry based on the notion of
5009:
4927:
4422:
4258:
4136:
3928:
3615:
3517:
3331:
223:
19777:
Brandenburg, Martin (2014-10-07). "Tensor categorical foundations of algebraic geometry".
14921:
11362:
10289:(the linked article does not discuss a loop group in algebraic geometry; for now see also
5043:
3311:
8:
19231:
18709:
18159:
14836:
12977:
11963:
11849:
11731:
10643:
10365:
9931:
9711:
8925:
8771:
8209:
8151:
7588:, the geometric meaning of flatness could roughly be described by saying that the fibers
5744:
4441:
4062:
3999:
3160:
2590:
19239:) if it is locally of finite type (resp. locally of finite presentation) and if for all
11458:
10323:
8624:
5816:
19874:
19778:
19733:
19700:
19679:
19658:
19637:
19262:
19242:
19212:
18909:
18889:
18042:
17989:
17969:
17946:
17812:
17730:
17708:
17688:
17650:
17600:
17580:
17516:
17426:
17364:
17236:
17216:
17193:
17173:
17153:
17133:
17086:
17066:
17046:
17026:
16456:
16436:
16412:
16406:
16113:
15159:
14955:
14531:
14462:
14358:
14338:
14304:
14146:
14080:
13463:
13413:
13393:
12784:
12730:
12710:
12597:
12573:
12485:
12408:
12235:
12144:
11977:
11915:
11891:
11859:
11828:
11777:
11713:
11660:
11633:
11613:
11593:
11573:
11553:
11530:
11506:
11472:
11090:
10161:
10141:
10027:
9046:
8798:
8185:
7571:
7529:
7225:
7205:
7091:
7053:
7033:
6919:
6869:
6849:
6735:
6651:
6019:
6003:
5726:
5706:
5089:
3979:
3504:. If the base field has characteristic zero instead of normality, then one may replace
2820:
2566:
2556:
2544:
2497:
2229:
2116:
2038:
20295:
20253:
20211:
15857:{\displaystyle \pi _{*}(e^{c_{1}(L)}(1-c_{1}(T^{*}X)/2))=\operatorname {deg} (L)-g+1.}
9518:. Using the correspondence of prime ideals and points in an affine scheme, this means
20363:
20343:
19972:
18599:
18419:
18066:
16997:
15999:
14604:
13431:
12818:
12415:
12132:
11909:
11822:
11498:
11315:
10868:
10203:
9951:
9876:
9870:
is local in the above sense, iff the corresponding property of rings is stable under
9753:
9668:
8533:
5418:
5391:
5037:
4235:
4227:
4159:
3667:
3609:
2869:
2832:
2674:
2432:
1846:
247:. For example, the point associated to the zero ideal for any integral affine scheme.
20291:
20249:
20207:
20165:
20123:
20081:
20039:
19997:
19510:
a scheme is weakly normal if any finite birational morphism to it is an isomorphism.
9837:
5812:
5700:
20416:
20329:
20309:
20267:
20225:
20183:
20141:
20099:
20057:
20015:
19962:
19930:
19531:
18705:
18673:
16483:
16088:
16023:
15876:
15075:{\displaystyle 2g(X)-2=\operatorname {deg} (\pi )(2g(Y)-2)+\sum _{y\in Y}(e_{y}-1)}
14942:(no wild ramification), for example, over a field of characteristic zero, then the
14620:
14569:
13528:
13520:
13163:
12972:
12490:
11971:
11853:
11366:
11070:
10342:
9852:
9833:
9672:
9586:
9550:
9538:
9209:
8755:
5666:
5463:
5364:
5348:
5085:
5073:
4816:
4593:
4545:
4023:
3956:
3007:
2881:
1739:
20169:
20127:
20085:
7864:
for a nodal curve in the projective plane says the genus of the curve is given as
7494:) if it is locally of finite presentation, quasi-compact, and quasi-separated. If
4454:
20353:
20317:
20275:
20233:
20191:
20149:
20107:
20065:
20043:
20023:
20001:
19982:
19958:
18611:
18415:
17679:
17575:
17021:
16544:
15648:
14939:
14515:
14376:
14330:
14101:
if it has no nonzero nilpotent elements, i.e., its nilradical is the zero ideal,
13388:
13330:
13118:
13114:
12963:
10513:
10194:
9892:
elements), then so are its localizations. An example for a non-local property is
9881:
9458:
9167:) endowed with the structure of reduced closed subscheme. But in general, unless
8537:
7944:
6887:
5619:
5483:
4505:
4492:
4425:
4293:
4086:
3962:
3924:
3679:
3651:
3512:
3387:
3050:
2983:
0-dimensional and Noetherian. The definition applies both to a scheme and a ring.
2972:
2808:
2658:
2586:
2505:
2478:
2186:
2061:
1293:
719:
17103:
is a property that is automatic or more common over algebraically closed fields
5747:, étale morphisms are precisely those inducing an isomorphism of tangent spaces
20421:
19657:
Deitmar, Anton (2006-05-16). "Remarks on zeta functions and K-theory over F1".
18662:
18011:
17682:
16855:
16215:
16019:
15948:{\displaystyle \operatorname {H} ^{1}(\mathbf {P} ^{n},T_{\mathbf {P} ^{n}})=0}
14628:
14527:
13591:
13524:
13501:
11350:
10525:
9971:
9594:
9462:
8141:{\displaystyle X_{E}=X\times _{\operatorname {Spec} k}{\operatorname {Spec} E}}
5970:
5662:
4467:
4301:
3643:
2470:
2459:
2166:
1967:
734:
19967:
12469:
3. Projective morphisms are defined similarly to affine morphisms:
10546:. For example, all regular schemes are normal, while singular curves are not.
9537:. (Rings possessing exactly one minimal prime ideal are therefore also called
20430:
18587:
18576:
16843:
16430:
16035:
14875:
14535:
13863:
13233:
12108:
12007:
12003:
11877:
10961:{\displaystyle \operatorname {Spec} _{X}(\oplus _{0}^{\infty }I^{n}/I^{n+1})}
10872:
10701:
10538:
10274:
9856:
9730:
8408:
7861:
7636:
6189:
5966:
5344:
5081:
5004:
4997:
4764:
4518:
3949:
2610:
2501:
2466:
1993:
1289:
244:
19001:{\displaystyle f^{\#}\colon {\mathcal {O}}_{X,f(y)}\to {\mathcal {O}}_{Y,y}}
11885:
9203:
are maps that factor through isomorphisms with subschemes. Specifically, an
4181:
over algebraically closed fields up to isomorphism. (a) In characteristic 0
19934:
19678:
Flores, Jaret (2015-03-08). "Homological Algebra for Commutative Monoids".
19141:{\displaystyle {\mathfrak {n}}=f^{\#}({\mathfrak {m}}){\mathcal {O}}_{Y,y}}
18135:
is a curve with some "mild" singularity, used to construct a good-behaving
18132:
16612:
16295:
14556:
14098:
11446:
11338:
10729:
10693:
10309:
9926:
be a covering of a scheme by open affine subschemes. For definiteness, let
9885:
9546:
8863:
8767:
8221:
7817:
amounts to defining the category of quasi-coherent sheaves on it. See also
7511:
6623:
6015:
5381:
4541:
4144:
4090:
3458:
2455:
is roughly a vector space where one has forgotten which point is the origin
2452:
2309:{\displaystyle \omega _{D}=(\omega _{X}\otimes {\mathcal {O}}_{X}(D))|_{D}}
1282:
786:
602:
20367:
16507:
1. The term "simple point" is an old term for a "smooth point".
10708:
is an integrally closed domain. This meaning is consistent with that of 2.
5982:
18059:
that is split in the sense of connected solvable linear algebraic groups.
14719:
13489:
12833:
12615:
11994:-valued points were a massive further step. As part of the predominating
9534:
9289:{\displaystyle f^{\sharp }:{\mathcal {O}}_{X}\to f_{*}{\mathcal {O}}_{Y}}
9115:
always contains (but is not necessarily equal to) the Zariski closure of
8212:
that is locally nonempty and in which two objects are locally isomorphic.
2547:
is a branch of mathematics that studies solutions to algebraic equations.
2512:
2483:
1725:
554:{\displaystyle F(D)=F\otimes _{{\mathcal {O}}_{X}}{\mathcal {O}}_{X}(D).}
427:{\displaystyle F(n)=F\otimes _{{\mathcal {O}}_{X}}{\mathcal {O}}_{X}(n).}
16788:{\displaystyle X_{\bar {y}}:=X\times _{Y}\mathrm {Spec} (k({\bar {y}}))}
15381:
2. The general version is due to Grothendieck and called the
5599:{\displaystyle H^{n-i}(X,F^{\vee }\otimes \omega )\simeq H^{i}(X,F)^{*}}
3915:
is the free abelian group generated by closed subvarieties of dimension
3308:
is an effective Cartier divisor, called the exceptional divisor and (2)
2519:
of a projective space is the Spec of the homogeneous coordinate ring of
20313:
20271:
20229:
20187:
20145:
20103:
20061:
20019:
19480:
18256:
16629:
is flat, locally of finite presentation, and for every geometric point
15672:
15434:
15370:. For example, the formula implies the degree of the canonical divisor
12100:{\displaystyle S^{\prime }\times _{S}{\textrm {Spec}}({\overline {K}})}
11386:
10290:
10286:
9401:
5631:
4533:
4027:
3873:
3172:
2064:
is a complete group variety. For example, consider the complex variety
17802:{\displaystyle B=B_{0}\supset B_{1}\supset \ldots \supset B_{t}=\{1\}}
14235:
are reduced rings. Equivalently X is reduced if, for each open subset
13492:
parametrizes quotients of locally free sheaves on a projective scheme.
9376:
may be homeomorphic but not isomorphic. This happens, for example, if
9213:
factors through an isomorphism with a closed subscheme. Equivalently,
9099:
This notion is distinct from that of the usual set-theoretic image of
5965:
is the projective space over a field and the last nonzero term is the
5317:{\displaystyle f_{*}(x)\in A_{0}(\operatorname {Spec} k)=\mathbb {Z} }
4162:
over an algebraically closed field up to rational equivalence. See an
64:
19663:
17008:
connected reductive) with an open dense orbit by a Borel subgroup of
9889:
8870:(of fixed genus) is roughly a vector bundle whose fiber over a curve
5431:
4253:
are defined to be those occurring in the following construction. Let
2549:
16084:
is the standard Borel; i.e., the group of upper triangular matrices.
12489:
if it factors as a closed immersion followed by the projection of a
11166:{\displaystyle \Gamma (X,L)^{\otimes n}\to \Gamma (X,L^{\otimes n})}
11042:{\displaystyle \operatorname {Spec} _{X}({\mathcal {S}}ym(I/I^{2}))}
10197:
rings. If in addition a finite number of such affine spectra covers
19684:
19642:
16474:
16384:
11848:
Geometric points are what in the most classical cases, for example
11289:
11211:
9934:
in the following. Most of the examples also work with the integers
9566:
7818:
4901:{\displaystyle X_{k}(f)=\{x\in X|\operatorname {rk} (f(x))\leq k\}}
4581:
19879:
19783:
19738:
19705:
19699:
Durov, Nikolai (2007-04-16). "New Approach to Arakelov Geometry".
19636:
Alain, Connes (2015-09-18). "An essay on the Riemann Hypothesis".
19518:
Another but more standard term for a "codimension-one cycle"; see
8607:
5819:, which is nowadays one of the cornerstones of algebraic geometry.
2571:
691:{\displaystyle |D|=\mathbf {P} (\Gamma (X,{\mathcal {O}}_{X}(D)))}
15675:
of the tangent bundle of a space, and, over the complex numbers,
11881:
8518:{\displaystyle (f_{*}{\mathcal {O}}_{X})^{G}={\mathcal {O}}_{Y}.}
6606:
commute; rather, it commutes up to natural isomorphism; i.e., it
5196:{\displaystyle \chi (L^{\otimes m})={d \over n!}m^{n}+O(m^{n-1})}
18371:{\displaystyle {\widetilde {Y}}\hookrightarrow {\widetilde {X}}}
18123:
has similar meanings for Lie theory and linear algebraic groups.
13711:
where the limit runs over all coordinates rings of open subsets
9461:. (Strictly speaking, this is not a local property, because the
7502:
is of finite presentation if, and only if, it is of finite type.
16662:(a morphism from the spectrum of an algebraically closed field
13519:
1. Over an algebraically closed field, a variety is
12941:) but differs from EGA and Hartshorne (they don't take a dual).
12399:
is linear with respect to the action of locally free sheaves.)
10207:. While it is true that the spectrum of a noetherian ring is a
3701:
and whose ideal sheaf is invertible (locally free of rank one).
2800:{\displaystyle \mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} }
2534:
From the preface to I.R. Shafarevich, Basic Algebraic Geometry.
16518:
as well as in stabilization for compactifying moduli problems.
11974:) to simplify the geometry by refining the basic objects. The
9647:
concerns the vanishing of higher cohomology of a flag variety.
3670:
of a canonical ring (assuming the ring is finitely generated.)
2649:. An irreducible algebraic set is called an algebraic variety.
2593:
in such a way the group operations are morphisms of varieties.
1618:(roughly differential forms with simple poles along a divisor
20002:"Éléments de géométrie algébrique: I. Le langage des schémas"
18855:{\displaystyle \pi :{\mathcal {C}}_{g}\to {\mathcal {M}}_{g}}
13981:{\displaystyle f_{*}({\mathcal {O}}_{X'})={\mathcal {O}}_{X}}
12590:. Note that this definition is more restrictive than that of
8935:(i.e., there is a linear system of dimension 1 and degree 2.)
8205:
18808:
with single marked points. In literature, the forgetful map
12975:. A deep property of proper morphisms is the existence of a
11321:(roughly an analog of torsion points of an abelian variety).
9743:
http://math.stanford.edu/~conrad/248BPage/handouts/level.pdf
9207:
factors through an isomorphism with an open subscheme and a
8592:
5442:
is dense if and only if the kernel of the corresponding map
4131:. The classification of smooth curves can be refined by the
3301:{\displaystyle \pi ^{-1}(Z)\hookrightarrow {\widetilde {X}}}
2843:
Algebraic geometry over the compactification of Spec of the
2717:
is algebraically closed causes some pathology; for example,
19553:
is a locally ringed space whose points are valuation rings.
19440:. This is the geometric version (and generalization) of an
16611:
factors as an étale morphism followed by the projection of
14062:{\displaystyle R^{i}f_{*}({\mathcal {O}}_{X'})=0,\,i\geq 1}
11345:
is the group of the isomorphism classes of line bundles on
7787:
6595:{\displaystyle \psi '\circ f(\alpha )=g(\beta )\circ \psi }
4517:
any irreducible scheme is connected but not vice versa. An
17150:
defined over a not necessarily algebraically closed field
14518:
if the canonical map to the second dual is an isomorphism.
12937:
Note this definition is standard nowadays (e.g., Fulton's
9880:
schemes, namely those which are covered by the spectra of
9412:
A locally free sheaf of a rank one. Equivalently, it is a
9221:
to a closed subset of the underlying topological space of
8770:
is a scheme whose sets of points have the structures of a
1738:
The notation is ambiguous. Its traditional meaning is the
13203:, if for some (equivalently: every) open affine cover of
9523:
4525:
4440:
A scheme is called Cohen-Macaulay if all local rings are
747:
219:
20366:, "Book on Moduli of Surfaces" available at his website
18162:
parametrizes sets of points together with automorphisms.
16365:
13523:
if it is birational to a projective space. For example,
12365:{\displaystyle f_{*}(F\otimes f^{*}E)=(f_{*}F)\otimes E}
11305:
9686:
2. The Kodaira dimension of a normal variety
9453:
A scheme that is both reduced and irreducible is called
7648:
6618:
One of Grothendieck's fundamental ideas is to emphasize
4815:-morphism between the total spaces of the bundles), the
4474:
is a quasi-coherent sheaf that is finitely generated as
4211:
naturally appear and are part of a wider classification.
2181:
is an effective Cartier divisor on an algebraic variety
2023:
1976:
1551:
1402:
1091:
996:
900:
802:
249:
18797:{\displaystyle {\mathcal {C}}_{g}={\mathcal {M}}_{g,1}}
16988:, which is called the index of speciality, is positive.
15620:{\displaystyle \pi _{!}=\sum _{i}(-1)^{i}R^{i}\pi _{*}}
11438:{\displaystyle \dim \Gamma (X,\omega _{X}^{\otimes n})}
8745:
5327:
3. For the degree of a finite morphism, see
4580:
A generalization of a homogeneous coordinate ring. See
4093:
after choosing any point on the curve as identity. (c)
19862:
Algebraic Geometry and Arithmetic Curves, exercise 2.3
11962:
it sets up. Historically there was a process by which
9404:
is an inductive limit of closed immersions of schemes.
4548:, and an example of a scheme that is not connected is
2327:
are smooth, then the formula is equivalent to saying:
1234:, the group of isomorphism classes of line bundles on
20286:
20244:
20202:
20160:
20118:
20076:
20034:
19992:
19917:
19848:
19821:
19764:
19752:
19719:
19346:
19309:
19285:
19265:
19245:
19215:
19178:
19154:
19084:
19038:
19014:
18932:
18926:, consider the corresponding morphism of local rings
18912:
18892:
18814:
18756:
18721:
18623:
18598:
A kind of a piecewise-linear algebraic geometry. See
18506:
18466:
18340:
18303:
18264:
18221:
18176:
18103:
18075:
18045:
18019:
17992:
17972:
17949:
17916:
17883:
17835:
17815:
17738:
17711:
17691:
17653:
17627:
17603:
17583:
17539:
17519:
17492:
17479:{\displaystyle k\subseteq L\subseteq {\overline {k}}}
17453:
17429:
17391:
17367:
17322:
17295:
17263:
17239:
17219:
17196:
17176:
17156:
17136:
17109:
17089:
17069:
17049:
17029:
16939:
16879:
16805:
16710:
16668:
16635:
16549:
The higher-dimensional analog of étale morphisms are
16489:
16459:
16439:
16415:
16356:{\displaystyle \oplus _{0}^{\infty }\Gamma (X,L^{n})}
16312:
16263:
16224:
16174:
16139:
16116:
16048:
15885:
15732:
15681:
15657:
15633:
15554:
15442:
15391:
15329:
15258:
15168:
15144:
15088:
14964:
14924:
14892:
14844:
14792:
14762:
14736:
14684:
14641:
14577:
14485:
14465:
14433:
14385:
14361:
14341:
14307:
14267:
14241:
14204:
14169:
14149:
14107:
14083:
13994:
13925:
13888:
13741:
13660:
13611:
13537:
13466:
13440:
13416:
13396:
13306:
is a locally closed subvariety of a projective space.
13268:
13065:
13035:
12999:
12846:
12787:
12753:
12733:
12713:
12689:
12658:
12634:
12600:
12576:
12498:
12432:
12378:
12298:
12274:
12243:
12216:
12185:
12153:
12053:
12020:
11980:
11941:
11918:
11894:
11862:
11831:
11800:
11780:
11740:
11716:
11683:
11663:
11636:
11616:
11596:
11576:
11556:
11533:
11509:
11475:
11395:
11243:
11103:
10982:
10892:
10742:
10663:
10598:
10555:
10463:
10417:
10381:
10217:
10164:
10144:
10119:
10075:
10050:
10030:
9906:
9797:
9602:
9422:
9348:
9314:
9231:
8952:
8880:
8811:
8691:
8627:
8571:
8549:
8536:
is a locally Noetherian scheme whose local rings are
8457:
8425:
8373:
8334:
8302:
8263:
8229:
8154:
8094:
7961:
7870:
7732:
7671:
7594:
7574:
7568:
as a family of schemes parametrized by the points of
7532:
7412:
7368:
7274:
7228:
7208:
7183:
7139:
7114:
7094:
7056:
7036:
7011:
6967:
6942:
6922:
6872:
6852:
6827:
6783:
6758:
6738:
6677:
6654:
6632:
6547:
6487:
6409:
6323:
6290:
6241:
6202:
6114:
6074:
6031:
5835:
5753:
5729:
5709:
5516:
5492:
5367:
on a normal variety is a reflexive sheaf of the form
5258:
5216:
5114:
5046:
5012:
4961:
4930:
4825:
4781:
4706:
4633:
4601:
4356:
4309:
4267:
4099:
4065:
4036:
4002:
3982:
3881:
3779:
3723:
3618:
3598:{\displaystyle {\mathcal {O}}_{X}(K_{X})=\omega _{X}}
3551:
3520:
3470:
3404:
3334:
3314:
3260:
3219:
3185:
3071:
3070:
2956:{\displaystyle (-1)^{r}(\chi ({\mathcal {O}}_{X})-1)}
2898:
2852:
2774:
2723:
2687:
2623:
2397:
2333:
2238:
2194:
2139:
2119:
2070:
1934:
1912:
1890:
1756:
1682:
1631:
1558:
1523:
1487:
1444:
1409:
1371:
1331:
1309:
1247:
1204:
1160:
1098:
1052:
1003:
960:
907:
873:
828:
754:
622:
485:
450:
358:
327:
283:
226:
17316:
is isomorphic to a product of multiplicative groups
8068:
The prime spectrum of an algebraically closed field.
6188:; it has the natural structure of a scheme over the
2681:
is an integral separated scheme of finite type over
18746:be the moduli of smooth projective curves of genus
16253:is the closure of the union of all secant lines to
12426:-scheme that factors through some projective space
11214:(closed, respectively), i.e. if open subschemes of
10349:
of algebraic varieties of dimension greater than 2.
10331:
8196:
is a good quotient such that the fibers are orbits.
7719:{\displaystyle V\subset H^{0}(C,{\mathcal {O}}(D))}
4218:
up to isomorphism over algebraically closed fields.
577:) by its reflexive hull (and call the result still
19626:is effective. The opposite direction is similar. □
19425:
19332:
19295:
19271:
19251:
19221:
19201:
19164:
19140:
19070:
19024:
19000:
18918:
18898:
18854:
18796:
18738:
18638:
18535:
18492:
18370:
18322:
18285:
18239:
18203:
18111:
18089:
18051:
18031:
17998:
17978:
17955:
17935:
17902:
17869:
17821:
17801:
17717:
17697:
17659:
17639:
17609:
17589:
17563:
17525:
17505:
17478:
17435:
17415:
17373:
17353:
17308:
17281:
17245:
17225:
17202:
17182:
17162:
17142:
17122:
17095:
17075:
17055:
17035:
16980:
16905:
16829:
16787:
16692:
16650:
16477:as a closed subscheme — in other words, the
16465:
16445:
16421:
16355:
16278:
16245:
16198:
16160:
16122:
16072:
15947:
15856:
15694:
15663:
15639:
15619:
15540:
15409:
15362:
15279:
15244:
15150:
15130:
15074:
14930:
14910:
14862:
14811:
14774:
14748:
14694:
14670:
14595:
14498:
14471:
14451:
14419:
14367:
14347:
14313:
14293:
14253:
14227:
14198:2. A scheme X is reduced if its stalks
14187:
14155:
14135:
14089:
14061:
13980:
13911:
13850:
13703:
13635:
13567:
13472:
13452:
13422:
13402:
13280:
13090:
13047:
13017:
12929:
12793:
12773:
12739:
12719:
12699:
12675:
12644:
12606:
12582:
12562:
12458:
12391:
12364:
12284:
12260:
12226:
12202:
12171:
12099:
12039:
11986:
11954:
11924:
11900:
11888:), which specialise to ordinary-sense points. The
11868:
11837:
11813:
11786:
11766:
11722:
11702:
11669:
11642:
11622:
11602:
11582:
11562:
11539:
11515:
11481:
11437:
11272:
11210:), if the underlying map of topological spaces is
11165:
11041:
10960:
10851:
10684:
10634:
10576:
10489:
10449:
10399:
10252:
10170:
10150:
10130:
10105:
10061:
10036:
9912:
9859:local properties are thus properties of the rings
9803:
9620:
9437:
9368:
9334:
9288:
8978:
8908:
8846:
8736:
8669:
8579:
8557:
8517:
8443:
8391:
8359:
8320:
8288:
8249:
8168:
8140:
8048:
7923:
7813:since, taking the theorem as an axiom, defining a
7749:
7718:
7619:
7580:
7538:
7438:
7394:
7286:
7234:
7214:
7194:
7169:
7125:
7100:
7062:
7042:
7022:
6997:
6953:
6928:
6878:
6858:
6838:
6813:
6769:
6744:
6698:
6660:
6640:
6594:
6533:
6446:
6359:
6309:
6253:
6227:
6180:
6092:
6052:
5975:
5951:
5803:
5735:
5715:
5598:
5498:
5316:
5240:
5195:
5052:
5025:
4988:
4943:
4900:
4799:
4751:'-scheme together with a pullback square in which
4723:
4669:
4619:
4411:
4334:
4284:
4111:
4077:
4051:
4014:
3988:
3903:
3826:
3755:
3639:denoted by the same symbol (and not well-defined.)
3631:
3597:
3533:
3488:
3445:
3347:
3320:
3300:
3246:
3197:
3143:
2955:
2860:
2799:
2760:
2705:
2641:
2617:is a reduced separated scheme of finite type over
2551:algebraic geometry over the field with one element
2423:
2383:
2308:
2220:
2154:
2125:
2106:{\displaystyle \mathbb {C} ^{n}/\mathbb {Z} ^{2n}}
2105:
1942:
1920:
1898:
1837:
1715:
1664:
1591:
1541:
1505:
1462:
1427:
1388:
1357:
1317:
1273:
1222:
1186:
1124:
1081:
1029:
986:
936:
890:
851:
818:but can also mean the self-intersection number of
774:
690:
553:
467:
426:
344:
309:
232:
20371:Martin's Olsson's course notes written by Anton,
18668:is represented by some scheme or algebraic space
17877:is isomorphic to either the multiplicative group
14671:{\displaystyle {\mathcal {I}}/{\mathcal {I}}^{2}}
13568:{\displaystyle \mathbb {P} ^{1},\mathbb {P} ^{2}}
12006:of a point. The other two are formed by creating
11876:of the underlying space include analogues of the
11856:, would be the ordinary-sense points. The points
11767:{\displaystyle {\textrm {Spec}}({\overline {K}})}
11677:is defined over (is equipped with a morphism to)
10277:. The notion is due to Fontaine-Illusie and Kato.
9988:
8611:
8238:
5329:morphism of varieties#Degree of a finite morphism
4627:between normal varieties is a morphism such that
4536:other than 0 and 1; such a ring is also called a
3703:
3378:is a Kähler metric whose Ricci curvature is zero.
763:
20428:
19832:
19830:
19487:can be embedded into a projective space so that
12683:is finitely generated and generates the algebra
9690:is the Kodaira dimension of its canonical sheaf.
9679:is the dimension of Proj of the section ring of
6602:. The resulting square with obvious projections
6447:{\displaystyle f(x){\overset {\sim }{\to }}g(y)}
5993:
4158:3. Classification of complete smooth
3446:{\displaystyle \omega _{X}=i_{*}\Omega _{U}^{n}}
3073:
20392:Glossary of arithmetic and Diophantine geometry
18712:would be another example of a universal object.
15252:is the divisor of the relative cotangent sheaf
12292:of finite rank, there is a natural isomorphism
4412:{\displaystyle (Z,({\mathcal {O}}_{X}/J)|_{Z})}
2037:with a certain weak topology; it is called the
1147:, then it is the inverse of the ideal sheaf of
48:glossary of arithmetic and Diophantine geometry
20402:Glossary of differential geometry and topology
18394:is a closed subscheme of an open subscheme of
16603:), respectively, such that the restriction of
14821:
13504:generalizes a quotient of a scheme or variety.
13294:if it is of finite type and has finite fibers.
13158:A quasi-coherent sheaf on a Noetherian scheme
11296:over the category of differentiable manifolds.
9940:
8565:-Gorenstein if the canonical divisor on it is
7300:if it is of finite type and has finite fibers.
6699:{\displaystyle {\textrm {Spec}}(\mathbb {Z} )}
6534:{\displaystyle \alpha :x\to x',\beta :y\to y'}
5804:{\displaystyle df:T_{y}Y\rightarrow T_{f(y)}X}
3020:
1952:
19872:
19827:
19491:is the restriction of Serre's twisting sheaf
17675:means a simple group that is split-reductive.
16837:in the sense of classical algebraic geometry.
13636:{\displaystyle \operatorname {Spec} (k)\to X}
12411:is a closed subvariety of a projective space.
9966:is a method of constructing a weaker form of
8543:2. A normal variety is said to be
7454:if it is finitely presented at all points of
7177:is covered by finitely many affine open sets
5671:
5618:is a smooth projective variety, then it is a
5333:
5241:{\displaystyle f:X\to \operatorname {Spec} k}
4089:, i.e. the curve is a complete 1-dimensional
2564:
46:. For the number-theoretic applications, see
19585:, then there is a nonzero rational function
18684:) that corresponds to the identity morphism
18334:is a closed immersion, then the induced map
17796:
17790:
13139:if each irreducible component has dimension
9692:
8847:{\displaystyle \chi ({\mathcal {O}}_{X}(s))}
8360:{\displaystyle \operatorname {Proj} (A^{G})}
8289:{\displaystyle \operatorname {Spec} (A^{G})}
6248:
6242:
6222:
6216:
6175:
6140:
5647:, a generalization of an algebraic variety.
4895:
4848:
4670:{\displaystyle f^{*}\omega _{Y}=\omega _{X}}
3355:with respect to the ideal sheaf determining
706:| and the set of effective Weil divisors on
20342:, vol. 52, New York: Springer-Verlag,
19776:
18443:
16906:{\displaystyle X\times _{k}{\overline {k}}}
16865:3. A smooth scheme over a field
12747:is an open subscheme of a projective space
11998:, there are three corresponding notions of
10704:; i.e., the homogeneous coordinate ring of
10536:1. An integral scheme is called
8782:An old term for a "smooth" algebraic group.
6261:has the natural structure of a scheme over
6181:{\displaystyle f^{-1}(y)=\{x\in X|f(x)=y\}}
4155:for details on the arithmetic implications.
3756:{\displaystyle f:\mathbf {P} _{S}^{n}\to S}
3612:is a representative of the canonical class
2768:is not a variety since the coordinate ring
1872:) for the Proj of the symmetric algebra of
810:An ambiguous notation. It usually means an
20407:Glossary of Riemannian and metric geometry
20328:
19893:
19836:
19797:
19731:
18804:that of smooth projective curves of genus
18204:{\displaystyle \pi :{\widetilde {X}}\to X}
16981:{\displaystyle h^{0}({\mathcal {O}}(K-D))}
14880:
13882:if there is a resolution of singularities
13434:if and only if it admits a Borel subgroup
12002:of a morphism: the first being the simple
11049:, the total space of the normal bundle to
9832:, if one needs to distinguish between the
9637:
9131:is any open (and not closed) subscheme of
8587:-Cartier (and need not be Cohen–Macaulay).
7334:) if there is an open affine neighborhood
5100:1. The degree of a line bundle
4989:{\displaystyle \pi :Y\to \mathbf {A} ^{1}}
3247:{\displaystyle \pi :{\widetilde {X}}\to X}
2813:
1089:. It is also called the hyperplane bundle.
698:. There is a bijection between the set of
19966:
19878:
19782:
19737:
19704:
19683:
19662:
19641:
18626:
18105:
18083:
17354:{\displaystyle G_{m,{\overline {k}}}^{n}}
16298:or the ring of sections of a line bundle
16266:
16246:{\displaystyle V\subset \mathbb {P} ^{r}}
16233:
16199:{\displaystyle n\in \mathbb {N} _{>1}}
16183:
16142:
15131:{\displaystyle K_{X}\sim \pi ^{*}K_{Y}+R}
14724:
14049:
13868:
13772:
13555:
13540:
12756:
12551:
12521:
12501:
12459:{\displaystyle \mathbf {P} _{S}^{N}\to S}
10692:is said to be projectively normal if the
10685:{\displaystyle X\subset \mathbf {P} ^{r}}
10577:{\displaystyle C\subset \mathbf {P} ^{r}}
10000:
9764:Most important properties of schemes are
9425:
8805:over a field is the Euler characteristic
8606:to higher direct image sheaves; see also
8600:Grauert–Riemenschneider vanishing theorem
8594:Grauert–Riemenschneider vanishing theorem
8573:
8551:
7734:
6689:
6634:
5339:An approach to algebraic geometry using (
5310:
4139:curves, in particular when restricted to
4039:
3033:; both formulas compute the trace of the
2854:
2835:if some tensor power of it is very ample.
2831:A line bundle on a projective variety is
2793:
2786:
2776:
2754:
2741:
2731:
2570:
2142:
2090:
2073:
1936:
1914:
1892:
1311:
20397:Glossary of classical algebraic geometry
19543:
16073:{\displaystyle \operatorname {Gr} (d,n)}
14420:{\displaystyle R_{u}(G_{\overline {k}})}
14188:{\displaystyle \operatorname {Spec} (R)}
13878:over a field of characteristic zero has
13725:
12836:of the symmetric algebra of the dual of
10976:, then the normal cone is isomorphic to
10497:are structure maps to the base category.
10253:{\displaystyle GL_{\infty }=\cup GL_{n}}
9836:and other possible topologies, like the
9369:{\displaystyle \operatorname {Spec} A/J}
9335:{\displaystyle \operatorname {Spec} A/I}
9302:Algebraic Geometry and Arithmetic Curves
9111:). For example, the underlying space of
8392:{\displaystyle X=\operatorname {Proj} A}
8321:{\displaystyle X=\operatorname {Spec} A}
7795:Gabriel–Rosenberg reconstruction theorem
7789:Gabriel–Rosenberg reconstruction theorem
6892:
4540:. Examples of connected schemes include
2706:{\displaystyle \operatorname {Spec} (k)}
2642:{\displaystyle \operatorname {Spec} (k)}
2169:is a (flat) family of abelian varieties.
1966:with Zariski topology; it is called the
1864:. In contrast, Hartshorne and EGA write
40:glossary of classical algebraic geometry
19656:
19148:be the ideal generated by the image of
19071:{\displaystyle {\mathcal {O}}_{X,f(y)}}
18565:Category:Theorems in algebraic geometry
18062:6. In the classification of
17966:5. A linear algebraic group
17213:2. A linear algebraic group
15293:
14946:relates the degree of π, the genera of
12614:to be projective if it is given by the
11451:
11273:{\displaystyle {\mathcal {O}}_{X}|_{U}}
10871:, it is locally free and is called the
10635:{\displaystyle |{\mathcal {O}}_{C}(k)|}
10588:-normal if the hypersurfaces of degree
10262:
10106:{\displaystyle f^{-1}({\text{Spec }}B)}
9956:
9621:{\displaystyle \operatorname {Pic} (X)}
7170:{\displaystyle f^{-1}({\text{Spec }}B)}
6998:{\displaystyle f^{-1}({\text{Spec }}B)}
6814:{\displaystyle f^{-1}({\text{Spec }}B)}
5983:http://www.math.ubc.ca/~behrend/cet.pdf
2589:is an algebraic variety that is also a
2496:-Algebras, defined by analogy with the
2221:{\displaystyle \omega _{D},\omega _{X}}
1716:{\displaystyle \Omega _{X}^{1}(\log D)}
1665:{\displaystyle \Omega _{X}^{p}(\log D)}
1592:{\displaystyle \Omega _{X}^{p}(\log D)}
1223:{\displaystyle \operatorname {Pic} (X)}
14:
20429:
20412:List of complex and algebraic surfaces
19949:
19677:
18536:{\displaystyle {\mathcal {O}}_{X}(-1)}
18406:An algebraic variety of dimension two.
18255:(also called proper transform) is the
16579:, there are open affine neighborhoods
15307:on a smooth projective curve of genus
14756:of stacks such that, for any morphism
14530:is a scheme where the local rings are
13715:of an (irreducible) algebraic variety
12943:
10407:of stacks (over, say, the category of
9844:and a cover by affine open subschemes
9171:is quasi-compact, the construction of
8909:{\displaystyle \Gamma (C,\omega _{C})}
7924:{\displaystyle g=(d-1)(d-2)/2-\delta }
7809:. The theorem is a starting point for
7801:can be recovered from the category of
7302:
4685:An algebraic variety of dimension one.
3496:is the sheaf of differential forms on
3153:
1962:The set of all prime ideals in a ring
1082:{\displaystyle {\mathcal {O}}_{X}(-1)}
937:{\displaystyle {\mathcal {O}}_{X}(-1)}
19698:
19635:
18493:{\displaystyle {\mathcal {O}}_{X}(1)}
18286:{\displaystyle {\widetilde {Y}}\to Y}
17130:. If this property holds already for
15975:to rigidify the geometric situation."
14918:between smooth projective curves, if
14379:if and only if the unipotent radical
14294:{\displaystyle {\mathcal {O}}_{X}(U)}
12137:
11075:
10524:An archaic term for "smooth" as in a
10180:
9752:Another term for the structure of an
8979:{\displaystyle {\mathcal {O}}_{X}(1)}
8791:
8685:and equipped with the multiplication
8178:
8070:
7845:
7439:{\displaystyle {\mathcal {O}}_{X}(U)}
7395:{\displaystyle {\mathcal {O}}_{Y}(V)}
5677:Localization of a finite type scheme.
4444:. For example, regular schemes, and
2538:
2171:
2033:The set of all valuations for a ring
1358:{\displaystyle {\mathcal {O}}_{X}(D)}
1292:). It need not be locally free, only
1274:{\displaystyle {\mathcal {O}}_{X}(D)}
1187:{\displaystyle {\mathcal {O}}_{X}(D)}
1125:{\displaystyle {\mathcal {O}}_{X}(D)}
1030:{\displaystyle {\mathcal {O}}_{X}(1)}
987:{\displaystyle {\mathcal {O}}_{X}(1)}
310:{\displaystyle {\mathcal {O}}_{X}(1)}
20301:Publications Mathématiques de l'IHÉS
20259:Publications Mathématiques de l'IHÉS
20217:Publications Mathématiques de l'IHÉS
20175:Publications Mathématiques de l'IHÉS
20133:Publications Mathématiques de l'IHÉS
20091:Publications Mathématiques de l'IHÉS
20049:Publications Mathématiques de l'IHÉS
20007:Publications Mathématiques de l'IHÉS
19333:{\displaystyle {\mathcal {O}}_{Y,y}}
19202:{\displaystyle {\mathcal {O}}_{Y,y}}
18592:
18408:
17063:there is the derived property split-
16990:
16512:simple normal crossing (snc) divisor
16395:
14596:{\displaystyle i:X\hookrightarrow Y}
14562:
14228:{\displaystyle {\mathcal {O}}_{X,x}}
13645:
12813:is a locally free sheaf on a scheme
12803:
12774:{\displaystyle \mathbb {P} _{A}^{n}}
12123:an embedding into a projective space
11547:of the underlying topological space;
11355:
9976:
9661:
8937:
8681:is a closed subvariety of a variety
7849:
6277:1. Another term for the "
5456:
5104:on a complete variety is an integer
4335:{\displaystyle {\mathcal {O}}_{X}/J}
4220:
2837:
2667:
2384:{\displaystyle K_{D}=(K_{X}+D)|_{D}}
852:{\displaystyle L={\mathcal {O}}_{X}}
19524:
19463:a synonym with "algebraic variety".
19418:
19383:
19288:
19157:
19110:
19087:
19017:
18164:
17829:such that each successive quotient
17385:if and only if it is isomorphic to
17020:1. In the context of an
16854:that is locally of finite type and
15664:{\displaystyle \operatorname {td} }
15640:{\displaystyle \operatorname {ch} }
15433:, then as equality in the rational
14708:
13296:
13104:
11908:-valued points are thought of, via
11703:{\displaystyle {\textrm {Spec}}(K)}
11055:
10592:cut out the complete linear series
10044:may be covered by affine open sets
9579:
9406:
7554:on stalks. When viewing a morphism
7108:may be covered by affine open sets
6936:may be covered by affine open sets
6752:may be covered by affine open sets
4769:
4562:
2992:
2874:
2595:
2476:4. A morphism is called
612:over an algebraically closed field
24:
19394:
19350:
19313:
19182:
19121:
19100:
19042:
18981:
18949:
18938:
18862:is often called a universal curve.
18841:
18824:
18777:
18760:
18739:{\displaystyle {\mathcal {M}}_{g}}
18725:
18569:
18510:
18470:
16955:
16754:
16751:
16748:
16745:
16497:Sheaf generated by global sections
16491:sheaf generated by global sections
16328:
16323:
16161:{\displaystyle \mathbb {P} ^{n+1}}
16091:is the closure of a Schubert cell.
15887:
15260:
15219:
15199:
14886:Given a finite separable morphism
14687:
14657:
14644:
14508:
14321:has no nonzero nilpotent sections.
14271:
14208:
14021:
13967:
13942:
13308:
13091:{\displaystyle F\times _{G}B\to B}
12889:
12692:
12676:{\displaystyle {\mathcal {S}}_{1}}
12662:
12637:
12546:
12543:
12540:
12537:
12277:
12261:{\displaystyle {\mathcal {O}}_{Y}}
12247:
12219:
12203:{\displaystyle {\mathcal {O}}_{X}}
12189:
12059:
12026:
11402:
11389:of a smooth projective variety is
11247:
11135:
11104:
11001:
10919:
10835:
10795:
10779:
10720:is a closed subscheme of a scheme
10657:3. A closed subvariety
10607:
10450:{\displaystyle P_{G}\circ f=P_{F}}
10226:
9735:
9275:
9248:
9237:
8956:
8881:
8821:
8501:
8474:
8415:with the action of a group scheme
8192:with the action of a group scheme
8062:
8032:
8008:
7981:
7968:
7937:
7699:
7416:
7372:
5871:
5845:
5468:
5450:is contained in the nilradical of
4509:as a topological space. Since the
4372:
4313:
4285:{\displaystyle {\mathcal {O}}_{X}}
4271:
3827:{\displaystyle R^{i}f_{*}F(r-i)=0}
3656:
3555:
3472:
3429:
3094:
3083:
2930:
2651:
2579:
2271:
1684:
1633:
1560:
1525:
1489:
1446:
1411:
1389:{\displaystyle {\mathcal {O}}_{X}}
1375:
1335:
1251:
1164:
1102:
1056:
1007:
964:
911:
891:{\displaystyle {\mathcal {O}}_{X}}
877:
863:, then it means the direct sum of
838:
662:
647:
528:
512:
468:{\displaystyle {\mathcal {O}}_{X}}
454:
401:
385:
345:{\displaystyle {\mathcal {O}}_{X}}
331:
287:
25:
20458:
19918:Grothendieck & Dieudonné 1964
19849:Grothendieck & Dieudonné 1960
19822:Grothendieck & Dieudonné 1960
19765:Grothendieck & Dieudonné 1964
19753:Grothendieck & Dieudonné 1964
19720:Grothendieck & Dieudonné 1960
17282:{\displaystyle G_{\overline {k}}}
16208:
15417:is a proper morphism with smooth
15383:Grothendieck–Riemann–Roch formula
14718:from an algebraic variety to the
14627:is a regular embedding, then the
14452:{\displaystyle G_{\overline {k}}}
14136:{\displaystyle {\sqrt {(0)}}=(0)}
14077:1. A commutative ring
13704:{\displaystyle k(X)=\varinjlim k}
13494:
13175:that is locally given by modules.
13152:
13129:
12707:. Both definitions coincide when
12010:of two morphisms. For example, a
11329:A linear system of dimension one.
11218:are mapped to open subschemes of
10411:-schemes) is a functor such that
7811:noncommutative algebraic geometry
7785:if there is such a linear system.
7342:and an open affine neighbourhood
6360:{\displaystyle f:F\to G,g:H\to G}
6228:{\displaystyle X\times _{Y}\{y\}}
5821:
5655:
5650:Séminaire de géométrie algébrique
5606:holds for any locally free sheaf
5210:is a cycle on a complete variety
4570:Nagata's compactification theorem
4460:
4434:
3933:
3508:by a resolution of singularities.
2661:is a quotient of a scheme by the
2500:. Important affine morphisms are
1154:2. Most of the times,
714:. The same definition is used if
19504:
18639:{\displaystyle \mathbb {G} _{m}}
18580:
18431:
16521:3. In the context of
16388:
16385:https://mathoverflow.net/q/22228
16279:{\displaystyle \mathbb {P} ^{r}}
15991:Éléments de géométrie algébrique
15924:
15904:
14555:
14538:over a field are regular, while
13759:
13744:
13177:
12874:
12871:
12868:
12865:
12848:
12435:
12107:. This makes the extension from
12040:{\displaystyle S^{\prime }\to S}
10672:
10564:
10361:is given locally by polynomials.
10267:
9776:if and only if for any cover of
9746:
9723:
9704:
9565:
9438:{\displaystyle \mathbb {G} _{m}}
9019:is any morphism of schemes, the
8918:
8776:
8752:Grothendieck's vanishing theorem
8747:Grothendieck's vanishing theorem
8401:
7854:
7830:
7819:https://mathoverflow.net/q/16257
7244:
6648:of integers; so that any scheme
6271:
6265:as Spec of the residue field of
6053:{\displaystyle \omega _{X}^{-1}}
5930:
5879:
5853:
5634:Éléments de géométrie algébrique
4976:
4819:is the (scheme-theoretic) locus
4453:
4421:closed subscheme defined by the
4214:7. Classification of
4196:5. Classification of
4173:4. Classification of
4052:{\displaystyle \mathbb {P} ^{1}}
3854:
3732:
2603:
2155:{\displaystyle \mathbb {F} _{q}}
1758:
1608:logarithmic Kähler differentials
1281:is the sheaf corresponding to a
728:
710:that are linearly equivalent to
640:
569:is reflexive, then one replaces
19923:
19911:
19899:
19887:
19866:
19854:
19842:
19815:
19803:
19791:
19770:
19512:
19296:{\displaystyle {\mathfrak {n}}}
19165:{\displaystyle {\mathfrak {m}}}
19025:{\displaystyle {\mathfrak {m}}}
17621:if and only if a maximal torus
17486:if and only if its base change
17309:{\displaystyle {\overline {k}}}
17123:{\displaystyle {\overline {k}}}
16381:Serre's conditions on normality
16288:
14499:{\displaystyle {\overline {k}}}
13288:is a finite set. A morphism is
13238:
12594:, II.5.5.2. The latter defines
12117:
11814:{\displaystyle {\overline {K}}}
11349:, the multiplication being the
11331:
10359:morphism of algebraic varieties
10326:, "Book on Moduli of Surfaces".
10131:{\displaystyle {\text{Spec }}A}
10113:is covered by affine open sets
10062:{\displaystyle {\text{Spec }}B}
9874:. For example, we can speak of
9851:. Using the dictionary between
9593:is the degree zero part of the
8856:
8760:
8608:https://arxiv.org/abs/1404.1827
8214:
7947:of a smooth projective variety
7750:{\displaystyle \mathbb {P} (V)}
7504:
7294:is a finite set. A morphism is
7195:{\displaystyle {\text{Spec }}A}
7126:{\displaystyle {\text{Spec }}B}
7023:{\displaystyle {\text{Spec }}A}
7005:is covered by affine open sets
6954:{\displaystyle {\text{Spec }}B}
6839:{\displaystyle {\text{Spec }}A}
6770:{\displaystyle {\text{Spec }}B}
5977:equivariant intersection theory
5827:The exact sequence of sheaves:
5434:. A morphism of affine schemes
4910:
4168:Enriques–Kodaira classification
3545:is the divisor class such that
3489:{\displaystyle \Omega _{U}^{n}}
2319:2. If, in addition,
1860:-points correspond to lines in
1542:{\displaystyle \Omega _{X}^{1}}
1506:{\displaystyle \Omega _{X}^{p}}
1463:{\displaystyle \Omega _{X}^{1}}
1428:{\displaystyle \Omega _{X}^{p}}
741:by an action of a group scheme
204:
67:
36:glossary of commutative algebra
19935:Tag 02G4 in the Stacks Project
19758:
19746:
19725:
19713:
19692:
19671:
19650:
19629:
19563:
19388:
19371:
19365:
19115:
19105:
19063:
19057:
18975:
18970:
18964:
18864:
18835:
18614:is a product of finitely many
18561:cohomology base change theorem
18530:
18521:
18487:
18481:
18353:
18277:
18231:
18195:
18148:moduli space of vector bundles
18090:{\displaystyle k=\mathbb {R} }
16975:
16972:
16960:
16950:
16873:that is geometrically smooth:
16824:
16818:
16809:
16782:
16779:
16773:
16764:
16758:
16721:
16687:
16681:
16672:
16642:
16350:
16331:
16067:
16055:
15936:
15899:
15839:
15833:
15821:
15818:
15807:
15791:
15772:
15767:
15761:
15743:
15588:
15578:
15535:
15532:
15526:
15514:
15508:
15499:
15483:
15477:
15465:
15449:
15401:
15363:{\displaystyle \chi (L)=d-g+1}
15339:
15333:
15236:
15215:
15069:
15050:
15028:
15019:
15013:
15004:
15001:
14995:
14977:
14971:
14902:
14854:
14766:
14740:
14695:{\displaystyle {\mathcal {I}}}
14611:has an affine neighborhood in
14587:
14414:
14396:
14288:
14282:
14182:
14176:
14130:
14124:
14116:
14110:
14037:
14015:
13958:
13936:
13903:
13845:
13791:
13788:
13785:
13773:
13754:
13721:function field (scheme theory)
13698:
13692:
13670:
13664:
13627:
13624:
13618:
13482:
13381:
13082:
13039:
13009:
12921:
12918:
12905:
12878:
12858:
12852:
12700:{\displaystyle {\mathcal {S}}}
12645:{\displaystyle {\mathcal {S}}}
12450:
12353:
12337:
12331:
12309:
12285:{\displaystyle {\mathcal {E}}}
12227:{\displaystyle {\mathcal {F}}}
12163:
12094:
12081:
12031:
11761:
11748:
11697:
11691:
11432:
11405:
11260:
11160:
11138:
11132:
11120:
11107:
11036:
11033:
11012:
10996:
10955:
10906:
10846:
10808:
10765:
10743:
10628:
10624:
10618:
10600:
10518:
10391:
10100:
10089:
9990:locally of finite presentation
9675:) of a semi-ample line bundle
9615:
9609:
9498:
9259:
9045:which satisfies the following
8973:
8967:
8903:
8884:
8841:
8838:
8832:
8815:
8728:
8716:
8710:
8704:
8698:
8692:
8664:
8652:
8646:
8640:
8634:
8628:
8619:Grothendieck ring of varieties
8613:Grothendieck ring of varieties
8486:
8458:
8435:
8354:
8341:
8283:
8270:
8043:
8020:
7995:
7971:
7904:
7892:
7889:
7877:
7744:
7738:
7713:
7710:
7704:
7688:
7614:
7608:
7452:locally of finite presentation
7433:
7427:
7389:
7383:
7164:
7153:
6992:
6981:
6808:
6797:
6693:
6685:
6583:
6577:
6568:
6562:
6520:
6497:
6441:
6435:
6424:
6419:
6413:
6351:
6333:
6166:
6160:
6153:
6134:
6128:
6100:between schemes, the fiber of
6084:
5940:
5922:
5917:
5905:
5898:
5891:
5865:
5839:
5793:
5787:
5776:
5587:
5574:
5558:
5533:
5303:
5291:
5275:
5269:
5226:
5190:
5171:
5134:
5118:
4971:
4886:
4883:
4877:
4871:
4861:
4842:
4836:
4791:
4710:
4694:
4611:
4406:
4396:
4391:
4366:
4357:
4164:overview of the classification
3898:
3892:
3815:
3803:
3747:
3711:Castelnuovo–Mumford regularity
3705:Castelnuovo–Mumford regularity
3579:
3566:
3283:
3280:
3274:
3238:
3116:
3097:
3080:
2965:
2950:
2941:
2924:
2918:
2909:
2899:
2700:
2694:
2636:
2630:
2371:
2366:
2347:
2296:
2291:
2288:
2282:
2252:
1878:
1832:
1829:
1816:
1807:
1795:
1792:
1786:
1780:
1768:
1762:
1710:
1698:
1659:
1647:
1586:
1574:
1352:
1346:
1268:
1262:
1217:
1211:
1181:
1175:
1119:
1113:
1076:
1067:
1024:
1018:
981:
975:
931:
922:
685:
682:
679:
673:
650:
644:
632:
624:
545:
539:
495:
489:
418:
412:
368:
362:
304:
298:
29:glossary of algebraic geometry
13:
1:
20340:Graduate Texts in Mathematics
19943:
19465:
18880:
17870:{\displaystyle B_{i}/B_{i+1}}
16830:{\displaystyle k({\bar {y}})}
16693:{\displaystyle k({\bar {y}})}
15875:is trivial. For example, the
15280:{\displaystyle \Omega _{X/Y}}
14812:{\displaystyle F\times _{G}B}
13375:is quasi-separated over Spec(
13262:if the fiber over each point
12983:
12401:
11375:
10549:2. A smooth curve
10279:
10069:such that each inverse image
9533:all have exactly one minimal
9416:for the multiplicative group
9394:
8526:
7726:, one says the linear system
7268:if the fiber over each point
7133:such that each inverse image
6961:such that each inverse image
6481:, ψ') is a pair of morphisms
6310:{\displaystyle F\times _{G}H}
5743:over an algebraically closed
5357:
4731:be a morphism of schemes and
4151:to an algebraic closure. See
4129:algebraic curves for examples
3862:
3368:
3000:weighted Euler characteristic
608:on a normal complete variety
199:
19495:(1) on the projective space.
18655:
18380:
18112:{\displaystyle \mathbb {C} }
18032:{\displaystyle B\subseteq G}
17667:is a split torus. Since any
17640:{\displaystyle T\subseteq G}
17471:
17339:
17301:
17273:
17115:
16898:
16399:
14491:
14443:
14408:
14323:
13578:2. Given a field
13453:{\displaystyle B\subseteq G}
13135:A scheme has pure dimension
12113:tensor product of R-algebras
12089:
11806:
11756:
10333:Mori's minimal model program
10209:noetherian topological space
9984:unique factorization domains
9529:is connected and the rings A
9181:
8580:{\displaystyle \mathbb {Q} }
8558:{\displaystyle \mathbb {Q} }
7665:on it and a vector subspace
7498:is locally Noetherian, then
7462:is locally Noetherian, then
6821:is affine — say of the form
6641:{\displaystyle \mathbb {Z} }
6108:is, as a set, the pre-image
5062:
5040:is a degeneration such that
4739:-scheme. Then a deformation
4497:
4191:here for curves and surfaces
4183:Hironaka's resolution result
3380:
3031:Grothendieck's trace formula
2861:{\displaystyle \mathbb {Z} }
2489:construction for sheaves of
2462:is a variety in affine space
1943:{\displaystyle \mathbb {Q} }
1921:{\displaystyle \mathbb {Q} }
1899:{\displaystyle \mathbb {Q} }
1851:ring of polynomial functions
1318:{\displaystyle \mathbb {Q} }
737:of, say, an algebraic space
274:is a projective scheme with
66:
7:
20385:
20381:worked out by many authors.
18557:theorem on formal functions
18378:is also a closed immersion.
18297:along the closed subscheme
17564:{\displaystyle G_{m,L}^{n}}
17447:over an intermediate field
17416:{\displaystyle G_{m,k}^{n}}
17257:if only if its base change
16042:-orbit on the Grassmannian
16028:
15961:
15714:is a smooth curve of genus
15706:. For example, if the base
15410:{\displaystyle \pi :X\to S}
15303:is a line bundle of degree
14911:{\displaystyle \pi :X\to Y}
14863:{\displaystyle \pi :Z\to X}
14829:resolution of singularities
14823:resolution of singularities
13862:= 3, it is also called the
13513:
13363:is quasi-compact. A scheme
13344:) if the diagonal morphism
12950:
12621:of a quasi-coherent graded
11282:
10972:is regularly embedded into
10490:{\displaystyle P_{F},P_{G}}
10351:
10158:is finitely generated as a
9968:resolution of singularities
9948:complete intersection rings
9942:local complete intersection
9447:
9139:is the inclusion map, then
9076:is any closed subscheme of
7822:
7489:is finitely presented over
7222:is finitely generated as a
7050:is finitely generated as a
6866:is finitely generated as a
6626:, the spectrum of the ring
5396:
4574:
4234:in algebraic geometry; see
3842:
3650:is the section ring of the
3064:is a line bundle such that
2977:
2424:{\displaystyle K_{D},K_{X}}
1606:is 1, this is the sheaf of
722:on a complete variety over
10:
20463:
19457:
19442:unramified field extension
18545:
18400:
16915:
16799:-dimensional variety over
16651:{\displaystyle {\bar {y}}}
16542:
14520:
14254:{\displaystyle U\subset X}
14071:
13059:a scheme, the base change
12147:says that, for a morphism
11912:, as a way of identifying
9159:is the Zariski closure of
7404:finitely presented algebra
7322:of finite presentation at
6025:whose anticanonical sheaf
5673:essentially of finite type
5353:derived algebraic geometry
5335:derived algebraic geometry
4775:Given a vector-bundle map
4586:
4452:) are Cohen–Macaulay, but
4026:curves, i.e. the curve is
3967:algebraically closed field
3672:
3198:{\displaystyle Z\subset X}
3165:
2663:étale equivalence relation
2054:
1241:3. In general,
20437:Glossaries of mathematics
19968:10.1007/978-1-4612-1700-8
19438:separable field extension
18211:along a closed subscheme
18146:is used to construct the
18125:
17423:without any base change.
17190:is said to satisfy split-
16535:
16501:
16093:
16004:
15873:infinitesimal deformation
14301:is a reduced ring, i.e.,
13912:{\displaystyle f:X'\to X}
13025:is said to have property
12955:
11323:
11188:1. A morphism
10710:
10544:integrally closed domains
10542:, if the local rings are
10530:
10375:3. A morphism
10347:birational classification
10302:
9700:Kodaira vanishing theorem
9694:Kodaira vanishing theorem
8604:Kodaira vanishing theorem
8419:is an invariant morphism
7629:
7620:{\displaystyle f^{-1}(x)}
6708:
6281:" in the category theory.
5094:
4240:
3693:is a closed subscheme of
3012:virtual fundamental class
2845:ring of rational integers
2555:One goal is to prove the
2515:over a closed subvariety
2473:of some commutative ring.
2445:
1141:effective Cartier divisor
859:, the structure sheaf on
440:is a Cartier divisor and
74:
18:Glossary of scheme theory
19557:
19303:is the maximal ideal of
19032:be the maximal ideal of
18604:
18451:tautological line bundle
18445:tautological line bundle
18390:, without qualifier, of
18240:{\displaystyle f:Y\to X}
18152:
18010:if and only if it has a
17289:to an algebraic closure
17014:
16516:strong desingularization
16218:to a projective variety
15865:
15704:integration along fibers
15695:{\displaystyle \pi _{*}}
14619:there is generated by a
14479:to an algebraic closure
13531:are those birational to
13339:is quasi-separated over
13304:quasi-projective variety
13018:{\displaystyle f:F\to G}
12172:{\displaystyle f:X\to Y}
11463:
11373:into a projective space.
11097:> 0, the natural map
11065:for normal crossing and
10400:{\displaystyle f:F\to G}
9758:
9553:are irreducible, while
8999:
8988:tautological line bundle
8986:. It is the dual of the
8737:{\displaystyle \cdot =.}
8444:{\displaystyle f:X\to Y}
8198:
8148:for any field extension
7838:
6612:
6093:{\displaystyle f:X\to Y}
6062:
5679:
4800:{\displaystyle f:E\to F}
4679:
4620:{\displaystyle f:X\to Y}
4485:
3904:{\displaystyle A_{k}(X)}
3767:is the smallest integer
3457:is the inclusion of the
2884:of a projective variety
2825:
2469:is a scheme that is the
2009:. It is also denoted by
1742:of a finite-dimensional
1396:of the integral part of
1045:tautological line bundle
1043:. It is the dual of the
949:tautological line bundle
795:action of a group scheme
194:
20288:Grothendieck, Alexandre
20246:Grothendieck, Alexandre
20204:Grothendieck, Alexandre
20162:Grothendieck, Alexandre
20120:Grothendieck, Alexandre
20078:Grothendieck, Alexandre
20036:Grothendieck, Alexandre
19994:Grothendieck, Alexandre
19446:algebraic number theory
18453:of a projective scheme
18323:{\displaystyle f^{-1}Z}
17936:{\displaystyle G_{m,a}}
17903:{\displaystyle G_{m,k}}
17685:linear algebraic group
17043:for certain properties
16704:), the geometric fiber
16523:linear algebraic groups
14944:Riemann–Hurwitz formula
14882:Riemann–Hurwitz formula
13500:Usually denoted by , a
12125:
12111:, where it is just the
11182:
10201:, the scheme is called
9888:(i.e., has no non-zero
9772:has a certain property
9645:Kempf vanishing theorem
9639:Kempf vanishing theorem
9300:and Q. Liu in his book
8801:of a projective scheme
8076:A property of a scheme
8058:Serre's duality theorem
8056:(where the equality is
7627:do not vary too wildly.
7520:
7070:-algebra. The morphism
6008:
5499:{\displaystyle \omega }
5078:equidimensional schemes
4955:) if there is a scheme
4916:1. A scheme
4724:{\displaystyle S\to S'}
4470:on a Noetherian scheme
4419:is a scheme called the
4112:{\displaystyle g\geq 2}
4030:to the projective line
3179:and a closed subscheme
3027:Behrend's trace formula
3022:Behrend's trace formula
2815:algebraic vector bundle
214:
44:glossary of ring theory
19427:
19334:
19297:
19273:
19253:
19223:
19203:
19166:
19142:
19072:
19026:
19002:
18920:
18900:
18856:
18798:
18740:
18640:
18553:Zariski's main theorem
18537:
18494:
18459:Serre's twisting sheaf
18372:
18324:
18287:
18241:
18205:
18137:moduli space of curves
18113:
18091:
18053:
18033:
18000:
17980:
17957:
17937:
17910:or the additive group
17904:
17871:
17823:
17803:
17729:if and only if it has
17719:
17699:
17661:
17641:
17611:
17591:
17565:
17527:
17507:
17480:
17437:
17417:
17375:
17355:
17310:
17283:
17247:
17227:
17204:
17184:
17164:
17144:
17124:
17097:
17077:
17057:
17037:
16982:
16907:
16831:
16789:
16694:
16652:
16467:
16447:
16423:
16357:
16280:
16247:
16200:
16162:
16124:
16101:rational normal scroll
16074:
15996:
15949:
15858:
15696:
15665:
15641:
15621:
15542:
15429:is a vector bundle on
15411:
15364:
15281:
15246:
15152:
15132:
15076:
14932:
14912:
14864:
14813:
14776:
14775:{\displaystyle B\to G}
14750:
14749:{\displaystyle F\to G}
14726:representable morphism
14702:is the ideal sheaf of
14696:
14672:
14597:
14500:
14473:
14453:
14421:
14369:
14349:
14334:linear algebraic group
14315:
14295:
14255:
14229:
14189:
14157:
14137:
14091:
14063:
13982:
13913:
13880:rational singularities
13870:rational singularities
13852:
13705:
13637:
13582:and a relative scheme
13569:
13474:
13454:
13424:
13404:
13282:
13281:{\displaystyle x\in X}
13125:linear algebraic group
13092:
13049:
13048:{\displaystyle B\to G}
13019:
12931:
12795:
12775:
12741:
12721:
12701:
12677:
12646:
12608:
12584:
12564:
12466:as a closed subscheme.
12460:
12393:
12366:
12286:
12262:
12228:
12204:
12173:
12101:
12041:
11988:
11956:
11926:
11902:
11870:
11839:
11815:
11788:
11768:
11724:
11704:
11671:
11644:
11624:
11604:
11584:
11564:
11541:
11517:
11501:, but the meanings of
11483:
11439:
11292:is often defined as a
11274:
11167:
11043:
10962:
10853:
10686:
10636:
10578:
10506:
10491:
10451:
10401:
10319:
10254:
10172:
10152:
10132:
10107:
10063:
10038:
10022:locally of finite type
10002:locally of finite type
9914:
9805:
9649:
9622:
9589:of a projective curve
9543:irreducible components
9478:irreducible polynomial
9439:
9370:
9336:
9290:
9225:, and if the morphism
9021:scheme-theoretic image
8980:
8945:Serre's twisting sheaf
8910:
8868:moduli space of curves
8848:
8738:
8671:
8581:
8559:
8519:
8445:
8393:
8361:
8322:
8290:
8251:
8250:{\displaystyle X/\!/G}
8170:
8142:
8050:
7925:
7803:quasi-coherent sheaves
7751:
7720:
7621:
7582:
7550:if it gives rise to a
7540:
7482:of finite presentation
7440:
7396:
7329:finitely presented at
7288:
7287:{\displaystyle x\in X}
7236:
7216:
7196:
7171:
7127:
7102:
7064:
7044:
7024:
6999:
6955:
6930:
6914:locally of finite type
6880:
6860:
6840:
6815:
6771:
6746:
6706:, and in a unique way.
6700:
6662:
6642:
6596:
6535:
6448:
6361:
6311:
6284:2. A stack
6255:
6229:
6182:
6094:
6054:
5953:
5805:
5737:
5717:
5600:
5500:
5318:
5242:
5197:
5054:
5027:
4990:
4945:
4902:
4801:
4725:
4671:
4621:
4515:irreducible components
4413:
4336:
4286:
4216:split reductive groups
4179:Zariski neighboorhoods
4125:curves of general type
4113:
4079:
4053:
4016:
3990:
3905:
3828:
3757:
3717:on a projective space
3633:
3599:
3535:
3490:
3447:
3349:
3322:
3302:
3248:
3199:
3145:
3043:
2957:
2862:
2801:
2762:
2707:
2643:
2561:field with one element
2531:
2425:
2385:
2310:
2222:
2156:
2127:
2107:
1944:
1922:
1900:
1839:
1717:
1676:-th exterior power of
1666:
1593:
1543:
1517:-th exterior power of
1507:
1464:
1429:
1390:
1359:
1319:
1275:
1224:
1188:
1126:
1083:
1041:Serre's twisting sheaf
1031:
988:
953:Serre's twisting sheaf
938:
892:
853:
776:
775:{\displaystyle X/\!/G}
692:
599:complete linear system
565:is a Weil divisor and
555:
469:
428:
346:
311:
276:Serre's twisting sheaf
234:
189:
184:
179:
174:
169:
164:
159:
154:
149:
144:
139:
134:
129:
124:
119:
114:
109:
104:
99:
94:
89:
84:
79:
19933:are unramified. See
19800:, Exercise II.3.11(d)
19573:be a Weil divisor on
19551:Zariski–Riemann space
19545:Zariski–Riemann space
19428:
19335:
19298:
19274:
19254:
19224:
19204:
19167:
19143:
19073:
19027:
19003:
18921:
18901:
18876:universally injective
18857:
18799:
18741:
18641:
18616:multiplicative groups
18538:
18495:
18439:Zariski tangent space
18373:
18325:
18288:
18242:
18206:
18114:
18092:
18054:
18034:
18001:
17986:defined over a field
17981:
17958:
17938:
17905:
17872:
17824:
17804:
17720:
17705:defined over a field
17700:
17662:
17642:
17612:
17597:defined over a field
17592:
17566:
17528:
17508:
17506:{\displaystyle G_{L}}
17481:
17438:
17418:
17376:
17356:
17311:
17284:
17248:
17233:defined over a field
17228:
17205:
17185:
17165:
17145:
17125:
17098:
17078:
17058:
17038:
16983:
16908:
16846:over a perfect field
16832:
16790:
16695:
16653:
16468:
16448:
16424:
16358:
16281:
16248:
16201:
16163:
16125:
16075:
15985:
15950:
15859:
15697:
15666:
15642:
15622:
15543:
15412:
15365:
15282:
15247:
15153:
15151:{\displaystyle \sim }
15133:
15077:
14933:
14913:
14865:
14814:
14777:
14751:
14697:
14673:
14615:so that the ideal of
14598:
14501:
14474:
14454:
14422:
14370:
14350:
14316:
14296:
14256:
14230:
14190:
14158:
14138:
14092:
14064:
13983:
13914:
13853:
13733:rational normal curve
13727:rational normal curve
13706:
13638:
13570:
13475:
13455:
13425:
13410:defined over a field
13405:
13283:
13093:
13050:
13020:
12932:
12796:
12776:
12742:
12722:
12702:
12678:
12647:
12609:
12585:
12565:
12461:
12394:
12392:{\displaystyle f_{*}}
12367:
12287:
12263:
12229:
12205:
12174:
12102:
12042:
11996:Grothendieck approach
11989:
11957:
11955:{\displaystyle h_{S}}
11934:representable functor
11927:
11903:
11871:
11840:
11816:
11789:
11769:
11734:, is a morphism from
11725:
11705:
11672:
11645:
11625:
11605:
11585:
11565:
11542:
11518:
11484:
11440:
11294:Deligne–Mumford stack
11275:
11237:with structure sheaf
11202:of schemes is called
11168:
11093:if, for each integer
11044:
10963:
10859:. If the embedded of
10854:
10687:
10637:
10579:
10492:
10452:
10402:
10370:locally ringed spaces
10339:minimal model program
10314:
10255:
10173:
10153:
10133:
10108:
10064:
10039:
9915:
9913:{\displaystyle \cup }
9806:
9804:{\displaystyle \cup }
9657:kawamata log terminal
9623:
9440:
9371:
9337:
9291:
9092:also factors through
8981:
8911:
8849:
8739:
8672:
8582:
8560:
8520:
8446:
8394:
8362:
8323:
8291:
8252:
8171:
8143:
8051:
7926:
7828:A principal G-bundle.
7815:noncommutative scheme
7752:
7721:
7622:
7583:
7541:
7441:
7397:
7289:
7237:
7217:
7197:
7172:
7128:
7103:
7065:
7045:
7025:
7000:
6956:
6931:
6894:finite type (locally)
6881:
6861:
6841:
6816:
6772:
6747:
6701:
6663:
6643:
6597:
6536:
6449:
6362:
6312:
6256:
6254:{\displaystyle \{y\}}
6230:
6196:as the fiber product
6183:
6095:
6055:
5954:
5815:and consequently the
5806:
5738:
5718:
5601:
5501:
5476:Cohen–Macaulay scheme
5319:
5252:, then its degree is
5243:
5198:
5055:
5028:
5026:{\displaystyle X_{0}}
4991:
4951:(called the limit of
4946:
4944:{\displaystyle X_{0}}
4903:
4802:
4726:
4672:
4622:
4414:
4337:
4287:
4209:terminal singularites
4205:minimal model program
4137:projectively embedded
4114:
4080:
4054:
4017:
3991:
3969:are classified up to
3929:rational equivalences
3906:
3829:
3758:
3634:
3632:{\displaystyle K_{X}}
3600:
3536:
3534:{\displaystyle K_{X}}
3491:
3448:
3350:
3348:{\displaystyle O_{X}}
3323:
3303:
3249:
3213:is a proper morphism
3200:
3146:
3010:is the degree of the
2958:
2863:
2802:
2763:
2713:. Note, not assuming
2708:
2644:
2526:
2426:
2386:
2311:
2223:
2157:
2128:
2113:or an elliptic curve
2108:
1945:
1923:
1901:
1840:
1718:
1667:
1594:
1544:
1508:
1465:
1430:
1391:
1360:
1320:
1276:
1225:
1189:
1127:
1084:
1032:
989:
939:
893:
854:
777:
702:-rational points of |
693:
556:
470:
429:
347:
312:
235:
233:{\displaystyle \eta }
19538:
19499:
19452:
19344:
19340:and the induced map
19307:
19283:
19263:
19243:
19213:
19176:
19152:
19082:
19036:
19012:
18930:
18910:
18890:
18872:universally catenary
18812:
18754:
18719:
18700:). If the values of
18650:
18621:
18504:
18464:
18426:
18338:
18301:
18262:
18219:
18174:
18144:stable vector bundle
18101:
18073:
18043:
18017:
17990:
17970:
17947:
17914:
17881:
17833:
17813:
17736:
17709:
17689:
17651:
17625:
17601:
17581:
17537:
17517:
17490:
17451:
17427:
17389:
17365:
17320:
17293:
17261:
17237:
17217:
17194:
17174:
17154:
17134:
17107:
17087:
17067:
17047:
17027:
16937:
16877:
16803:
16708:
16666:
16633:
16457:
16437:
16413:
16310:
16261:
16222:
16172:
16137:
16114:
16046:
16016:locally ringed space
15979:
15883:
15730:
15679:
15655:
15631:
15552:
15440:
15389:
15327:
15317:Euler characteristic
15313:Riemann–Roch formula
15295:Riemann–Roch formula
15256:
15166:
15142:
15086:
14962:
14956:ramification indices
14931:{\displaystyle \pi }
14922:
14890:
14842:
14790:
14760:
14734:
14682:
14639:
14575:
14514:A coherent sheaf is
14483:
14463:
14431:
14383:
14359:
14339:
14305:
14265:
14239:
14202:
14195:is a reduced scheme.
14167:
14147:
14105:
14081:
13992:
13923:
13886:
13739:
13658:
13609:
13535:
13508:
13464:
13438:
13414:
13394:
13266:
13147:
13063:
13033:
12997:
12967:if it is separated,
12844:
12785:
12751:
12731:
12711:
12687:
12656:
12632:
12598:
12574:
12496:
12430:
12376:
12296:
12272:
12241:
12214:
12183:
12151:
12051:
12018:
11978:
11939:
11916:
11892:
11860:
11829:
11798:
11778:
11738:
11714:
11681:
11661:
11634:
11614:
11594:
11574:
11554:
11531:
11507:
11491:locally ringed space
11473:
11453:Poincaré residue map
11393:
11371:Grassmannian variety
11300:
11241:
11177:
11101:
10980:
10890:
10740:
10661:
10596:
10553:
10501:
10461:
10415:
10379:
10297:
10264:logarithmic geometry
10215:
10162:
10142:
10117:
10073:
10048:
10028:
9982:The local rings are
9964:local uniformization
9958:local uniformization
9946:The local rings are
9904:
9840:. Consider a scheme
9795:
9718:
9632:
9600:
9574:
9445:(i.e., line bundle).
9420:
9346:
9312:
9229:
8994:
8950:
8878:
8874:is the vector space
8809:
8786:
8689:
8625:
8569:
8547:
8455:
8423:
8371:
8332:
8300:
8261:
8227:
8152:
8092:
7959:
7868:
7730:
7669:
7643:
7592:
7572:
7530:
7410:
7366:
7316:, then the morphism
7272:
7226:
7206:
7181:
7137:
7112:
7092:
7054:
7034:
7009:
6965:
6940:
6920:
6870:
6850:
6825:
6781:
6756:
6736:
6675:
6652:
6630:
6545:
6485:
6407:
6403:), ψ an isomorphism
6321:
6288:
6239:
6200:
6112:
6072:
6029:
5988:
5833:
5751:
5727:
5707:
5626:
5514:
5490:
5486:is a coherent sheaf
5256:
5214:
5112:
5053:{\displaystyle \pi }
5044:
5010:
4959:
4928:
4823:
4779:
4704:
4689:
4631:
4599:
4511:connected components
4354:
4307:
4265:
4097:
4063:
4034:
4000:
3980:
3971:rational equivalence
3911:of a smooth variety
3879:
3777:
3721:
3713:of a coherent sheaf
3646:of a normal variety
3616:
3549:
3541:on a normal variety
3518:
3468:
3402:
3390:on a normal variety
3363:
3332:
3321:{\displaystyle \pi }
3312:
3258:
3217:
3183:
3068:
3006:with respect to the
2987:
2971:Another term for an
2896:
2850:
2772:
2721:
2685:
2621:
2395:
2331:
2236:
2192:
2137:
2133:over a finite field
2117:
2068:
2049:
1932:
1910:
1906:-factorial if every
1888:
1884:A normal variety is
1754:
1680:
1629:
1556:
1521:
1485:
1472:Kähler differentials
1442:
1407:
1369:
1329:
1307:
1245:
1202:
1158:
1096:
1050:
1001:
958:
951:. It is the dual of
905:
871:
826:
814:-th tensor power of
752:
620:
483:
448:
356:
325:
281:
224:
19955:Intersection theory
18710:tautological bundle
18661:1. If a
17560:
17412:
17350:
16367:Serre's conditions
16327:
16306:is the graded ring
15957:Kodaira–Spencer map
14837:birational morphism
14532:regular local rings
14427:of the base change
12978:Stein factorization
12945:projectively normal
12939:Intersection theory
12770:
12515:
12449:
11966:added more points (
11964:projective geometry
11850:algebraic varieties
11590:is a morphism from
11431:
10923:
10644:projectively normal
10366:morphism of schemes
9996:finite presentation
9853:(commutative) rings
9780:by open subschemes
9712:Kuranishi structure
9480:is integral, while
8169:{\displaystyle E/k}
7994:
7304:finite presentation
6049:
4811:(that is, a scheme
4763:' is assumed to be
4755:is the pullback of
4342:is a closed subset
4200:in small dimension.
4078:{\displaystyle g=1}
4015:{\displaystyle g=0}
3746:
3485:
3442:
3161:birational morphism
3155:birational morphism
1697:
1646:
1573:
1538:
1502:
1459:
1424:
68:Contents:
20442:Algebraic geometry
20335:Algebraic Geometry
20314:10.1007/bf02732123
20272:10.1007/bf02684343
20230:10.1007/bf02684322
20188:10.1007/bf02684747
20146:10.1007/bf02684890
20104:10.1007/bf02684274
20062:10.1007/bf02699291
20020:10.1007/bf02684778
19810:The Stacks Project
19423:
19330:
19293:
19269:
19249:
19219:
19199:
19162:
19138:
19068:
19022:
18998:
18916:
18896:
18852:
18794:
18736:
18715:2. Let
18636:
18575:An old term for a
18533:
18490:
18368:
18320:
18283:
18237:
18201:
18109:
18087:
18067:split Lie algebras
18049:
18029:
17996:
17976:
17953:
17933:
17900:
17867:
17819:
17799:
17731:composition series
17715:
17695:
17673:split simple group
17657:
17637:
17607:
17587:
17561:
17540:
17523:
17503:
17476:
17433:
17413:
17392:
17371:
17351:
17323:
17306:
17279:
17243:
17223:
17200:
17180:
17160:
17140:
17120:
17093:
17073:
17053:
17033:
16978:
16925:on a smooth curve
16903:
16827:
16785:
16690:
16648:
16463:
16453:with itself along
16443:
16419:
16407:separated morphism
16353:
16313:
16276:
16243:
16196:
16158:
16120:
16070:
16018:that is locally a
15945:
15854:
15692:
15661:
15637:
15617:
15577:
15538:
15407:
15360:
15277:
15242:
15190:
15160:linear equivalence
15148:
15128:
15072:
15049:
14928:
14908:
14860:
14809:
14786:, the base change
14772:
14746:
14706:, is locally free.
14692:
14668:
14593:
14496:
14469:
14449:
14417:
14365:
14345:
14311:
14291:
14251:
14225:
14185:
14153:
14133:
14087:
14059:
13978:
13909:
13848:
13701:
13684:
13651:An element in the
13633:
13565:
13470:
13450:
13420:
13400:
13278:
13117:in the context of
13088:
13045:
13015:
12969:universally closed
12927:
12791:
12771:
12754:
12737:
12717:
12697:
12673:
12642:
12604:
12580:
12560:
12499:
12456:
12433:
12409:projective variety
12389:
12362:
12282:
12258:
12224:
12200:
12169:
12145:projection formula
12139:projection formula
12097:
12037:
11984:
11952:
11922:
11898:
11866:
11835:
11811:
11784:
11764:
11720:
11700:
11667:
11640:
11620:
11600:
11580:
11560:
11537:
11513:
11479:
11435:
11414:
11270:
11233:is an open subset
11163:
11091:normally generated
11077:normally generated
11039:
10958:
10909:
10878:2. The
10849:
10682:
10632:
10574:
10487:
10447:
10397:
10250:
10182:locally Noetherian
10168:
10148:
10128:
10103:
10059:
10034:
9910:
9877:locally Noetherian
9801:
9667:1. The
9655:Abbreviation for "
9618:
9435:
9380:is the radical of
9366:
9332:
9298:Algebraic Geometry
9286:
9143:is different from
9047:universal property
8990:(whence the term).
8976:
8906:
8844:
8799:Hilbert polynomial
8793:Hilbert polynomial
8734:
8670:{\displaystyle =+}
8667:
8577:
8555:
8515:
8441:
8389:
8357:
8318:
8286:
8247:
8186:geometric quotient
8180:geometric quotient
8166:
8138:
8088:" if it holds for
8072:geometric property
8046:
7980:
7921:
7747:
7716:
7617:
7578:
7536:
7436:
7392:
7284:
7232:
7212:
7192:
7167:
7123:
7098:
7060:
7040:
7020:
6995:
6951:
6926:
6876:
6856:
6846:— and furthermore
6836:
6811:
6767:
6742:
6696:
6658:
6638:
6592:
6531:
6462:); an arrow from (
6444:
6357:
6307:
6251:
6225:
6178:
6090:
6050:
6032:
6020:projective variety
6004:Frobenius morphism
5981:See Chapter II of
5949:
5801:
5733:
5713:
5614:; for example, if
5596:
5496:
5478:of pure dimension
5314:
5238:
5193:
5090:algebraic surfaces
5050:
5023:
4986:
4941:
4898:
4797:
4721:
4667:
4617:
4409:
4332:
4282:
4203:6. The
4153:Faltings's theorem
4109:
4075:
4049:
4012:
3986:
3901:
3824:
3753:
3730:
3697:that is flat over
3642:4. The
3629:
3608:3. The
3595:
3531:
3511:2. The
3486:
3471:
3443:
3428:
3386:1. The
3345:
3318:
3298:
3244:
3195:
3141:
3140:
3087:
3002:of a (nice) stack
2953:
2858:
2821:locally free sheaf
2797:
2758:
2703:
2639:
2557:Riemann hypothesis
2545:Algebraic geometry
2540:algebraic geometry
2511:5. The
2498:spectrum of a ring
2433:canonical divisors
2421:
2381:
2306:
2230:adjunction formula
2218:
2173:adjunction formula
2152:
2123:
2103:
2039:Berkovich spectrum
1940:
1918:
1896:
1835:
1713:
1683:
1662:
1632:
1589:
1559:
1539:
1524:
1503:
1488:
1460:
1445:
1425:
1410:
1386:
1355:
1315:
1271:
1220:
1184:
1122:
1079:
1027:
984:
934:
888:
849:
772:
688:
551:
465:
424:
342:
307:
230:
57:and a morphism an
20349:978-0-387-90244-9
20330:Hartshorne, Robin
19978:978-3-540-62046-4
19937:for more details.
19931:closed immersions
19812:, Chapter 21, §4.
19722:, 4.1.2 and 4.1.3
19272:{\displaystyle Y}
19252:{\displaystyle y}
19222:{\displaystyle f}
18919:{\displaystyle Y}
18899:{\displaystyle y}
18676:is an element of
18600:tropical geometry
18594:tropical geometry
18420:symmetric variety
18410:symmetric variety
18365:
18350:
18274:
18192:
18064:real Lie algebras
18052:{\displaystyle k}
17999:{\displaystyle k}
17979:{\displaystyle G}
17956:{\displaystyle k}
17822:{\displaystyle k}
17718:{\displaystyle k}
17698:{\displaystyle G}
17660:{\displaystyle k}
17610:{\displaystyle k}
17590:{\displaystyle G}
17533:is isomorphic to
17526:{\displaystyle L}
17474:
17436:{\displaystyle G}
17374:{\displaystyle G}
17342:
17304:
17276:
17246:{\displaystyle k}
17226:{\displaystyle G}
17203:{\displaystyle P}
17183:{\displaystyle G}
17163:{\displaystyle k}
17143:{\displaystyle G}
17118:
17096:{\displaystyle P}
17076:{\displaystyle P}
17056:{\displaystyle P}
17036:{\displaystyle G}
16998:spherical variety
16992:spherical variety
16901:
16821:
16776:
16724:
16684:
16645:
16527:semisimple groups
16479:diagonal morphism
16466:{\displaystyle f}
16446:{\displaystyle f}
16422:{\displaystyle f}
16123:{\displaystyle n}
15722:is a line bundle
15568:
15299:1. If
15175:
15034:
14607:if each point of
14605:regular embedding
14564:regular embedding
14494:
14472:{\displaystyle G}
14446:
14411:
14368:{\displaystyle k}
14348:{\displaystyle G}
14314:{\displaystyle X}
14156:{\displaystyle R}
14119:
14090:{\displaystyle R}
13677:
13647:rational function
13529:rational surfaces
13473:{\displaystyle k}
13423:{\displaystyle k}
13403:{\displaystyle G}
12819:projective bundle
12805:projective bundle
12794:{\displaystyle A}
12740:{\displaystyle X}
12720:{\displaystyle X}
12607:{\displaystyle f}
12583:{\displaystyle X}
12416:projective scheme
12133:Proj construction
12092:
12078:
12047:is thought of as
11987:{\displaystyle T}
11925:{\displaystyle S}
11901:{\displaystyle T}
11880:(in the sense of
11869:{\displaystyle P}
11854:complex manifolds
11838:{\displaystyle K}
11823:algebraic closure
11809:
11787:{\displaystyle S}
11759:
11745:
11723:{\displaystyle K}
11688:
11670:{\displaystyle S}
11643:{\displaystyle T}
11630:, for any scheme
11623:{\displaystyle S}
11603:{\displaystyle T}
11583:{\displaystyle S}
11570:-valued point of
11563:{\displaystyle T}
11540:{\displaystyle P}
11516:{\displaystyle S}
11499:topological space
11482:{\displaystyle S}
11363:Plücker embedding
11357:Plücker embedding
11225:2. An
10724:with ideal sheaf
10716:1. If
10368:is a morphism of
10171:{\displaystyle B}
10151:{\displaystyle A}
10123:
10095:
10054:
10037:{\displaystyle X}
9978:locally factorial
9952:regular embedding
9824:has the property
9754:equivariant sheaf
9671:(also called the
9669:Kodaira dimension
9663:Kodaira dimension
9155:is reduced, then
8943:Another term for
8939:hyperplane bundle
8534:Gorenstein scheme
7846:#arithmetic genus
7581:{\displaystyle X}
7539:{\displaystyle f}
7518:of vector spaces.
7235:{\displaystyle B}
7215:{\displaystyle A}
7187:
7159:
7118:
7101:{\displaystyle X}
7063:{\displaystyle B}
7043:{\displaystyle A}
7015:
6987:
6946:
6929:{\displaystyle X}
6879:{\displaystyle B}
6859:{\displaystyle A}
6831:
6803:
6762:
6745:{\displaystyle X}
6682:
6661:{\displaystyle S}
6430:
6367:: an object over
5736:{\displaystyle Y}
5716:{\displaystyle X}
5458:dualizing complex
5392:divisorial scheme
5349:commutative rings
5206:2. If
5153:
5038:flat degeneration
4247:Closed subschemes
4236:classifying stack
4228:classifying space
4222:classifying stack
4177:resp. associated
4121:Hyperbolic curves
3989:{\displaystyle g}
3610:canonical divisor
3376:Calabi–Yau metric
3295:
3235:
3205:, the blow-up of
3072:
3041:-adic cohomology.
2870:Arakelov geometry
2839:Arakelov geometry
2823:of a finite rank.
2675:algebraic variety
2669:algebraic variety
2465:3. An
2458:2. An
2187:dualizing sheaves
2185:, both admitting
2177:1. If
2165:2. An
2126:{\displaystyle E}
2060:1. An
2017:) or simply Spec(
1928:-Weil divisor is
1602:1. If
1299:4. If
1135:1. If
479:arbitrary), then
436:2. If
270:1. If
16:(Redirected from
20454:
20417:List of surfaces
20360:
20325:
20283:
20241:
20199:
20157:
20115:
20073:
20031:
19989:
19970:
19938:
19927:
19921:
19915:
19909:
19903:
19897:
19891:
19885:
19884:
19882:
19870:
19864:
19858:
19852:
19846:
19840:
19834:
19825:
19819:
19813:
19807:
19801:
19795:
19789:
19788:
19786:
19774:
19768:
19762:
19756:
19750:
19744:
19743:
19741:
19729:
19723:
19717:
19711:
19710:
19708:
19696:
19690:
19689:
19687:
19675:
19669:
19668:
19666:
19654:
19648:
19647:
19645:
19633:
19627:
19609:is a section of
19567:
19532:Weil reciprocity
19526:Weil reciprocity
19432:
19430:
19429:
19424:
19422:
19421:
19415:
19410:
19409:
19398:
19397:
19387:
19386:
19380:
19375:
19374:
19354:
19353:
19339:
19337:
19336:
19331:
19329:
19328:
19317:
19316:
19302:
19300:
19299:
19294:
19292:
19291:
19278:
19276:
19275:
19270:
19258:
19256:
19255:
19250:
19228:
19226:
19225:
19220:
19208:
19206:
19205:
19200:
19198:
19197:
19186:
19185:
19171:
19169:
19168:
19163:
19161:
19160:
19147:
19145:
19144:
19139:
19137:
19136:
19125:
19124:
19114:
19113:
19104:
19103:
19091:
19090:
19077:
19075:
19074:
19069:
19067:
19066:
19046:
19045:
19031:
19029:
19028:
19023:
19021:
19020:
19007:
19005:
19004:
18999:
18997:
18996:
18985:
18984:
18974:
18973:
18953:
18952:
18942:
18941:
18925:
18923:
18922:
18917:
18905:
18903:
18902:
18897:
18861:
18859:
18858:
18853:
18851:
18850:
18845:
18844:
18834:
18833:
18828:
18827:
18803:
18801:
18800:
18795:
18793:
18792:
18781:
18780:
18770:
18769:
18764:
18763:
18745:
18743:
18742:
18737:
18735:
18734:
18729:
18728:
18674:universal object
18645:
18643:
18642:
18637:
18635:
18634:
18629:
18542:
18540:
18539:
18534:
18520:
18519:
18514:
18513:
18499:
18497:
18496:
18491:
18480:
18479:
18474:
18473:
18377:
18375:
18374:
18369:
18367:
18366:
18358:
18352:
18351:
18343:
18329:
18327:
18326:
18321:
18316:
18315:
18292:
18290:
18289:
18284:
18276:
18275:
18267:
18249:strict transform
18246:
18244:
18243:
18238:
18210:
18208:
18207:
18202:
18194:
18193:
18185:
18170:Given a blow-up
18166:strict transform
18142:2. A
18131:1. A
18118:
18116:
18115:
18110:
18108:
18096:
18094:
18093:
18088:
18086:
18058:
18056:
18055:
18050:
18038:
18036:
18035:
18030:
18005:
18003:
18002:
17997:
17985:
17983:
17982:
17977:
17962:
17960:
17959:
17954:
17942:
17940:
17939:
17934:
17932:
17931:
17909:
17907:
17906:
17901:
17899:
17898:
17876:
17874:
17873:
17868:
17866:
17865:
17850:
17845:
17844:
17828:
17826:
17825:
17820:
17808:
17806:
17805:
17800:
17786:
17785:
17767:
17766:
17754:
17753:
17724:
17722:
17721:
17716:
17704:
17702:
17701:
17696:
17678:4. A
17666:
17664:
17663:
17658:
17646:
17644:
17643:
17638:
17616:
17614:
17613:
17608:
17596:
17594:
17593:
17588:
17574:3. A
17570:
17568:
17567:
17562:
17559:
17554:
17532:
17530:
17529:
17524:
17512:
17510:
17509:
17504:
17502:
17501:
17485:
17483:
17482:
17477:
17475:
17467:
17442:
17440:
17439:
17434:
17422:
17420:
17419:
17414:
17411:
17406:
17380:
17378:
17377:
17372:
17360:
17358:
17357:
17352:
17349:
17344:
17343:
17335:
17315:
17313:
17312:
17307:
17305:
17297:
17288:
17286:
17285:
17280:
17278:
17277:
17269:
17252:
17250:
17249:
17244:
17232:
17230:
17229:
17224:
17209:
17207:
17206:
17201:
17189:
17187:
17186:
17181:
17169:
17167:
17166:
17161:
17149:
17147:
17146:
17141:
17129:
17127:
17126:
17121:
17119:
17111:
17102:
17100:
17099:
17094:
17082:
17080:
17079:
17074:
17062:
17060:
17059:
17054:
17042:
17040:
17039:
17034:
16987:
16985:
16984:
16979:
16959:
16958:
16949:
16948:
16912:
16910:
16909:
16904:
16902:
16894:
16892:
16891:
16842:2. A
16836:
16834:
16833:
16828:
16823:
16822:
16814:
16794:
16792:
16791:
16786:
16778:
16777:
16769:
16757:
16743:
16742:
16727:
16726:
16725:
16717:
16699:
16697:
16696:
16691:
16686:
16685:
16677:
16657:
16655:
16654:
16649:
16647:
16646:
16638:
16566:
16551:smooth morphisms
16510:2. A
16484:closed immersion
16472:
16470:
16469:
16464:
16452:
16450:
16449:
16444:
16428:
16426:
16425:
16420:
16396:#dualizing sheaf
16362:
16360:
16359:
16354:
16349:
16348:
16326:
16321:
16285:
16283:
16282:
16277:
16275:
16274:
16269:
16252:
16250:
16249:
16244:
16242:
16241:
16236:
16205:
16203:
16202:
16197:
16195:
16194:
16186:
16167:
16165:
16164:
16159:
16157:
16156:
16145:
16132:projective space
16129:
16127:
16126:
16121:
16089:Schubert variety
16087:2. A
16079:
16077:
16076:
16071:
16034:1. A
16024:commutative ring
16001:
15969:level structures
15954:
15952:
15951:
15946:
15935:
15934:
15933:
15932:
15927:
15913:
15912:
15907:
15895:
15894:
15877:projective space
15863:
15861:
15860:
15855:
15814:
15803:
15802:
15790:
15789:
15771:
15770:
15760:
15759:
15742:
15741:
15701:
15699:
15698:
15693:
15691:
15690:
15670:
15668:
15667:
15662:
15646:
15644:
15643:
15638:
15626:
15624:
15623:
15618:
15616:
15615:
15606:
15605:
15596:
15595:
15576:
15564:
15563:
15547:
15545:
15544:
15539:
15498:
15497:
15461:
15460:
15416:
15414:
15413:
15408:
15369:
15367:
15366:
15361:
15286:
15284:
15283:
15278:
15276:
15275:
15271:
15251:
15249:
15248:
15243:
15235:
15234:
15230:
15211:
15210:
15209:
15208:
15203:
15202:
15189:
15157:
15155:
15154:
15149:
15137:
15135:
15134:
15129:
15121:
15120:
15111:
15110:
15098:
15097:
15081:
15079:
15078:
15073:
15062:
15061:
15048:
14937:
14935:
14934:
14929:
14917:
14915:
14914:
14909:
14869:
14867:
14866:
14861:
14818:
14816:
14815:
14810:
14805:
14804:
14781:
14779:
14778:
14773:
14755:
14753:
14752:
14747:
14710:regular function
14701:
14699:
14698:
14693:
14691:
14690:
14677:
14675:
14674:
14669:
14667:
14666:
14661:
14660:
14653:
14648:
14647:
14621:regular sequence
14602:
14600:
14599:
14594:
14570:closed immersion
14559:
14536:smooth varieties
14534:. For example,
14505:
14503:
14502:
14497:
14495:
14487:
14478:
14476:
14475:
14470:
14458:
14456:
14455:
14450:
14448:
14447:
14439:
14426:
14424:
14423:
14418:
14413:
14412:
14404:
14395:
14394:
14374:
14372:
14371:
14366:
14354:
14352:
14351:
14346:
14320:
14318:
14317:
14312:
14300:
14298:
14297:
14292:
14281:
14280:
14275:
14274:
14260:
14258:
14257:
14252:
14234:
14232:
14231:
14226:
14224:
14223:
14212:
14211:
14194:
14192:
14191:
14186:
14162:
14160:
14159:
14154:
14143:. Equivalently,
14142:
14140:
14139:
14134:
14120:
14109:
14096:
14094:
14093:
14088:
14068:
14066:
14065:
14060:
14036:
14035:
14034:
14025:
14024:
14014:
14013:
14004:
14003:
13987:
13985:
13984:
13979:
13977:
13976:
13971:
13970:
13957:
13956:
13955:
13946:
13945:
13935:
13934:
13918:
13916:
13915:
13910:
13902:
13857:
13855:
13854:
13849:
13844:
13843:
13822:
13821:
13803:
13802:
13768:
13767:
13762:
13753:
13752:
13747:
13735:is the image of
13710:
13708:
13707:
13702:
13685:
13642:
13640:
13639:
13634:
13574:
13572:
13571:
13566:
13564:
13563:
13558:
13549:
13548:
13543:
13479:
13477:
13476:
13471:
13459:
13457:
13456:
13451:
13429:
13427:
13426:
13421:
13409:
13407:
13406:
13401:
13362:
13327:
13298:quasi-projective
13287:
13285:
13284:
13279:
13257:
13221:, the preimages
13196:
13106:pseudo-reductive
13097:
13095:
13094:
13089:
13078:
13077:
13054:
13052:
13051:
13046:
13024:
13022:
13021:
13016:
12973:complete variety
12936:
12934:
12933:
12928:
12917:
12916:
12901:
12900:
12899:
12898:
12893:
12892:
12877:
12851:
12800:
12798:
12797:
12792:
12780:
12778:
12777:
12772:
12769:
12764:
12759:
12746:
12744:
12743:
12738:
12726:
12724:
12723:
12718:
12706:
12704:
12703:
12698:
12696:
12695:
12682:
12680:
12679:
12674:
12672:
12671:
12666:
12665:
12651:
12649:
12648:
12643:
12641:
12640:
12613:
12611:
12610:
12605:
12589:
12587:
12586:
12581:
12569:
12567:
12566:
12561:
12556:
12555:
12554:
12549:
12530:
12529:
12524:
12514:
12509:
12504:
12491:projective space
12482:
12465:
12463:
12462:
12457:
12448:
12443:
12438:
12414:2. A
12407:1. A
12398:
12396:
12395:
12390:
12388:
12387:
12371:
12369:
12368:
12363:
12349:
12348:
12327:
12326:
12308:
12307:
12291:
12289:
12288:
12283:
12281:
12280:
12267:
12265:
12264:
12259:
12257:
12256:
12251:
12250:
12233:
12231:
12230:
12225:
12223:
12222:
12209:
12207:
12206:
12201:
12199:
12198:
12193:
12192:
12178:
12176:
12175:
12170:
12106:
12104:
12103:
12098:
12093:
12085:
12080:
12079:
12076:
12073:
12072:
12063:
12062:
12046:
12044:
12043:
12038:
12030:
12029:
11993:
11991:
11990:
11985:
11972:line at infinity
11970:complex points,
11961:
11959:
11958:
11953:
11951:
11950:
11931:
11929:
11928:
11923:
11907:
11905:
11904:
11899:
11875:
11873:
11872:
11867:
11844:
11842:
11841:
11836:
11820:
11818:
11817:
11812:
11810:
11802:
11793:
11791:
11790:
11785:
11773:
11771:
11770:
11765:
11760:
11752:
11747:
11746:
11743:
11729:
11727:
11726:
11721:
11709:
11707:
11706:
11701:
11690:
11689:
11686:
11676:
11674:
11673:
11668:
11649:
11647:
11646:
11641:
11629:
11627:
11626:
11621:
11609:
11607:
11606:
11601:
11589:
11587:
11586:
11581:
11569:
11567:
11566:
11561:
11546:
11544:
11543:
11538:
11522:
11520:
11519:
11514:
11488:
11486:
11485:
11480:
11459:Poincaré residue
11444:
11442:
11441:
11436:
11430:
11422:
11367:closed embedding
11319:-divisible group
11310:-divisible group
11279:
11277:
11276:
11271:
11269:
11268:
11263:
11257:
11256:
11251:
11250:
11201:
11172:
11170:
11169:
11164:
11159:
11158:
11131:
11130:
11071:normal crossings
11057:normal crossings
11048:
11046:
11045:
11040:
11032:
11031:
11022:
11005:
11004:
10992:
10991:
10967:
10965:
10964:
10959:
10954:
10953:
10938:
10933:
10932:
10922:
10917:
10902:
10901:
10858:
10856:
10855:
10850:
10845:
10844:
10839:
10838:
10828:
10827:
10818:
10807:
10806:
10805:
10804:
10799:
10798:
10783:
10782:
10773:
10772:
10763:
10762:
10753:
10691:
10689:
10688:
10683:
10681:
10680:
10675:
10650:-normal for all
10641:
10639:
10638:
10633:
10631:
10617:
10616:
10611:
10610:
10603:
10583:
10581:
10580:
10575:
10573:
10572:
10567:
10496:
10494:
10493:
10488:
10486:
10485:
10473:
10472:
10456:
10454:
10453:
10448:
10446:
10445:
10427:
10426:
10406:
10404:
10403:
10398:
10364:2. A
10357:1. A
10343:research program
10327:
10308:See for example
10259:
10257:
10256:
10251:
10249:
10248:
10230:
10229:
10177:
10175:
10174:
10169:
10157:
10155:
10154:
10149:
10137:
10135:
10134:
10129:
10124:
10121:
10112:
10110:
10109:
10104:
10096:
10093:
10088:
10087:
10068:
10066:
10065:
10060:
10055:
10052:
10043:
10041:
10040:
10035:
10019:
9919:
9917:
9916:
9911:
9882:Noetherian rings
9834:Zariski topology
9810:
9808:
9807:
9802:
9768:, i.e. a scheme
9673:Iitaka dimension
9627:
9625:
9624:
9619:
9587:Jacobian variety
9581:Jacobian variety
9569:
9551:projective space
9459:integral domains
9444:
9442:
9441:
9436:
9434:
9433:
9428:
9408:invertible sheaf
9375:
9373:
9372:
9367:
9362:
9341:
9339:
9338:
9333:
9328:
9295:
9293:
9292:
9287:
9285:
9284:
9279:
9278:
9271:
9270:
9258:
9257:
9252:
9251:
9241:
9240:
9210:closed immersion
9202:
9175:is not local on
9084:factors through
9075:
9055:factors through
9044:
9018:
8985:
8983:
8982:
8977:
8966:
8965:
8960:
8959:
8915:
8913:
8912:
8907:
8902:
8901:
8853:
8851:
8850:
8845:
8831:
8830:
8825:
8824:
8756:local cohomology
8743:
8741:
8740:
8735:
8676:
8674:
8673:
8668:
8586:
8584:
8583:
8578:
8576:
8564:
8562:
8561:
8556:
8554:
8538:Gorenstein rings
8532:1. A
8524:
8522:
8521:
8516:
8511:
8510:
8505:
8504:
8494:
8493:
8484:
8483:
8478:
8477:
8470:
8469:
8450:
8448:
8447:
8442:
8398:
8396:
8395:
8390:
8366:
8364:
8363:
8358:
8353:
8352:
8327:
8325:
8324:
8319:
8295:
8293:
8292:
8287:
8282:
8281:
8256:
8254:
8253:
8248:
8243:
8237:
8175:
8173:
8172:
8167:
8162:
8147:
8145:
8144:
8139:
8137:
8126:
8125:
8104:
8103:
8055:
8053:
8052:
8047:
8042:
8041:
8036:
8035:
8016:
8015:
7993:
7988:
7930:
7928:
7927:
7922:
7911:
7850:#geometric genus
7797:states a scheme
7756:
7754:
7753:
7748:
7737:
7725:
7723:
7722:
7717:
7703:
7702:
7687:
7686:
7626:
7624:
7623:
7618:
7607:
7606:
7587:
7585:
7584:
7579:
7567:
7545:
7543:
7542:
7537:
7479:
7445:
7443:
7442:
7437:
7426:
7425:
7420:
7419:
7401:
7399:
7398:
7393:
7382:
7381:
7376:
7375:
7293:
7291:
7290:
7285:
7263:
7241:
7239:
7238:
7233:
7221:
7219:
7218:
7213:
7201:
7199:
7198:
7193:
7188:
7185:
7176:
7174:
7173:
7168:
7160:
7157:
7152:
7151:
7132:
7130:
7129:
7124:
7119:
7116:
7107:
7105:
7104:
7099:
7083:
7069:
7067:
7066:
7061:
7049:
7047:
7046:
7041:
7029:
7027:
7026:
7021:
7016:
7013:
7004:
7002:
7001:
6996:
6988:
6985:
6980:
6979:
6960:
6958:
6957:
6952:
6947:
6944:
6935:
6933:
6932:
6927:
6911:
6885:
6883:
6882:
6877:
6865:
6863:
6862:
6857:
6845:
6843:
6842:
6837:
6832:
6829:
6820:
6818:
6817:
6812:
6804:
6801:
6796:
6795:
6776:
6774:
6773:
6768:
6763:
6760:
6751:
6749:
6748:
6743:
6727:
6705:
6703:
6702:
6697:
6692:
6684:
6683:
6680:
6667:
6665:
6664:
6659:
6647:
6645:
6644:
6639:
6637:
6601:
6599:
6598:
6593:
6555:
6540:
6538:
6537:
6532:
6530:
6507:
6480:
6453:
6451:
6450:
6445:
6431:
6423:
6366:
6364:
6363:
6358:
6316:
6314:
6313:
6308:
6303:
6302:
6260:
6258:
6257:
6252:
6234:
6232:
6231:
6226:
6215:
6214:
6187:
6185:
6184:
6179:
6156:
6127:
6126:
6099:
6097:
6096:
6091:
6059:
6057:
6056:
6051:
6048:
6040:
5969:, is called the
5958:
5956:
5955:
5950:
5939:
5938:
5933:
5921:
5920:
5890:
5889:
5888:
5887:
5882:
5875:
5874:
5864:
5863:
5862:
5861:
5856:
5849:
5848:
5817:étale cohomology
5810:
5808:
5807:
5802:
5797:
5796:
5772:
5771:
5742:
5740:
5739:
5734:
5722:
5720:
5719:
5714:
5698:
5667:projective curve
5605:
5603:
5602:
5597:
5595:
5594:
5573:
5572:
5551:
5550:
5532:
5531:
5505:
5503:
5502:
5497:
5474:On a projective
5464:Coherent duality
5415:
5390:2. A
5365:divisorial sheaf
5363:1. A
5323:
5321:
5320:
5315:
5313:
5290:
5289:
5268:
5267:
5247:
5245:
5244:
5239:
5202:
5200:
5199:
5194:
5189:
5188:
5164:
5163:
5154:
5152:
5141:
5133:
5132:
5086:algebraic curves
5080:in dimension 0:
5074:Global dimension
5059:
5057:
5056:
5051:
5036:2. A
5032:
5030:
5029:
5024:
5022:
5021:
4995:
4993:
4992:
4987:
4985:
4984:
4979:
4950:
4948:
4947:
4942:
4940:
4939:
4907:
4905:
4904:
4899:
4864:
4835:
4834:
4817:degeneracy locus
4806:
4804:
4803:
4798:
4771:degeneracy locus
4730:
4728:
4727:
4722:
4720:
4676:
4674:
4673:
4668:
4666:
4665:
4653:
4652:
4643:
4642:
4626:
4624:
4623:
4618:
4594:crepant morphism
4568:See for example
4564:compactification
4546:projective space
4457:
4418:
4416:
4415:
4410:
4405:
4404:
4399:
4387:
4382:
4381:
4376:
4375:
4341:
4339:
4338:
4333:
4328:
4323:
4322:
4317:
4316:
4291:
4289:
4288:
4283:
4281:
4280:
4275:
4274:
4118:
4116:
4115:
4110:
4084:
4082:
4081:
4076:
4058:
4056:
4055:
4050:
4048:
4047:
4042:
4021:
4019:
4018:
4013:
3995:
3993:
3992:
3987:
3910:
3908:
3907:
3902:
3891:
3890:
3860:A special fiber.
3833:
3831:
3830:
3825:
3799:
3798:
3789:
3788:
3762:
3760:
3759:
3754:
3745:
3740:
3735:
3638:
3636:
3635:
3630:
3628:
3627:
3604:
3602:
3601:
3596:
3594:
3593:
3578:
3577:
3565:
3564:
3559:
3558:
3540:
3538:
3537:
3532:
3530:
3529:
3495:
3493:
3492:
3487:
3484:
3479:
3452:
3450:
3449:
3444:
3441:
3436:
3427:
3426:
3414:
3413:
3354:
3352:
3351:
3346:
3344:
3343:
3327:
3325:
3324:
3319:
3307:
3305:
3304:
3299:
3297:
3296:
3288:
3273:
3272:
3253:
3251:
3250:
3245:
3237:
3236:
3228:
3204:
3202:
3201:
3196:
3150:
3148:
3147:
3142:
3133:
3132:
3123:
3115:
3114:
3086:
3008:Behrend function
2994:Behrend function
2962:
2960:
2959:
2954:
2940:
2939:
2934:
2933:
2917:
2916:
2882:arithmetic genus
2876:arithmetic genus
2867:
2865:
2864:
2859:
2857:
2806:
2804:
2803:
2798:
2796:
2791:
2790:
2789:
2779:
2767:
2765:
2764:
2759:
2757:
2746:
2745:
2744:
2734:
2712:
2710:
2709:
2704:
2648:
2646:
2645:
2640:
2597:algebraic scheme
2576:
2574:
2535:
2506:finite morphisms
2430:
2428:
2427:
2422:
2420:
2419:
2407:
2406:
2390:
2388:
2387:
2382:
2380:
2379:
2374:
2359:
2358:
2343:
2342:
2315:
2313:
2312:
2307:
2305:
2304:
2299:
2281:
2280:
2275:
2274:
2264:
2263:
2248:
2247:
2227:
2225:
2224:
2219:
2217:
2216:
2204:
2203:
2161:
2159:
2158:
2153:
2151:
2150:
2145:
2132:
2130:
2129:
2124:
2112:
2110:
2109:
2104:
2102:
2101:
2093:
2087:
2082:
2081:
2076:
1949:
1947:
1946:
1941:
1939:
1927:
1925:
1924:
1919:
1917:
1905:
1903:
1902:
1897:
1895:
1844:
1842:
1841:
1836:
1828:
1827:
1761:
1740:projectivization
1722:
1720:
1719:
1714:
1696:
1691:
1671:
1669:
1668:
1663:
1645:
1640:
1598:
1596:
1595:
1590:
1572:
1567:
1548:
1546:
1545:
1540:
1537:
1532:
1512:
1510:
1509:
1504:
1501:
1496:
1470:is the sheaf of
1469:
1467:
1466:
1461:
1458:
1453:
1434:
1432:
1431:
1426:
1423:
1418:
1395:
1393:
1392:
1387:
1385:
1384:
1379:
1378:
1364:
1362:
1361:
1356:
1345:
1344:
1339:
1338:
1324:
1322:
1321:
1316:
1314:
1280:
1278:
1277:
1272:
1261:
1260:
1255:
1254:
1229:
1227:
1226:
1221:
1194:is the image of
1193:
1191:
1190:
1185:
1174:
1173:
1168:
1167:
1131:
1129:
1128:
1123:
1112:
1111:
1106:
1105:
1088:
1086:
1085:
1080:
1066:
1065:
1060:
1059:
1036:
1034:
1033:
1028:
1017:
1016:
1011:
1010:
993:
991:
990:
985:
974:
973:
968:
967:
943:
941:
940:
935:
921:
920:
915:
914:
897:
895:
894:
889:
887:
886:
881:
880:
858:
856:
855:
850:
848:
847:
842:
841:
781:
779:
778:
773:
768:
762:
697:
695:
694:
689:
672:
671:
666:
665:
643:
635:
627:
560:
558:
557:
552:
538:
537:
532:
531:
524:
523:
522:
521:
516:
515:
474:
472:
471:
466:
464:
463:
458:
457:
433:
431:
430:
425:
411:
410:
405:
404:
397:
396:
395:
394:
389:
388:
351:
349:
348:
343:
341:
340:
335:
334:
316:
314:
313:
308:
297:
296:
291:
290:
239:
237:
236:
231:
69:
21:
20462:
20461:
20457:
20456:
20455:
20453:
20452:
20451:
20427:
20426:
20388:
20350:
20292:Dieudonné, Jean
20250:Dieudonné, Jean
20208:Dieudonné, Jean
20166:Dieudonné, Jean
20124:Dieudonné, Jean
20082:Dieudonné, Jean
20040:Dieudonné, Jean
19998:Dieudonné, Jean
19979:
19959:Springer-Verlag
19951:Fulton, William
19946:
19941:
19928:
19924:
19916:
19912:
19908:, II.5.5.4(ii).
19904:
19900:
19894:Hartshorne 1977
19892:
19888:
19871:
19867:
19859:
19855:
19847:
19843:
19837:Hartshorne 1977
19835:
19828:
19820:
19816:
19808:
19804:
19798:Hartshorne 1977
19796:
19792:
19775:
19771:
19763:
19759:
19751:
19747:
19730:
19726:
19718:
19714:
19697:
19693:
19676:
19672:
19655:
19651:
19634:
19630:
19617:
19568:
19564:
19560:
19546:
19541:
19527:
19515:
19507:
19502:
19468:
19460:
19455:
19417:
19416:
19411:
19399:
19393:
19392:
19391:
19382:
19381:
19376:
19355:
19349:
19348:
19347:
19345:
19342:
19341:
19318:
19312:
19311:
19310:
19308:
19305:
19304:
19287:
19286:
19284:
19281:
19280:
19264:
19261:
19260:
19244:
19241:
19240:
19214:
19211:
19210:
19209:. The morphism
19187:
19181:
19180:
19179:
19177:
19174:
19173:
19156:
19155:
19153:
19150:
19149:
19126:
19120:
19119:
19118:
19109:
19108:
19099:
19095:
19086:
19085:
19083:
19080:
19079:
19047:
19041:
19040:
19039:
19037:
19034:
19033:
19016:
19015:
19013:
19010:
19009:
18986:
18980:
18979:
18978:
18954:
18948:
18947:
18946:
18937:
18933:
18931:
18928:
18927:
18911:
18908:
18907:
18891:
18888:
18887:
18883:
18867:
18846:
18840:
18839:
18838:
18829:
18823:
18822:
18821:
18813:
18810:
18809:
18782:
18776:
18775:
18774:
18765:
18759:
18758:
18757:
18755:
18752:
18751:
18730:
18724:
18723:
18722:
18720:
18717:
18716:
18706:universal curve
18658:
18653:
18630:
18625:
18624:
18622:
18619:
18618:
18607:
18595:
18583:
18572:
18571:torus embedding
18548:
18515:
18509:
18508:
18507:
18505:
18502:
18501:
18475:
18469:
18468:
18467:
18465:
18462:
18461:
18457:is the dual of
18446:
18434:
18429:
18416:symmetric space
18414:An analog of a
18411:
18403:
18383:
18357:
18356:
18342:
18341:
18339:
18336:
18335:
18308:
18304:
18302:
18299:
18298:
18266:
18265:
18263:
18260:
18259:
18220:
18217:
18216:
18215:and a morphism
18184:
18183:
18175:
18172:
18171:
18167:
18155:
18128:
18104:
18102:
18099:
18098:
18082:
18074:
18071:
18070:
18044:
18041:
18040:
18018:
18015:
18014:
17991:
17988:
17987:
17971:
17968:
17967:
17948:
17945:
17944:
17921:
17917:
17915:
17912:
17911:
17888:
17884:
17882:
17879:
17878:
17855:
17851:
17846:
17840:
17836:
17834:
17831:
17830:
17814:
17811:
17810:
17781:
17777:
17762:
17758:
17749:
17745:
17737:
17734:
17733:
17710:
17707:
17706:
17690:
17687:
17686:
17671:is reductive a
17652:
17649:
17648:
17626:
17623:
17622:
17619:split-reductive
17602:
17599:
17598:
17582:
17579:
17578:
17576:reductive group
17555:
17544:
17538:
17535:
17534:
17518:
17515:
17514:
17497:
17493:
17491:
17488:
17487:
17466:
17452:
17449:
17448:
17428:
17425:
17424:
17407:
17396:
17390:
17387:
17386:
17366:
17363:
17362:
17345:
17334:
17327:
17321:
17318:
17317:
17296:
17294:
17291:
17290:
17268:
17264:
17262:
17259:
17258:
17238:
17235:
17234:
17218:
17215:
17214:
17195:
17192:
17191:
17175:
17172:
17171:
17155:
17152:
17151:
17135:
17132:
17131:
17110:
17108:
17105:
17104:
17088:
17085:
17084:
17068:
17065:
17064:
17048:
17045:
17044:
17028:
17025:
17024:
17022:algebraic group
17017:
16993:
16954:
16953:
16944:
16940:
16938:
16935:
16934:
16918:
16893:
16887:
16883:
16878:
16875:
16874:
16813:
16812:
16804:
16801:
16800:
16768:
16767:
16744:
16738:
16734:
16716:
16715:
16711:
16709:
16706:
16705:
16676:
16675:
16667:
16664:
16663:
16637:
16636:
16634:
16631:
16630:
16554:
16547:
16545:smooth morphism
16538:
16504:
16492:
16458:
16455:
16454:
16438:
16435:
16434:
16414:
16411:
16410:
16402:
16391:
16376:
16375:
16344:
16340:
16322:
16317:
16311:
16308:
16307:
16291:
16270:
16265:
16264:
16262:
16259:
16258:
16237:
16232:
16231:
16223:
16220:
16219:
16211:
16187:
16182:
16181:
16173:
16170:
16169:
16146:
16141:
16140:
16138:
16135:
16134:
16115:
16112:
16111:
16096:
16047:
16044:
16043:
16031:
16007:
16003:
15998:
15982:
15964:
15955:(and using the
15928:
15923:
15922:
15921:
15917:
15908:
15903:
15902:
15890:
15886:
15884:
15881:
15880:
15879:is rigid since
15868:
15810:
15798:
15794:
15785:
15781:
15755:
15751:
15750:
15746:
15737:
15733:
15731:
15728:
15727:
15686:
15682:
15680:
15677:
15676:
15656:
15653:
15652:
15649:Chern character
15632:
15629:
15628:
15611:
15607:
15601:
15597:
15591:
15587:
15572:
15559:
15555:
15553:
15550:
15549:
15493:
15489:
15456:
15452:
15441:
15438:
15437:
15390:
15387:
15386:
15328:
15325:
15324:
15296:
15267:
15263:
15259:
15257:
15254:
15253:
15226:
15222:
15218:
15204:
15198:
15197:
15196:
15195:
15191:
15179:
15167:
15164:
15163:
15143:
15140:
15139:
15116:
15112:
15106:
15102:
15093:
15089:
15087:
15084:
15083:
15057:
15053:
15038:
14963:
14960:
14959:
14940:tamely ramified
14923:
14920:
14919:
14891:
14888:
14887:
14883:
14843:
14840:
14839:
14824:
14800:
14796:
14791:
14788:
14787:
14761:
14758:
14757:
14735:
14732:
14731:
14727:
14711:
14686:
14685:
14683:
14680:
14679:
14662:
14656:
14655:
14654:
14649:
14643:
14642:
14640:
14637:
14636:
14576:
14573:
14572:
14565:
14523:
14511:
14510:reflexive sheaf
14486:
14484:
14481:
14480:
14464:
14461:
14460:
14438:
14434:
14432:
14429:
14428:
14403:
14399:
14390:
14386:
14384:
14381:
14380:
14377:reductive group
14360:
14357:
14356:
14340:
14337:
14336:
14326:
14306:
14303:
14302:
14276:
14270:
14269:
14268:
14266:
14263:
14262:
14240:
14237:
14236:
14213:
14207:
14206:
14205:
14203:
14200:
14199:
14168:
14165:
14164:
14148:
14145:
14144:
14108:
14106:
14103:
14102:
14082:
14079:
14078:
14074:
14027:
14026:
14020:
14019:
14018:
14009:
14005:
13999:
13995:
13993:
13990:
13989:
13972:
13966:
13965:
13964:
13948:
13947:
13941:
13940:
13939:
13930:
13926:
13924:
13921:
13920:
13895:
13887:
13884:
13883:
13871:
13839:
13835:
13811:
13807:
13798:
13794:
13763:
13758:
13757:
13748:
13743:
13742:
13740:
13737:
13736:
13728:
13676:
13659:
13656:
13655:
13648:
13610:
13607:
13606:
13595:-rational point
13559:
13554:
13553:
13544:
13539:
13538:
13536:
13533:
13532:
13525:rational curves
13516:
13511:
13497:
13485:
13465:
13462:
13461:
13439:
13436:
13435:
13415:
13412:
13411:
13395:
13392:
13391:
13389:reductive group
13384:
13369:quasi-separated
13358:
13345:
13331:quasi-separated
13315:
13311:
13310:quasi-separated
13299:
13267:
13264:
13263:
13245:
13241:
13230:
13219:
13212:
13184:
13180:
13172:
13155:
13150:
13132:
13111:Pseudoreductive
13107:
13073:
13069:
13064:
13061:
13060:
13034:
13031:
13030:
12998:
12995:
12994:
12986:
12958:
12946:
12912:
12908:
12894:
12888:
12887:
12886:
12885:
12881:
12864:
12847:
12845:
12842:
12841:
12806:
12786:
12783:
12782:
12765:
12760:
12755:
12752:
12749:
12748:
12732:
12729:
12728:
12712:
12709:
12708:
12691:
12690:
12688:
12685:
12684:
12667:
12661:
12660:
12659:
12657:
12654:
12653:
12636:
12635:
12633:
12630:
12629:
12626:
12599:
12596:
12595:
12575:
12572:
12571:
12550:
12536:
12535:
12531:
12525:
12520:
12519:
12510:
12505:
12500:
12497:
12494:
12493:
12470:
12444:
12439:
12434:
12431:
12428:
12427:
12404:
12383:
12379:
12377:
12374:
12373:
12344:
12340:
12322:
12318:
12303:
12299:
12297:
12294:
12293:
12276:
12275:
12273:
12270:
12269:
12252:
12246:
12245:
12244:
12242:
12239:
12238:
12218:
12217:
12215:
12212:
12211:
12194:
12188:
12187:
12186:
12184:
12181:
12180:
12179:of schemes, an
12152:
12149:
12148:
12140:
12128:
12120:
12084:
12075:
12074:
12068:
12064:
12058:
12054:
12052:
12049:
12048:
12025:
12021:
12019:
12016:
12015:
12012:geometric fiber
11979:
11976:
11975:
11946:
11942:
11940:
11937:
11936:
11917:
11914:
11913:
11893:
11890:
11889:
11861:
11858:
11857:
11830:
11827:
11826:
11801:
11799:
11796:
11795:
11779:
11776:
11775:
11751:
11742:
11741:
11739:
11736:
11735:
11715:
11712:
11711:
11685:
11684:
11682:
11679:
11678:
11662:
11659:
11658:
11655:geometric point
11635:
11632:
11631:
11615:
11612:
11611:
11595:
11592:
11591:
11575:
11572:
11571:
11555:
11552:
11551:
11532:
11529:
11528:
11524:are threefold:
11508:
11505:
11504:
11474:
11471:
11470:
11466:
11454:
11423:
11418:
11394:
11391:
11390:
11378:
11358:
11334:
11326:
11311:
11303:
11285:
11264:
11259:
11258:
11252:
11246:
11245:
11244:
11242:
11239:
11238:
11189:
11185:
11180:
11151:
11147:
11123:
11119:
11102:
11099:
11098:
11078:
11058:
11027:
11023:
11018:
11000:
10999:
10987:
10983:
10981:
10978:
10977:
10943:
10939:
10934:
10928:
10924:
10918:
10913:
10897:
10893:
10891:
10888:
10887:
10840:
10834:
10833:
10832:
10823:
10819:
10814:
10800:
10794:
10793:
10792:
10791:
10787:
10778:
10777:
10768:
10764:
10758:
10754:
10749:
10741:
10738:
10737:
10713:
10676:
10671:
10670:
10662:
10659:
10658:
10627:
10612:
10606:
10605:
10604:
10599:
10597:
10594:
10593:
10568:
10563:
10562:
10554:
10551:
10550:
10533:
10521:
10514:nef line bundle
10509:
10504:
10481:
10477:
10468:
10464:
10462:
10459:
10458:
10441:
10437:
10422:
10418:
10416:
10413:
10412:
10380:
10377:
10376:
10354:
10334:
10329:
10322:Kollár, János,
10321:
10305:
10300:
10282:
10270:
10265:
10244:
10240:
10225:
10221:
10216:
10213:
10212:
10191:
10183:
10163:
10160:
10159:
10143:
10140:
10139:
10120:
10118:
10115:
10114:
10092:
10080:
10076:
10074:
10071:
10070:
10051:
10049:
10046:
10045:
10029:
10026:
10025:
10007:
10003:
9991:
9979:
9972:valuation rings
9959:
9943:
9924:
9905:
9902:
9901:
9864:
9849:
9822:
9815:
9796:
9793:
9792:
9785:
9766:local in nature
9761:
9756:/vector bundle.
9749:
9738:
9737:level structure
9726:
9721:
9707:
9695:
9664:
9652:
9640:
9635:
9601:
9598:
9597:
9582:
9577:
9532:
9522:is irreducible
9501:
9450:
9429:
9424:
9423:
9421:
9418:
9417:
9409:
9397:
9358:
9347:
9344:
9343:
9324:
9313:
9310:
9309:
9280:
9274:
9273:
9272:
9266:
9262:
9253:
9247:
9246:
9245:
9236:
9232:
9230:
9227:
9226:
9190:
9184:
9063:
9032:
9006:
9002:
8997:
8961:
8955:
8954:
8953:
8951:
8948:
8947:
8940:
8934:
8921:
8897:
8893:
8879:
8876:
8875:
8859:
8826:
8820:
8819:
8818:
8810:
8807:
8806:
8794:
8789:
8779:
8763:
8748:
8690:
8687:
8686:
8626:
8623:
8622:
8614:
8595:
8572:
8570:
8567:
8566:
8550:
8548:
8545:
8544:
8529:
8506:
8500:
8499:
8498:
8489:
8485:
8479:
8473:
8472:
8471:
8465:
8461:
8456:
8453:
8452:
8424:
8421:
8420:
8404:
8372:
8369:
8368:
8348:
8344:
8333:
8330:
8329:
8301:
8298:
8297:
8277:
8273:
8262:
8259:
8258:
8239:
8233:
8228:
8225:
8224:
8217:
8208:is (roughly) a
8201:
8181:
8158:
8153:
8150:
8149:
8127:
8115:
8111:
8099:
8095:
8093:
8090:
8089:
8073:
8065:
8064:geometric point
8037:
8031:
8030:
8029:
8011:
8007:
7989:
7984:
7960:
7957:
7956:
7945:geometric genus
7940:
7939:geometric genus
7907:
7869:
7866:
7865:
7857:
7841:
7833:
7825:
7790:
7784:
7760:
7733:
7731:
7728:
7727:
7698:
7697:
7682:
7678:
7670:
7667:
7666:
7654:
7653:
7646:
7632:
7599:
7595:
7593:
7590:
7589:
7573:
7570:
7569:
7555:
7531:
7528:
7527:
7523:
7514:parametrizes a
7507:
7467:
7446:. The morphism
7421:
7415:
7414:
7413:
7411:
7408:
7407:
7377:
7371:
7370:
7369:
7367:
7364:
7363:
7305:
7273:
7270:
7269:
7251:
7247:
7227:
7224:
7223:
7207:
7204:
7203:
7184:
7182:
7179:
7178:
7156:
7144:
7140:
7138:
7135:
7134:
7115:
7113:
7110:
7109:
7093:
7090:
7089:
7071:
7055:
7052:
7051:
7035:
7032:
7031:
7012:
7010:
7007:
7006:
6984:
6972:
6968:
6966:
6963:
6962:
6943:
6941:
6938:
6937:
6921:
6918:
6917:
6899:
6895:
6888:finite morphism
6871:
6868:
6867:
6851:
6848:
6847:
6828:
6826:
6823:
6822:
6800:
6788:
6784:
6782:
6779:
6778:
6777:such that each
6759:
6757:
6754:
6753:
6737:
6734:
6733:
6715:
6711:
6688:
6679:
6678:
6676:
6673:
6672:
6653:
6650:
6649:
6633:
6631:
6628:
6627:
6615:
6548:
6546:
6543:
6542:
6523:
6500:
6486:
6483:
6482:
6478:
6422:
6408:
6405:
6404:
6322:
6319:
6318:
6298:
6294:
6289:
6286:
6285:
6274:
6240:
6237:
6236:
6210:
6206:
6201:
6198:
6197:
6152:
6119:
6115:
6113:
6110:
6109:
6073:
6070:
6069:
6065:
6041:
6036:
6030:
6027:
6026:
6011:
5999:
5991:
5978:
5934:
5929:
5928:
5901:
5897:
5883:
5878:
5877:
5876:
5870:
5869:
5868:
5857:
5852:
5851:
5850:
5844:
5843:
5842:
5834:
5831:
5830:
5824:
5783:
5779:
5767:
5763:
5752:
5749:
5748:
5728:
5725:
5724:
5708:
5705:
5704:
5686:
5682:
5674:
5658:
5636:
5629:
5620:canonical sheaf
5590:
5586:
5568:
5564:
5546:
5542:
5521:
5517:
5515:
5512:
5511:
5491:
5488:
5487:
5484:dualizing sheaf
5471:
5470:dualizing sheaf
5459:
5422:, if the image
5403:
5399:
5375:
5360:
5336:
5309:
5285:
5281:
5263:
5259:
5257:
5254:
5253:
5215:
5212:
5211:
5178:
5174:
5159:
5155:
5145:
5140:
5125:
5121:
5113:
5110:
5109:
5097:
5065:
5045:
5042:
5041:
5017:
5013:
5011:
5008:
5007:
4980:
4975:
4974:
4960:
4957:
4956:
4935:
4931:
4929:
4926:
4925:
4913:
4860:
4830:
4826:
4824:
4821:
4820:
4807:over a variety
4780:
4777:
4776:
4772:
4713:
4705:
4702:
4701:
4697:
4692:
4682:
4661:
4657:
4648:
4644:
4638:
4634:
4632:
4629:
4628:
4600:
4597:
4596:
4589:
4577:
4565:
4500:
4493:algebraic curve
4488:
4482:
4463:
4437:
4426:sheaf of ideals
4400:
4395:
4394:
4383:
4377:
4371:
4370:
4369:
4355:
4352:
4351:
4324:
4318:
4312:
4311:
4310:
4308:
4305:
4304:
4276:
4270:
4269:
4268:
4266:
4263:
4262:
4243:
4226:An analog of a
4223:
4187:here for curves
4098:
4095:
4094:
4087:Elliptic curves
4064:
4061:
4060:
4043:
4038:
4037:
4035:
4032:
4031:
4001:
3998:
3997:
3981:
3978:
3977:
3936:
3886:
3882:
3880:
3877:
3876:
3865:
3857:
3845:
3794:
3790:
3784:
3780:
3778:
3775:
3774:
3741:
3736:
3731:
3722:
3719:
3718:
3706:
3680:Cartier divisor
3675:
3664:canonical model
3659:
3658:canonical model
3652:canonical sheaf
3623:
3619:
3617:
3614:
3613:
3589:
3585:
3573:
3569:
3560:
3554:
3553:
3552:
3550:
3547:
3546:
3525:
3521:
3519:
3516:
3515:
3513:canonical class
3480:
3475:
3469:
3466:
3465:
3437:
3432:
3422:
3418:
3409:
3405:
3403:
3400:
3399:
3388:canonical sheaf
3383:
3371:
3366:
3339:
3335:
3333:
3330:
3329:
3313:
3310:
3309:
3287:
3286:
3265:
3261:
3259:
3256:
3255:
3227:
3226:
3218:
3215:
3214:
3184:
3181:
3180:
3168:
3156:
3128:
3124:
3119:
3110:
3106:
3076:
3069:
3066:
3065:
3051:big line bundle
3046:
3023:
2995:
2990:
2980:
2973:algebraic stack
2968:
2935:
2929:
2928:
2927:
2912:
2908:
2897:
2894:
2893:
2877:
2853:
2851:
2848:
2847:
2840:
2828:
2816:
2809:integral domain
2792:
2785:
2784:
2780:
2775:
2773:
2770:
2769:
2753:
2740:
2739:
2735:
2730:
2722:
2719:
2718:
2686:
2683:
2682:
2670:
2659:algebraic space
2654:
2653:algebraic space
2622:
2619:
2618:
2606:
2598:
2587:algebraic group
2582:
2581:algebraic group
2559:. See also the
2552:
2541:
2537:
2533:
2494:
2448:
2415:
2411:
2402:
2398:
2396:
2393:
2392:
2375:
2370:
2369:
2354:
2350:
2338:
2334:
2332:
2329:
2328:
2300:
2295:
2294:
2276:
2270:
2269:
2268:
2259:
2255:
2243:
2239:
2237:
2234:
2233:
2212:
2208:
2199:
2195:
2193:
2190:
2189:
2174:
2146:
2141:
2140:
2138:
2135:
2134:
2118:
2115:
2114:
2094:
2089:
2088:
2083:
2077:
2072:
2071:
2069:
2066:
2065:
2062:abelian variety
2057:
2052:
2030:
2004:
1989:
1983:
1959:
1935:
1933:
1930:
1929:
1913:
1911:
1908:
1907:
1891:
1889:
1886:
1885:
1881:
1823:
1819:
1757:
1755:
1752:
1751:
1735:
1692:
1687:
1681:
1678:
1677:
1641:
1636:
1630:
1627:
1626:
1599:
1568:
1563:
1557:
1554:
1553:
1533:
1528:
1522:
1519:
1518:
1497:
1492:
1486:
1483:
1482:
1454:
1449:
1443:
1440:
1439:
1435:
1419:
1414:
1408:
1405:
1404:
1380:
1374:
1373:
1372:
1370:
1367:
1366:
1340:
1334:
1333:
1332:
1330:
1327:
1326:
1325:-divisor, then
1310:
1308:
1305:
1304:
1256:
1250:
1249:
1248:
1246:
1243:
1242:
1203:
1200:
1199:
1169:
1163:
1162:
1161:
1159:
1156:
1155:
1132:
1107:
1101:
1100:
1099:
1097:
1094:
1093:
1061:
1055:
1054:
1053:
1051:
1048:
1047:
1037:
1012:
1006:
1005:
1004:
1002:
999:
998:
969:
963:
962:
961:
959:
956:
955:
944:
916:
910:
909:
908:
906:
903:
902:
882:
876:
875:
874:
872:
869:
868:
843:
837:
836:
835:
827:
824:
823:
807:
782:
764:
758:
753:
750:
749:
730:
720:Cartier divisor
667:
661:
660:
659:
639:
631:
623:
621:
618:
617:
594:
533:
527:
526:
525:
517:
511:
510:
509:
508:
504:
484:
481:
480:
459:
453:
452:
451:
449:
446:
445:
406:
400:
399:
398:
390:
384:
383:
382:
381:
377:
357:
354:
353:
336:
330:
329:
328:
326:
323:
322:
292:
286:
285:
284:
282:
279:
278:
267:
240:
225:
222:
221:
217:
212:
211:
210:
209:
70:
23:
22:
15:
12:
11:
5:
20460:
20450:
20449:
20444:
20439:
20425:
20424:
20422:List of curves
20419:
20414:
20409:
20404:
20399:
20394:
20387:
20384:
20383:
20382:
20375:
20369:
20361:
20348:
20326:
20284:
20242:
20200:
20158:
20116:
20074:
20032:
19990:
19977:
19945:
19942:
19940:
19939:
19922:
19910:
19898:
19886:
19865:
19853:
19841:
19826:
19814:
19802:
19790:
19769:
19757:
19745:
19724:
19712:
19691:
19670:
19649:
19628:
19613:
19561:
19559:
19556:
19555:
19554:
19547:
19544:
19540:
19537:
19536:
19535:
19528:
19525:
19523:
19516:
19513:
19511:
19508:
19505:
19501:
19498:
19497:
19496:
19471:A line bundle
19469:
19466:
19464:
19461:
19458:
19454:
19451:
19450:
19449:
19420:
19414:
19408:
19405:
19402:
19396:
19390:
19385:
19379:
19373:
19370:
19367:
19364:
19361:
19358:
19352:
19327:
19324:
19321:
19315:
19290:
19268:
19248:
19218:
19196:
19193:
19190:
19184:
19159:
19135:
19132:
19129:
19123:
19117:
19112:
19107:
19102:
19098:
19094:
19089:
19065:
19062:
19059:
19056:
19053:
19050:
19044:
19019:
18995:
18992:
18989:
18983:
18977:
18972:
18969:
18966:
18963:
18960:
18957:
18951:
18945:
18940:
18936:
18915:
18895:
18884:
18881:
18879:
18868:
18865:
18863:
18849:
18843:
18837:
18832:
18826:
18820:
18817:
18791:
18788:
18785:
18779:
18773:
18768:
18762:
18733:
18727:
18713:
18663:moduli functor
18659:
18656:
18652:
18649:
18648:
18647:
18633:
18628:
18608:
18605:
18603:
18596:
18593:
18591:
18584:
18581:
18579:
18573:
18570:
18568:
18549:
18546:
18544:
18532:
18529:
18526:
18523:
18518:
18512:
18489:
18486:
18483:
18478:
18472:
18447:
18444:
18442:
18435:
18432:
18428:
18425:
18424:
18423:
18412:
18409:
18407:
18404:
18401:
18399:
18384:
18381:
18379:
18364:
18361:
18355:
18349:
18346:
18319:
18314:
18311:
18307:
18282:
18279:
18273:
18270:
18236:
18233:
18230:
18227:
18224:
18200:
18197:
18191:
18188:
18182:
18179:
18168:
18165:
18163:
18156:
18153:
18151:
18140:
18129:
18126:
18124:
18107:
18085:
18081:
18078:
18060:
18048:
18028:
18025:
18022:
18012:Borel subgroup
17995:
17975:
17964:
17952:
17930:
17927:
17924:
17920:
17897:
17894:
17891:
17887:
17864:
17861:
17858:
17854:
17849:
17843:
17839:
17818:
17798:
17795:
17792:
17789:
17784:
17780:
17776:
17773:
17770:
17765:
17761:
17757:
17752:
17748:
17744:
17741:
17714:
17694:
17676:
17656:
17636:
17633:
17630:
17606:
17586:
17572:
17558:
17553:
17550:
17547:
17543:
17522:
17500:
17496:
17473:
17470:
17465:
17462:
17459:
17456:
17432:
17410:
17405:
17402:
17399:
17395:
17370:
17348:
17341:
17338:
17333:
17330:
17326:
17303:
17300:
17275:
17272:
17267:
17242:
17222:
17211:
17199:
17179:
17159:
17139:
17117:
17114:
17092:
17072:
17052:
17032:
17018:
17015:
17013:
16994:
16991:
16989:
16977:
16974:
16971:
16968:
16965:
16962:
16957:
16952:
16947:
16943:
16919:
16916:
16914:
16900:
16897:
16890:
16886:
16882:
16863:
16840:
16839:
16838:
16826:
16820:
16817:
16811:
16808:
16784:
16781:
16775:
16772:
16766:
16763:
16760:
16756:
16753:
16750:
16747:
16741:
16737:
16733:
16730:
16723:
16720:
16714:
16689:
16683:
16680:
16674:
16671:
16644:
16641:
16624:
16543:Main article:
16541:1.
16539:
16536:
16534:
16519:
16508:
16505:
16502:
16500:
16493:
16490:
16488:
16462:
16442:
16429:such that the
16418:
16409:is a morphism
16403:
16400:
16398:
16392:
16389:
16387:
16377:
16371:
16366:
16364:
16352:
16347:
16343:
16339:
16336:
16333:
16330:
16325:
16320:
16316:
16292:
16289:
16287:
16273:
16268:
16240:
16235:
16230:
16227:
16216:secant variety
16212:
16210:secant variety
16209:
16207:
16193:
16190:
16185:
16180:
16177:
16155:
16152:
16149:
16144:
16119:
16097:
16094:
16092:
16085:
16069:
16066:
16063:
16060:
16057:
16054:
16051:
16032:
16029:
16027:
16020:prime spectrum
16008:
16005:
15984:
15981:
15978:
15977:
15976:
15965:
15962:
15960:
15944:
15941:
15938:
15931:
15926:
15920:
15916:
15911:
15906:
15901:
15898:
15893:
15889:
15869:
15866:
15864:
15853:
15850:
15847:
15844:
15841:
15838:
15835:
15832:
15829:
15826:
15823:
15820:
15817:
15813:
15809:
15806:
15801:
15797:
15793:
15788:
15784:
15780:
15777:
15774:
15769:
15766:
15763:
15758:
15754:
15749:
15745:
15740:
15736:
15689:
15685:
15660:
15636:
15614:
15610:
15604:
15600:
15594:
15590:
15586:
15583:
15580:
15575:
15571:
15567:
15562:
15558:
15537:
15534:
15531:
15528:
15525:
15522:
15519:
15516:
15513:
15510:
15507:
15504:
15501:
15496:
15492:
15488:
15485:
15482:
15479:
15476:
15473:
15470:
15467:
15464:
15459:
15455:
15451:
15448:
15445:
15406:
15403:
15400:
15397:
15394:
15385:. It says: if
15379:
15359:
15356:
15353:
15350:
15347:
15344:
15341:
15338:
15335:
15332:
15297:
15294:
15292:
15274:
15270:
15266:
15262:
15241:
15238:
15233:
15229:
15225:
15221:
15217:
15214:
15207:
15201:
15194:
15188:
15185:
15182:
15178:
15174:
15171:
15147:
15127:
15124:
15119:
15115:
15109:
15105:
15101:
15096:
15092:
15071:
15068:
15065:
15060:
15056:
15052:
15047:
15044:
15041:
15037:
15033:
15030:
15027:
15024:
15021:
15018:
15015:
15012:
15009:
15006:
15003:
15000:
14997:
14994:
14991:
14988:
14985:
14982:
14979:
14976:
14973:
14970:
14967:
14927:
14907:
14904:
14901:
14898:
14895:
14884:
14881:
14879:
14859:
14856:
14853:
14850:
14847:
14825:
14822:
14820:
14808:
14803:
14799:
14795:
14782:from a scheme
14771:
14768:
14765:
14745:
14742:
14739:
14728:
14725:
14723:
14712:
14709:
14707:
14689:
14665:
14659:
14652:
14646:
14629:conormal sheaf
14592:
14589:
14586:
14583:
14580:
14566:
14563:
14561:
14528:regular scheme
14524:
14521:
14519:
14512:
14509:
14507:
14493:
14490:
14468:
14445:
14442:
14437:
14416:
14410:
14407:
14402:
14398:
14393:
14389:
14364:
14344:
14327:
14324:
14322:
14310:
14290:
14287:
14284:
14279:
14273:
14250:
14247:
14244:
14222:
14219:
14216:
14210:
14196:
14184:
14181:
14178:
14175:
14172:
14163:is reduced if
14152:
14132:
14129:
14126:
14123:
14118:
14115:
14112:
14086:
14075:
14072:
14070:
14058:
14055:
14052:
14048:
14045:
14042:
14039:
14033:
14030:
14023:
14017:
14012:
14008:
14002:
13998:
13975:
13969:
13963:
13960:
13954:
13951:
13944:
13938:
13933:
13929:
13908:
13905:
13901:
13898:
13894:
13891:
13872:
13869:
13867:
13847:
13842:
13838:
13834:
13831:
13828:
13825:
13820:
13817:
13814:
13810:
13806:
13801:
13797:
13793:
13790:
13787:
13784:
13781:
13778:
13775:
13771:
13766:
13761:
13756:
13751:
13746:
13729:
13726:
13724:
13700:
13697:
13694:
13691:
13688:
13683:
13680:
13675:
13672:
13669:
13666:
13663:
13653:function field
13649:
13646:
13644:
13632:
13629:
13626:
13623:
13620:
13617:
13614:
13576:
13562:
13557:
13552:
13547:
13542:
13517:
13514:
13510:
13507:
13506:
13505:
13502:quotient stack
13498:
13496:quotient stack
13495:
13493:
13486:
13483:
13481:
13469:
13449:
13446:
13443:
13419:
13399:
13385:
13382:
13380:
13354:
13312:
13309:
13307:
13300:
13297:
13295:
13277:
13274:
13271:
13242:
13239:
13237:
13228:
13217:
13210:
13181:
13178:
13176:
13168:
13156:
13154:quasi-coherent
13153:
13149:
13146:
13145:
13144:
13133:
13131:pure dimension
13130:
13128:
13108:
13105:
13103:
13087:
13084:
13081:
13076:
13072:
13068:
13044:
13041:
13038:
13014:
13011:
13008:
13005:
13002:
12987:
12984:
12982:
12961:A morphism is
12959:
12956:
12954:
12947:
12944:
12942:
12926:
12923:
12920:
12915:
12911:
12907:
12904:
12897:
12891:
12884:
12880:
12876:
12873:
12870:
12867:
12863:
12860:
12857:
12854:
12850:
12807:
12804:
12802:
12790:
12768:
12763:
12758:
12736:
12716:
12694:
12670:
12664:
12639:
12624:
12603:
12579:
12559:
12553:
12548:
12545:
12542:
12539:
12534:
12528:
12523:
12518:
12513:
12508:
12503:
12467:
12455:
12452:
12447:
12442:
12437:
12418:over a scheme
12412:
12405:
12402:
12400:
12386:
12382:
12361:
12358:
12355:
12352:
12347:
12343:
12339:
12336:
12333:
12330:
12325:
12321:
12317:
12314:
12311:
12306:
12302:
12279:
12255:
12249:
12221:
12197:
12191:
12168:
12165:
12162:
12159:
12156:
12141:
12138:
12136:
12129:
12126:
12124:
12121:
12118:
12116:
12109:affine schemes
12096:
12091:
12088:
12083:
12071:
12067:
12061:
12057:
12036:
12033:
12028:
12024:
12014:of a morphism
12008:fiber products
11983:
11949:
11945:
11921:
11910:Yoneda's lemma
11897:
11884:, not that of
11878:generic points
11865:
11847:
11846:
11834:
11808:
11805:
11783:
11763:
11758:
11755:
11750:
11719:
11699:
11696:
11693:
11666:
11651:
11639:
11619:
11599:
11579:
11559:
11548:
11536:
11512:
11478:
11467:
11464:
11462:
11455:
11452:
11450:
11434:
11429:
11426:
11421:
11417:
11413:
11410:
11407:
11404:
11401:
11398:
11379:
11376:
11374:
11359:
11356:
11354:
11351:tensor product
11335:
11332:
11330:
11327:
11324:
11322:
11312:
11306:
11302:
11299:
11298:
11297:
11286:
11283:
11281:
11267:
11262:
11255:
11249:
11227:open subscheme
11223:
11186:
11183:
11179:
11176:
11175:
11174:
11173:is surjective.
11162:
11157:
11154:
11150:
11146:
11143:
11140:
11137:
11134:
11129:
11126:
11122:
11118:
11115:
11112:
11109:
11106:
11089:is said to be
11081:A line bundle
11079:
11076:
11074:
11061:Abbreviations
11059:
11056:
11054:
11038:
11035:
11030:
11026:
11021:
11017:
11014:
11011:
11008:
11003:
10998:
10995:
10990:
10986:
10957:
10952:
10949:
10946:
10942:
10937:
10931:
10927:
10921:
10916:
10912:
10908:
10905:
10900:
10896:
10876:
10848:
10843:
10837:
10831:
10826:
10822:
10817:
10813:
10810:
10803:
10797:
10790:
10786:
10781:
10776:
10771:
10767:
10761:
10757:
10752:
10748:
10745:
10714:
10711:
10709:
10679:
10674:
10669:
10666:
10655:
10630:
10626:
10623:
10620:
10615:
10609:
10602:
10584:is said to be
10571:
10566:
10561:
10558:
10547:
10534:
10531:
10529:
10526:smooth variety
10522:
10519:
10517:
10510:
10507:
10503:
10500:
10499:
10498:
10484:
10480:
10476:
10471:
10467:
10444:
10440:
10436:
10433:
10430:
10425:
10421:
10396:
10393:
10390:
10387:
10384:
10373:
10362:
10355:
10352:
10350:
10335:
10332:
10330:
10313:
10306:
10303:
10299:
10296:
10295:
10294:
10283:
10280:
10278:
10271:
10268:
10266:
10263:
10261:
10247:
10243:
10239:
10236:
10233:
10228:
10224:
10220:
10189:
10184:
10181:
10179:
10167:
10147:
10127:
10102:
10099:
10091:
10086:
10083:
10079:
10058:
10033:
10004:
10001:
9999:
9992:
9989:
9987:
9980:
9977:
9975:
9960:
9957:
9955:
9944:
9941:
9939:
9922:
9909:
9862:
9857:affine schemes
9847:
9838:étale topology
9820:
9813:
9800:
9783:
9762:
9759:
9757:
9750:
9747:
9745:
9739:
9736:
9734:
9727:
9724:
9720:
9717:
9716:
9715:
9708:
9705:
9703:
9696:
9693:
9691:
9684:
9665:
9662:
9660:
9653:
9650:
9648:
9641:
9638:
9634:
9631:
9630:
9629:
9617:
9614:
9611:
9608:
9605:
9595:Picard variety
9583:
9580:
9576:
9573:
9572:
9571:
9530:
9508:is said to be
9502:
9499:
9497:
9463:disjoint union
9451:
9448:
9446:
9432:
9427:
9410:
9407:
9405:
9398:
9395:
9393:
9365:
9361:
9357:
9354:
9351:
9331:
9327:
9323:
9320:
9317:
9283:
9277:
9269:
9265:
9261:
9256:
9250:
9244:
9239:
9235:
9205:open immersion
9185:
9182:
9180:
9098:
9097:
9060:
9027:is the unique
9003:
9000:
8996:
8993:
8992:
8991:
8975:
8972:
8969:
8964:
8958:
8941:
8938:
8936:
8932:
8922:
8919:
8917:
8905:
8900:
8896:
8892:
8889:
8886:
8883:
8860:
8857:
8855:
8843:
8840:
8837:
8834:
8829:
8823:
8817:
8814:
8795:
8792:
8788:
8785:
8784:
8783:
8780:
8777:
8775:
8764:
8761:
8759:
8749:
8746:
8744:
8733:
8730:
8727:
8724:
8721:
8718:
8715:
8712:
8709:
8706:
8703:
8700:
8697:
8694:
8666:
8663:
8660:
8657:
8654:
8651:
8648:
8645:
8642:
8639:
8636:
8633:
8630:
8615:
8612:
8610:
8596:
8593:
8591:
8588:
8575:
8553:
8541:
8530:
8527:
8525:
8514:
8509:
8503:
8497:
8492:
8488:
8482:
8476:
8468:
8464:
8460:
8440:
8437:
8434:
8431:
8428:
8405:
8402:
8400:
8388:
8385:
8382:
8379:
8376:
8356:
8351:
8347:
8343:
8340:
8337:
8317:
8314:
8311:
8308:
8305:
8285:
8280:
8276:
8272:
8269:
8266:
8246:
8242:
8236:
8232:
8218:
8215:
8213:
8202:
8199:
8197:
8182:
8179:
8177:
8165:
8161:
8157:
8136:
8133:
8130:
8124:
8121:
8118:
8114:
8110:
8107:
8102:
8098:
8074:
8071:
8069:
8066:
8063:
8061:
8045:
8040:
8034:
8028:
8025:
8022:
8019:
8014:
8010:
8006:
8003:
8000:
7997:
7992:
7987:
7983:
7979:
7976:
7973:
7970:
7967:
7964:
7941:
7938:
7936:
7920:
7917:
7914:
7910:
7906:
7903:
7900:
7897:
7894:
7891:
7888:
7885:
7882:
7879:
7876:
7873:
7858:
7855:
7853:
7842:
7839:
7837:
7836:A dense point.
7834:
7831:
7829:
7826:
7823:
7821:
7791:
7788:
7786:
7782:
7765:has dimension
7758:
7746:
7743:
7740:
7736:
7715:
7712:
7709:
7706:
7701:
7696:
7693:
7690:
7685:
7681:
7677:
7674:
7657:Given a curve
7655:
7651:
7649:
7645:
7642:
7641:
7640:
7633:
7630:
7628:
7616:
7613:
7610:
7605:
7602:
7598:
7577:
7535:
7524:
7521:
7519:
7508:
7505:
7503:
7435:
7432:
7429:
7424:
7418:
7391:
7388:
7385:
7380:
7374:
7358:) ⊆
7312:is a point of
7306:
7303:
7301:
7283:
7280:
7277:
7248:
7245:
7243:
7231:
7211:
7191:
7166:
7163:
7155:
7150:
7147:
7143:
7122:
7097:
7086:of finite type
7059:
7039:
7019:
6994:
6991:
6983:
6978:
6975:
6971:
6950:
6925:
6896:
6893:
6891:
6886:-module. See
6875:
6855:
6835:
6810:
6807:
6799:
6794:
6791:
6787:
6766:
6741:
6712:
6709:
6707:
6695:
6691:
6687:
6657:
6636:
6616:
6613:
6611:
6591:
6588:
6585:
6582:
6579:
6576:
6573:
6570:
6567:
6564:
6561:
6558:
6554:
6551:
6529:
6526:
6522:
6519:
6516:
6513:
6510:
6506:
6503:
6499:
6496:
6493:
6490:
6443:
6440:
6437:
6434:
6429:
6426:
6421:
6418:
6415:
6412:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6335:
6332:
6329:
6326:
6306:
6301:
6297:
6293:
6282:
6275:
6272:
6270:
6250:
6247:
6244:
6224:
6221:
6218:
6213:
6209:
6205:
6177:
6174:
6171:
6168:
6165:
6162:
6159:
6155:
6151:
6148:
6145:
6142:
6139:
6136:
6133:
6130:
6125:
6122:
6118:
6089:
6086:
6083:
6080:
6077:
6066:
6063:
6061:
6047:
6044:
6039:
6035:
6012:
6009:
6007:
6000:
5994:
5990:
5987:
5986:
5985:
5979:
5976:
5974:
5971:Euler sequence
5960:
5959:
5948:
5945:
5942:
5937:
5932:
5927:
5924:
5919:
5916:
5913:
5910:
5907:
5904:
5900:
5896:
5893:
5886:
5881:
5873:
5867:
5860:
5855:
5847:
5841:
5838:
5825:
5823:Euler sequence
5822:
5820:
5813:étale topology
5800:
5795:
5792:
5789:
5786:
5782:
5778:
5775:
5770:
5766:
5762:
5759:
5756:
5732:
5712:
5683:
5680:
5678:
5675:
5672:
5670:
5663:elliptic curve
5659:
5657:elliptic curve
5656:
5654:
5637:
5632:
5628:
5625:
5624:
5623:
5593:
5589:
5585:
5582:
5579:
5576:
5571:
5567:
5563:
5560:
5557:
5554:
5549:
5545:
5541:
5538:
5535:
5530:
5527:
5524:
5520:
5495:
5472:
5469:
5467:
5460:
5457:
5455:
5400:
5397:
5395:
5388:
5371:
5361:
5358:
5356:
5337:
5334:
5332:
5325:
5312:
5308:
5305:
5302:
5299:
5296:
5293:
5288:
5284:
5280:
5277:
5274:
5271:
5266:
5262:
5237:
5234:
5231:
5228:
5225:
5222:
5219:
5204:
5192:
5187:
5184:
5181:
5177:
5173:
5170:
5167:
5162:
5158:
5151:
5148:
5144:
5139:
5136:
5131:
5128:
5124:
5120:
5117:
5098:
5095:
5093:
5066:
5063:
5061:
5049:
5034:
5020:
5016:
4983:
4978:
4973:
4970:
4967:
4964:
4938:
4934:
4914:
4911:
4909:
4897:
4894:
4891:
4888:
4885:
4882:
4879:
4876:
4873:
4870:
4867:
4863:
4859:
4856:
4853:
4850:
4847:
4844:
4841:
4838:
4833:
4829:
4796:
4793:
4790:
4787:
4784:
4773:
4770:
4768:
4719:
4716:
4712:
4709:
4698:
4695:
4691:
4688:
4687:
4686:
4683:
4680:
4678:
4664:
4660:
4656:
4651:
4647:
4641:
4637:
4616:
4613:
4610:
4607:
4604:
4590:
4587:
4585:
4578:
4575:
4573:
4566:
4563:
4561:
4538:connected ring
4503:The scheme is
4501:
4498:
4496:
4495:of degree two.
4489:
4486:
4484:
4478:
4468:coherent sheaf
4464:
4462:coherent sheaf
4461:
4459:
4442:Cohen-Macaulay
4438:
4436:Cohen–Macaulay
4435:
4433:
4423:quasi-coherent
4408:
4403:
4398:
4393:
4390:
4386:
4380:
4374:
4368:
4365:
4362:
4359:
4331:
4327:
4321:
4315:
4302:quotient sheaf
4279:
4273:
4259:quasi-coherent
4244:
4241:
4239:
4224:
4221:
4219:
4212:
4201:
4198:Fano varieties
4194:
4171:
4156:
4123:, also called
4108:
4105:
4102:
4074:
4071:
4068:
4046:
4041:
4011:
4008:
4005:
3985:
3955:2.
3953:
3941:Classification
3939:1.
3937:
3935:classification
3934:
3932:
3900:
3897:
3894:
3889:
3885:
3866:
3863:
3861:
3858:
3855:
3853:
3846:
3843:
3841:
3835:
3834:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3797:
3793:
3787:
3783:
3763:over a scheme
3752:
3749:
3744:
3739:
3734:
3729:
3726:
3707:
3704:
3702:
3676:
3673:
3671:
3660:
3657:
3655:
3644:canonical ring
3640:
3626:
3622:
3606:
3592:
3588:
3584:
3581:
3576:
3572:
3568:
3563:
3557:
3528:
3524:
3509:
3483:
3478:
3474:
3440:
3435:
3431:
3425:
3421:
3417:
3412:
3408:
3384:
3381:
3379:
3372:
3369:
3365:
3362:
3361:
3360:
3342:
3338:
3317:
3294:
3291:
3285:
3282:
3279:
3276:
3271:
3268:
3264:
3254:such that (1)
3243:
3240:
3234:
3231:
3225:
3222:
3194:
3191:
3188:
3169:
3166:
3164:
3157:
3154:
3152:
3139:
3136:
3131:
3127:
3122:
3118:
3113:
3109:
3105:
3102:
3099:
3096:
3093:
3090:
3085:
3082:
3079:
3075:
3074:lim sup
3047:
3044:
3042:
3024:
3021:
3019:
2996:
2993:
2989:
2986:
2985:
2984:
2981:
2978:
2976:
2969:
2966:
2964:
2952:
2949:
2946:
2943:
2938:
2932:
2926:
2923:
2920:
2915:
2911:
2907:
2904:
2901:
2878:
2875:
2873:
2856:
2841:
2838:
2836:
2829:
2826:
2824:
2817:
2814:
2812:
2795:
2788:
2783:
2778:
2756:
2752:
2749:
2743:
2738:
2733:
2729:
2726:
2702:
2699:
2696:
2693:
2690:
2671:
2668:
2666:
2655:
2652:
2650:
2638:
2635:
2632:
2629:
2626:
2607:
2604:
2602:
2599:
2596:
2594:
2583:
2580:
2578:
2553:
2550:
2548:
2542:
2539:
2525:
2524:
2509:
2502:vector bundles
2492:
2474:
2471:prime spectrum
2463:
2460:affine variety
2456:
2451:1.
2449:
2446:
2444:
2418:
2414:
2410:
2405:
2401:
2378:
2373:
2368:
2365:
2362:
2357:
2353:
2349:
2346:
2341:
2337:
2317:
2303:
2298:
2293:
2290:
2287:
2284:
2279:
2273:
2267:
2262:
2258:
2254:
2251:
2246:
2242:
2215:
2211:
2207:
2202:
2198:
2175:
2172:
2170:
2167:abelian scheme
2163:
2149:
2144:
2122:
2100:
2097:
2092:
2086:
2080:
2075:
2058:
2055:
2051:
2048:
2047:
2046:
2031:
2024:
2022:
2000:
1990:
1979:
1977:
1975:
1968:prime spectrum
1960:
1953:
1951:
1938:
1916:
1894:
1882:
1879:
1877:
1834:
1831:
1826:
1822:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1764:
1760:
1746:-vector space
1736:
1726:
1724:
1712:
1709:
1706:
1703:
1700:
1695:
1690:
1686:
1661:
1658:
1655:
1652:
1649:
1644:
1639:
1635:
1625:2.
1623:
1600:
1588:
1585:
1582:
1579:
1576:
1571:
1566:
1562:
1552:
1550:
1536:
1531:
1527:
1500:
1495:
1491:
1481:2.
1479:
1457:
1452:
1448:
1438:1.
1436:
1422:
1417:
1413:
1403:
1401:
1383:
1377:
1354:
1351:
1348:
1343:
1337:
1313:
1297:
1270:
1267:
1264:
1259:
1253:
1239:
1219:
1216:
1213:
1210:
1207:
1183:
1180:
1177:
1172:
1166:
1152:
1133:
1121:
1118:
1115:
1110:
1104:
1092:
1090:
1078:
1075:
1072:
1069:
1064:
1058:
1038:
1026:
1023:
1020:
1015:
1009:
997:
995:
983:
980:
977:
972:
966:
945:
933:
930:
927:
924:
919:
913:
901:
899:
885:
879:
846:
840:
834:
831:
808:
803:
801:
783:
771:
767:
761:
757:
748:
746:
735:quotient stack
731:
729:
727:
687:
684:
681:
678:
675:
670:
664:
658:
655:
652:
649:
646:
642:
638:
634:
630:
626:
595:
588:
586:
550:
547:
544:
541:
536:
530:
520:
514:
507:
503:
500:
497:
494:
491:
488:
462:
456:
434:
423:
420:
417:
414:
409:
403:
393:
387:
380:
376:
373:
370:
367:
364:
361:
352:-module, then
339:
333:
306:
303:
300:
295:
289:
268:
250:
248:
241:
229:
220:
216:
213:
208:
207:
202:
197:
192:
187:
182:
177:
172:
167:
162:
157:
152:
147:
142:
137:
132:
127:
122:
117:
112:
107:
102:
97:
92:
87:
82:
77:
71:
65:
63:
9:
6:
4:
3:
2:
20459:
20448:
20447:Scheme theory
20445:
20443:
20440:
20438:
20435:
20434:
20432:
20423:
20420:
20418:
20415:
20413:
20410:
20408:
20405:
20403:
20400:
20398:
20395:
20393:
20390:
20389:
20380:
20376:
20374:
20370:
20368:
20365:
20364:Kollár, János
20362:
20359:
20355:
20351:
20345:
20341:
20337:
20336:
20331:
20327:
20323:
20319:
20315:
20311:
20307:
20303:
20302:
20297:
20293:
20289:
20285:
20281:
20277:
20273:
20269:
20265:
20261:
20260:
20255:
20251:
20247:
20243:
20239:
20235:
20231:
20227:
20223:
20219:
20218:
20213:
20209:
20205:
20201:
20197:
20193:
20189:
20185:
20181:
20177:
20176:
20171:
20167:
20163:
20159:
20155:
20151:
20147:
20143:
20139:
20135:
20134:
20129:
20125:
20121:
20117:
20113:
20109:
20105:
20101:
20097:
20093:
20092:
20087:
20083:
20079:
20075:
20071:
20067:
20063:
20059:
20055:
20051:
20050:
20045:
20041:
20037:
20033:
20029:
20025:
20021:
20017:
20013:
20009:
20008:
20003:
19999:
19995:
19991:
19988:
19984:
19980:
19974:
19969:
19964:
19960:
19956:
19952:
19948:
19947:
19936:
19932:
19926:
19919:
19914:
19907:
19902:
19895:
19890:
19881:
19876:
19869:
19863:
19857:
19850:
19845:
19838:
19833:
19831:
19823:
19818:
19811:
19806:
19799:
19794:
19785:
19780:
19773:
19766:
19761:
19754:
19749:
19740:
19735:
19728:
19721:
19716:
19707:
19702:
19695:
19686:
19681:
19674:
19665:
19660:
19653:
19644:
19639:
19632:
19625:
19621:
19616:
19612:
19608:
19604:
19600:
19596:
19592:
19588:
19584:
19580:
19576:
19572:
19566:
19562:
19552:
19548:
19542:
19533:
19529:
19521:
19517:
19509:
19506:weakly normal
19503:
19494:
19490:
19486:
19482:
19478:
19475:on a variety
19474:
19470:
19462:
19456:
19447:
19443:
19439:
19436:
19412:
19406:
19403:
19400:
19377:
19368:
19362:
19359:
19356:
19325:
19322:
19319:
19266:
19246:
19238:
19234:
19233:
19216:
19194:
19191:
19188:
19133:
19130:
19127:
19096:
19092:
19060:
19054:
19051:
19048:
18993:
18990:
18987:
18967:
18961:
18958:
18955:
18943:
18934:
18913:
18893:
18885:
18877:
18873:
18869:
18847:
18830:
18818:
18815:
18807:
18789:
18786:
18783:
18771:
18766:
18749:
18731:
18714:
18711:
18707:
18703:
18699:
18695:
18692:(which is an
18691:
18687:
18683:
18679:
18675:
18671:
18667:
18664:
18660:
18654:
18631:
18617:
18613:
18609:
18601:
18597:
18589:
18588:toric variety
18585:
18582:toric variety
18578:
18577:toric variety
18574:
18566:
18562:
18558:
18554:
18550:
18527:
18524:
18516:
18484:
18476:
18460:
18456:
18452:
18448:
18440:
18436:
18433:tangent space
18430:
18421:
18417:
18413:
18405:
18397:
18393:
18389:
18385:
18362:
18359:
18347:
18344:
18333:
18317:
18312:
18309:
18305:
18296:
18280:
18271:
18268:
18258:
18254:
18250:
18234:
18228:
18225:
18222:
18214:
18198:
18189:
18186:
18180:
18177:
18169:
18161:
18157:
18149:
18145:
18141:
18138:
18134:
18130:
18122:
18079:
18076:
18068:
18065:
18061:
18046:
18039:defined over
18026:
18023:
18020:
18013:
18009:
17993:
17973:
17965:
17950:
17928:
17925:
17922:
17918:
17895:
17892:
17889:
17885:
17862:
17859:
17856:
17852:
17847:
17841:
17837:
17816:
17809:defined over
17793:
17787:
17782:
17778:
17774:
17771:
17768:
17763:
17759:
17755:
17750:
17746:
17742:
17739:
17732:
17728:
17712:
17692:
17684:
17681:
17677:
17674:
17670:
17654:
17647:defined over
17634:
17631:
17628:
17620:
17604:
17584:
17577:
17573:
17556:
17551:
17548:
17545:
17541:
17520:
17498:
17494:
17468:
17463:
17460:
17457:
17454:
17446:
17430:
17408:
17403:
17400:
17397:
17393:
17384:
17368:
17346:
17336:
17331:
17328:
17324:
17298:
17270:
17265:
17256:
17240:
17220:
17212:
17197:
17177:
17157:
17137:
17112:
17090:
17070:
17050:
17030:
17023:
17019:
17011:
17007:
17003:
16999:
16995:
16969:
16966:
16963:
16945:
16941:
16932:
16928:
16924:
16920:
16895:
16888:
16884:
16880:
16872:
16868:
16864:
16861:
16857:
16853:
16849:
16845:
16844:smooth scheme
16841:
16815:
16806:
16798:
16770:
16761:
16739:
16735:
16731:
16728:
16718:
16712:
16703:
16678:
16669:
16661:
16639:
16628:
16625:
16622:
16618:
16616:
16610:
16606:
16602:
16598:
16594:
16590:
16586:
16582:
16578:
16574:
16570:
16569:
16568:
16565:
16561:
16557:
16552:
16546:
16540:
16532:
16531:simple groups
16528:
16524:
16520:
16517:
16513:
16509:
16506:
16498:
16494:
16486:
16485:
16480:
16476:
16460:
16440:
16432:
16431:fiber product
16416:
16408:
16404:
16397:
16393:
16390:Serre duality
16386:
16382:
16378:
16374:
16370:
16345:
16341:
16337:
16334:
16318:
16314:
16305:
16301:
16297:
16293:
16271:
16256:
16238:
16228:
16225:
16217:
16213:
16191:
16188:
16178:
16175:
16153:
16150:
16147:
16133:
16117:
16110:
16106:
16105:ruled surface
16102:
16098:
16090:
16086:
16083:
16064:
16061:
16058:
16052:
16049:
16041:
16037:
16036:Schubert cell
16033:
16025:
16021:
16017:
16013:
16009:
16002:
16000:
15995:
15993:
15992:
15983:
15974:
15973:marked points
15970:
15966:
15958:
15942:
15939:
15929:
15918:
15914:
15909:
15896:
15891:
15878:
15874:
15870:
15851:
15848:
15845:
15842:
15836:
15830:
15827:
15824:
15815:
15811:
15804:
15799:
15795:
15786:
15782:
15778:
15775:
15764:
15756:
15752:
15747:
15738:
15734:
15725:
15721:
15717:
15713:
15709:
15705:
15687:
15683:
15674:
15658:
15650:
15634:
15612:
15608:
15602:
15598:
15592:
15584:
15581:
15573:
15569:
15565:
15560:
15556:
15529:
15523:
15520:
15517:
15511:
15505:
15502:
15494:
15490:
15486:
15480:
15474:
15471:
15468:
15462:
15457:
15453:
15446:
15443:
15436:
15432:
15428:
15424:
15420:
15404:
15398:
15395:
15392:
15384:
15380:
15377:
15373:
15357:
15354:
15351:
15348:
15345:
15342:
15336:
15330:
15322:
15318:
15315:computes the
15314:
15310:
15306:
15302:
15298:
15290:
15272:
15268:
15264:
15239:
15231:
15227:
15223:
15212:
15205:
15192:
15186:
15183:
15180:
15176:
15172:
15169:
15161:
15145:
15125:
15122:
15117:
15113:
15107:
15103:
15099:
15094:
15090:
15066:
15063:
15058:
15054:
15045:
15042:
15039:
15035:
15031:
15025:
15022:
15016:
15010:
15007:
14998:
14992:
14989:
14986:
14983:
14980:
14974:
14968:
14965:
14957:
14953:
14949:
14945:
14941:
14925:
14905:
14899:
14896:
14893:
14885:
14877:
14873:
14857:
14851:
14848:
14845:
14838:
14834:
14830:
14826:
14806:
14801:
14797:
14793:
14785:
14769:
14763:
14743:
14737:
14729:
14721:
14717:
14713:
14705:
14663:
14650:
14634:
14630:
14626:
14622:
14618:
14614:
14610:
14606:
14590:
14584:
14581:
14578:
14571:
14567:
14558:
14553:
14549:
14545:
14541:
14537:
14533:
14529:
14525:
14517:
14513:
14488:
14466:
14440:
14435:
14405:
14400:
14391:
14387:
14378:
14362:
14355:over a field
14342:
14335:
14332:
14328:
14308:
14285:
14277:
14248:
14245:
14242:
14220:
14217:
14214:
14197:
14179:
14173:
14170:
14150:
14127:
14121:
14113:
14100:
14084:
14076:
14056:
14053:
14050:
14046:
14043:
14040:
14031:
14028:
14010:
14006:
14000:
13996:
13973:
13961:
13952:
13949:
13931:
13927:
13906:
13899:
13896:
13892:
13889:
13881:
13877:
13873:
13865:
13864:twisted cubic
13861:
13840:
13836:
13832:
13829:
13826:
13823:
13818:
13815:
13812:
13808:
13804:
13799:
13795:
13782:
13779:
13776:
13769:
13764:
13749:
13734:
13730:
13722:
13718:
13714:
13695:
13689:
13686:
13681:
13678:
13673:
13667:
13661:
13654:
13650:
13630:
13621:
13615:
13612:
13604:
13600:
13596:
13594:
13589:
13585:
13581:
13577:
13560:
13550:
13545:
13530:
13526:
13522:
13518:
13512:
13503:
13499:
13491:
13487:
13467:
13460:defined over
13447:
13444:
13441:
13433:
13417:
13397:
13390:
13386:
13378:
13374:
13370:
13366:
13361:
13357:
13352:
13348:
13343:
13342:
13338:
13333:
13332:
13326:
13322:
13318:
13313:
13305:
13301:
13293:
13292:
13275:
13272:
13269:
13261:
13260:finite fibers
13256:
13252:
13248:
13244:The morphism
13243:
13235:
13234:quasi-compact
13231:
13224:
13220:
13213:
13206:
13202:
13201:
13200:quasi-compact
13195:
13191:
13187:
13182:
13179:quasi-compact
13174:
13171:
13167:
13161:
13157:
13151:
13142:
13138:
13134:
13126:
13123:
13120:
13116:
13112:
13109:
13101:
13098:has property
13085:
13079:
13074:
13070:
13066:
13058:
13042:
13036:
13028:
13012:
13006:
13003:
13000:
12992:
12988:
12980:
12979:
12974:
12970:
12966:
12965:
12960:
12952:
12948:
12940:
12924:
12913:
12909:
12902:
12895:
12882:
12861:
12855:
12839:
12835:
12831:
12827:
12823:
12820:
12816:
12812:
12808:
12788:
12766:
12761:
12734:
12714:
12668:
12627:
12620:
12619:
12601:
12593:
12577:
12557:
12532:
12526:
12516:
12511:
12506:
12492:
12488:
12487:
12481:
12477:
12473:
12468:
12453:
12445:
12440:
12425:
12421:
12417:
12413:
12410:
12406:
12384:
12380:
12359:
12356:
12350:
12345:
12341:
12334:
12328:
12323:
12319:
12315:
12312:
12304:
12300:
12253:
12237:
12195:
12166:
12160:
12157:
12154:
12146:
12142:
12134:
12130:
12122:
12114:
12110:
12086:
12069:
12065:
12055:
12034:
12022:
12013:
12009:
12005:
12004:inverse image
12001:
11997:
11981:
11973:
11969:
11965:
11947:
11943:
11935:
11919:
11911:
11895:
11887:
11883:
11879:
11863:
11855:
11851:
11832:
11824:
11803:
11781:
11753:
11733:
11717:
11694:
11664:
11656:
11652:
11637:
11617:
11597:
11577:
11557:
11549:
11534:
11526:
11525:
11523:
11510:
11500:
11496:
11492:
11476:
11468:
11460:
11456:
11448:
11427:
11424:
11419:
11415:
11411:
11408:
11399:
11396:
11388:
11384:
11380:
11372:
11368:
11364:
11360:
11352:
11348:
11344:
11340:
11336:
11328:
11320:
11318:
11313:
11309:
11304:
11295:
11291:
11287:
11265:
11253:
11236:
11232:
11228:
11224:
11221:
11217:
11213:
11209:
11205:
11200:
11196:
11192:
11187:
11181:
11155:
11152:
11148:
11144:
11141:
11127:
11124:
11116:
11113:
11110:
11096:
11092:
11088:
11085:on a variety
11084:
11080:
11072:
11068:
11064:
11060:
11052:
11028:
11024:
11019:
11015:
11009:
11006:
10993:
10988:
10984:
10975:
10971:
10950:
10947:
10944:
10940:
10935:
10929:
10925:
10914:
10910:
10903:
10898:
10894:
10885:
10881:
10877:
10874:
10873:normal bundle
10870:
10866:
10862:
10841:
10829:
10824:
10820:
10815:
10811:
10801:
10788:
10784:
10774:
10769:
10759:
10755:
10750:
10746:
10735:
10731:
10727:
10723:
10719:
10715:
10707:
10703:
10702:normal scheme
10699:
10695:
10677:
10667:
10664:
10656:
10653:
10649:
10645:
10621:
10613:
10591:
10587:
10569:
10559:
10556:
10548:
10545:
10541:
10540:
10535:
10527:
10523:
10515:
10511:
10505:
10482:
10478:
10474:
10469:
10465:
10442:
10438:
10434:
10431:
10428:
10423:
10419:
10410:
10394:
10388:
10385:
10382:
10374:
10371:
10367:
10363:
10360:
10356:
10348:
10345:aiming to do
10344:
10340:
10336:
10328:
10325:
10318:
10311:
10307:
10301:
10292:
10288:
10284:
10276:
10275:log structure
10272:
10269:log structure
10245:
10241:
10237:
10234:
10231:
10222:
10218:
10210:
10206:
10205:
10200:
10196:
10192:
10185:
10165:
10145:
10125:
10097:
10084:
10081:
10077:
10056:
10031:
10023:
10018:
10014:
10010:
10006:The morphism
10005:
9997:
9993:
9985:
9981:
9973:
9969:
9965:
9961:
9953:
9949:
9945:
9937:
9933:
9929:
9925:
9907:
9899:
9895:
9894:separatedness
9891:
9887:
9883:
9879:
9878:
9873:
9869:
9866:. A property
9865:
9858:
9854:
9850:
9843:
9839:
9835:
9831:
9830:Zariski-local
9827:
9823:
9816:
9798:
9790:
9786:
9779:
9775:
9771:
9767:
9763:
9755:
9751:
9748:linearization
9744:
9740:
9732:
9731:Lelong number
9728:
9725:Lelong number
9722:
9713:
9709:
9706:Kuranishi map
9701:
9697:
9689:
9685:
9682:
9678:
9674:
9670:
9666:
9658:
9654:
9646:
9642:
9636:
9612:
9606:
9603:
9596:
9592:
9588:
9584:
9578:
9568:
9563:
9559:
9556:
9552:
9548:
9544:
9540:
9536:
9528:
9525:
9521:
9517:
9513:
9512:
9507:
9503:
9495:
9491:
9487:
9483:
9479:
9476:
9472:
9468:
9464:
9460:
9456:
9452:
9430:
9415:
9411:
9403:
9399:
9391:
9387:
9383:
9379:
9363:
9359:
9355:
9352:
9349:
9329:
9325:
9321:
9318:
9315:
9307:
9303:
9299:
9281:
9267:
9263:
9254:
9242:
9233:
9224:
9220:
9216:
9212:
9211:
9206:
9201:
9197:
9193:
9189:
9186:
9178:
9174:
9170:
9166:
9162:
9158:
9154:
9150:
9146:
9142:
9138:
9134:
9130:
9126:
9122:
9118:
9114:
9110:
9106:
9102:
9095:
9091:
9087:
9083:
9079:
9074:
9070:
9066:
9061:
9058:
9054:
9051:
9050:
9048:
9043:
9039:
9035:
9030:
9026:
9022:
9017:
9013:
9009:
9004:
8998:
8989:
8970:
8962:
8946:
8942:
8931:
8927:
8926:hyperelliptic
8923:
8920:hyperelliptic
8898:
8894:
8890:
8887:
8873:
8869:
8865:
8861:
8835:
8827:
8812:
8804:
8800:
8796:
8790:
8781:
8778:group variety
8773:
8769:
8765:
8757:
8753:
8750:
8731:
8725:
8722:
8719:
8713:
8707:
8701:
8695:
8684:
8680:
8661:
8658:
8655:
8649:
8643:
8637:
8631:
8620:
8616:
8609:
8605:
8601:
8597:
8589:
8542:
8539:
8535:
8531:
8512:
8507:
8495:
8490:
8480:
8466:
8462:
8438:
8432:
8429:
8426:
8418:
8414:
8410:
8409:good quotient
8406:
8403:good quotient
8386:
8383:
8380:
8377:
8374:
8349:
8345:
8338:
8335:
8315:
8312:
8309:
8306:
8303:
8278:
8274:
8267:
8264:
8244:
8240:
8234:
8230:
8223:
8219:
8211:
8207:
8203:
8195:
8191:
8187:
8183:
8163:
8159:
8155:
8134:
8131:
8128:
8122:
8119:
8116:
8112:
8108:
8105:
8100:
8096:
8087:
8083:
8080:over a field
8079:
8075:
8067:
8059:
8038:
8026:
8023:
8017:
8012:
8004:
8001:
7998:
7990:
7985:
7977:
7974:
7965:
7962:
7954:
7951:of dimension
7950:
7946:
7942:
7934:
7918:
7915:
7912:
7908:
7901:
7898:
7895:
7886:
7883:
7880:
7874:
7871:
7863:
7862:genus formula
7859:
7856:genus formula
7851:
7847:
7843:
7835:
7832:generic point
7827:
7820:
7816:
7812:
7808:
7804:
7800:
7796:
7792:
7780:
7776:
7772:
7768:
7764:
7741:
7707:
7694:
7691:
7683:
7679:
7675:
7672:
7664:
7660:
7656:
7647:
7638:
7637:formal scheme
7634:
7611:
7603:
7600:
7596:
7575:
7566:
7562:
7558:
7553:
7549:
7533:
7525:
7517:
7513:
7509:
7501:
7497:
7493:
7492:
7488:
7483:
7478:
7474:
7470:
7465:
7461:
7457:
7453:
7449:
7430:
7422:
7405:
7386:
7378:
7361:
7357:
7353:
7349:
7345:
7341:
7337:
7333:
7332:
7326:
7325:
7319:
7315:
7311:
7307:
7299:
7298:
7281:
7278:
7275:
7267:
7266:finite fibers
7262:
7258:
7254:
7250:The morphism
7249:
7246:finite fibers
7229:
7209:
7189:
7161:
7148:
7145:
7141:
7120:
7095:
7087:
7082:
7078:
7074:
7057:
7037:
7017:
6989:
6976:
6973:
6969:
6948:
6923:
6915:
6910:
6906:
6902:
6898:The morphism
6897:
6889:
6873:
6853:
6833:
6805:
6792:
6789:
6785:
6764:
6739:
6731:
6726:
6722:
6718:
6714:The morphism
6713:
6671:
6655:
6625:
6621:
6617:
6609:
6605:
6589:
6586:
6580:
6574:
6571:
6565:
6559:
6556:
6552:
6549:
6527:
6524:
6517:
6514:
6511:
6508:
6504:
6501:
6494:
6491:
6488:
6477:
6473:
6469:
6465:
6461:
6457:
6438:
6432:
6427:
6416:
6410:
6402:
6398:
6394:
6390:
6386:
6382:
6378:
6374:
6371:is a triple (
6370:
6354:
6348:
6345:
6342:
6339:
6336:
6330:
6327:
6324:
6304:
6299:
6295:
6291:
6283:
6280:
6276:
6273:fiber product
6268:
6264:
6245:
6219:
6211:
6207:
6203:
6195:
6191:
6190:residue field
6172:
6169:
6163:
6157:
6149:
6146:
6143:
6137:
6131:
6123:
6120:
6116:
6107:
6103:
6087:
6081:
6078:
6075:
6067:
6045:
6042:
6037:
6033:
6024:
6021:
6017:
6013:
6005:
6001:
5997:
5992:
5984:
5980:
5972:
5968:
5967:tangent sheaf
5964:
5946:
5943:
5935:
5925:
5914:
5911:
5908:
5902:
5894:
5884:
5858:
5836:
5829:
5828:
5826:
5818:
5814:
5798:
5790:
5784:
5780:
5773:
5768:
5764:
5760:
5757:
5754:
5746:
5730:
5710:
5702:
5697:
5693:
5689:
5684:
5676:
5669:of genus one.
5668:
5664:
5660:
5652:
5651:
5646:
5642:
5638:
5635:
5630:
5621:
5617:
5613:
5609:
5591:
5583:
5580:
5577:
5569:
5565:
5561:
5555:
5552:
5547:
5543:
5539:
5536:
5528:
5525:
5522:
5518:
5509:
5493:
5485:
5481:
5477:
5473:
5465:
5461:
5453:
5449:
5445:
5441:
5437:
5433:
5429:
5425:
5421:
5420:
5414:
5410:
5406:
5401:
5393:
5389:
5386:
5383:
5379:
5374:
5370:
5366:
5362:
5354:
5350:
5346:
5342:
5338:
5330:
5326:
5306:
5300:
5297:
5294:
5286:
5282:
5278:
5272:
5264:
5260:
5251:
5248:over a field
5235:
5232:
5229:
5223:
5220:
5217:
5209:
5205:
5185:
5182:
5179:
5175:
5168:
5165:
5160:
5156:
5149:
5146:
5142:
5137:
5129:
5126:
5122:
5115:
5107:
5103:
5099:
5091:
5087:
5083:
5079:
5075:
5071:
5067:
5047:
5039:
5035:
5018:
5014:
5006:
5005:special fiber
5002:
4999:
4998:generic fiber
4981:
4968:
4965:
4962:
4954:
4936:
4932:
4923:
4919:
4915:
4892:
4889:
4880:
4874:
4868:
4865:
4857:
4854:
4851:
4845:
4839:
4831:
4827:
4818:
4814:
4810:
4794:
4788:
4785:
4782:
4774:
4766:
4762:
4759:' (typically
4758:
4754:
4750:
4746:
4742:
4738:
4734:
4717:
4714:
4707:
4699:
4693:
4684:
4662:
4658:
4654:
4649:
4645:
4639:
4635:
4614:
4608:
4605:
4602:
4595:
4591:
4583:
4579:
4571:
4567:
4559:
4555:
4551:
4547:
4543:
4539:
4535:
4532:possesses no
4531:
4527:
4524:is connected
4523:
4520:
4519:affine scheme
4516:
4512:
4508:
4507:
4502:
4494:
4490:
4481:
4477:
4473:
4469:
4465:
4456:
4451:
4447:
4443:
4439:
4431:
4430:
4427:
4424:
4401:
4388:
4384:
4378:
4363:
4360:
4349:
4345:
4329:
4325:
4319:
4303:
4299:
4295:
4277:
4260:
4256:
4252:
4248:
4245:
4237:
4233:
4229:
4225:
4217:
4213:
4210:
4206:
4202:
4199:
4195:
4192:
4188:
4184:
4180:
4176:
4175:singularities
4172:
4169:
4165:
4161:
4157:
4154:
4150:
4146:
4142:
4138:
4134:
4130:
4126:
4122:
4106:
4103:
4100:
4092:
4088:
4072:
4069:
4066:
4044:
4029:
4025:
4009:
4006:
4003:
3983:
3976:
3972:
3968:
3964:
3961:
3958:
3954:
3951:
3950:moduli spaces
3946:
3942:
3938:
3930:
3926:
3922:
3918:
3914:
3895:
3887:
3883:
3875:
3871:
3867:
3859:
3856:central fiber
3851:
3847:
3839:
3821:
3818:
3812:
3809:
3806:
3800:
3795:
3791:
3785:
3781:
3773:
3772:
3770:
3766:
3750:
3742:
3737:
3727:
3724:
3716:
3712:
3708:
3700:
3696:
3692:
3688:
3684:
3681:
3678:An effective
3677:
3669:
3665:
3661:
3653:
3649:
3645:
3641:
3624:
3620:
3611:
3607:
3590:
3586:
3582:
3574:
3570:
3561:
3544:
3526:
3522:
3514:
3510:
3507:
3503:
3499:
3481:
3476:
3463:
3460:
3456:
3438:
3433:
3423:
3419:
3415:
3410:
3406:
3397:
3394:of dimension
3393:
3389:
3385:
3377:
3373:
3367:
3358:
3340:
3336:
3315:
3292:
3289:
3277:
3269:
3266:
3262:
3241:
3232:
3229:
3223:
3220:
3212:
3208:
3192:
3189:
3186:
3178:
3174:
3170:
3162:
3158:
3137:
3134:
3129:
3125:
3120:
3111:
3107:
3103:
3100:
3091:
3088:
3077:
3063:
3060:of dimension
3059:
3055:
3052:
3048:
3040:
3036:
3032:
3028:
3025:
3017:
3013:
3009:
3005:
3001:
2997:
2991:
2982:
2974:
2970:
2947:
2944:
2936:
2921:
2913:
2905:
2902:
2891:
2888:of dimension
2887:
2883:
2879:
2871:
2846:
2842:
2834:
2830:
2822:
2818:
2810:
2781:
2750:
2747:
2736:
2727:
2724:
2716:
2697:
2691:
2688:
2680:
2677:over a field
2676:
2672:
2664:
2660:
2656:
2633:
2627:
2624:
2616:
2613:over a field
2612:
2611:algebraic set
2608:
2605:algebraic set
2600:
2592:
2588:
2584:
2577:as well as .
2573:
2568:
2562:
2558:
2554:
2546:
2543:
2536:
2530:
2522:
2518:
2514:
2510:
2507:
2503:
2499:
2495:
2488:
2487:
2481:
2480:
2475:
2472:
2468:
2467:affine scheme
2464:
2461:
2457:
2454:
2450:
2442:
2438:
2434:
2416:
2412:
2408:
2403:
2399:
2376:
2363:
2360:
2355:
2351:
2344:
2339:
2335:
2326:
2322:
2318:
2301:
2285:
2277:
2265:
2260:
2256:
2249:
2244:
2240:
2231:
2213:
2209:
2205:
2200:
2196:
2188:
2184:
2180:
2176:
2168:
2164:
2147:
2120:
2098:
2095:
2084:
2078:
2063:
2059:
2053:
2044:
2040:
2036:
2032:
2028:
2020:
2016:
2012:
2008:
2003:
1999:
1995:
1994:relative Spec
1991:
1987:
1982:
1973:
1969:
1965:
1961:
1957:
1883:
1875:
1871:
1867:
1863:
1859:
1855:
1852:
1848:
1824:
1820:
1813:
1810:
1804:
1801:
1798:
1789:
1783:
1777:
1774:
1771:
1765:
1749:
1745:
1741:
1737:
1733:
1729:
1707:
1704:
1701:
1693:
1688:
1675:
1656:
1653:
1650:
1642:
1637:
1624:
1621:
1617:
1613:
1609:
1605:
1601:
1583:
1580:
1577:
1569:
1564:
1534:
1529:
1516:
1498:
1493:
1480:
1477:
1473:
1455:
1450:
1437:
1420:
1415:
1399:
1381:
1349:
1341:
1302:
1298:
1295:
1291:
1290:normal scheme
1287:
1284:
1265:
1257:
1240:
1237:
1233:
1214:
1208:
1205:
1197:
1178:
1170:
1153:
1150:
1146:
1142:
1138:
1134:
1116:
1108:
1073:
1070:
1062:
1046:
1042:
1039:
1021:
1013:
978:
970:
954:
950:
946:
928:
925:
917:
883:
866:
862:
844:
832:
829:
821:
817:
813:
809:
806:
799:
796:
792:
788:
784:
769:
765:
759:
755:
744:
740:
736:
732:
725:
721:
717:
713:
709:
705:
701:
676:
668:
656:
653:
636:
628:
615:
611:
607:
604:
600:
596:
592:
584:
580:
576:
572:
568:
564:
548:
542:
534:
518:
505:
501:
498:
492:
486:
478:
460:
443:
439:
435:
421:
415:
407:
391:
378:
374:
371:
365:
359:
337:
320:
301:
293:
277:
273:
269:
265:
261:
257:
253:
246:
245:generic point
242:
227:
218:
206:
203:
201:
198:
196:
193:
191:
188:
186:
183:
181:
178:
176:
173:
171:
168:
166:
163:
161:
158:
156:
153:
151:
148:
146:
143:
141:
138:
136:
133:
131:
128:
126:
123:
121:
118:
116:
113:
111:
108:
106:
103:
101:
98:
96:
93:
91:
88:
86:
83:
81:
78:
76:
73:
72:
62:
60:
56:
51:
49:
45:
41:
37:
32:
30:
19:
20333:
20305:
20299:
20263:
20257:
20221:
20215:
20179:
20173:
20137:
20131:
20095:
20089:
20053:
20047:
20011:
20005:
19954:
19925:
19913:
19901:
19889:
19868:
19861:
19856:
19844:
19817:
19805:
19793:
19772:
19760:
19748:
19727:
19715:
19694:
19673:
19664:math/0605429
19652:
19631:
19623:
19619:
19614:
19610:
19606:
19602:
19598:
19594:
19590:
19586:
19582:
19578:
19574:
19570:
19565:
19514:Weil divisor
19492:
19488:
19484:
19476:
19472:
19237:G-unramified
19236:
19230:
18886:For a point
18805:
18747:
18701:
18697:
18693:
18689:
18685:
18681:
18677:
18669:
18665:
18454:
18395:
18391:
18387:
18331:
18294:
18252:
18212:
18133:stable curve
18120:
18007:
17726:
17672:
17669:simple group
17618:
17444:
17382:
17009:
17005:
17001:
17000:is a normal
16926:
16922:
16870:
16869:is a scheme
16866:
16859:
16851:
16850:is a scheme
16847:
16796:
16795:is a smooth
16701:
16659:
16626:
16620:
16614:
16608:
16604:
16600:
16596:
16592:
16588:
16584:
16580:
16576:
16572:
16563:
16559:
16555:
16550:
16548:
16482:
16372:
16368:
16303:
16302:on a scheme
16299:
16296:section ring
16290:section ring
16254:
16107:which is of
16081:
16039:
15997:
15989:
15986:
15723:
15719:
15715:
15711:
15710:is a point,
15707:
15430:
15426:
15422:
15418:
15375:
15371:
15320:
15308:
15304:
15300:
15287:(called the
14951:
14947:
14871:
14835:is a proper
14832:
14831:of a scheme
14783:
14703:
14632:
14624:
14616:
14612:
14608:
14551:
14547:
14543:
14539:
13875:
13859:
13716:
13712:
13602:
13598:
13592:
13587:
13583:
13579:
13376:
13372:
13368:
13364:
13359:
13355:
13350:
13346:
13340:
13336:
13335:
13329:
13324:
13320:
13316:
13291:quasi-finite
13289:
13259:
13254:
13250:
13246:
13240:quasi-finite
13226:
13222:
13215:
13208:
13204:
13198:
13193:
13189:
13185:
13169:
13165:
13159:
13140:
13136:
13113:generalizes
13099:
13056:
13029:if, for any
13026:
12990:
12976:
12968:
12962:
12938:
12837:
12829:
12825:
12821:
12814:
12810:
12781:over a ring
12622:
12617:
12484:
12479:
12475:
12471:
12423:
12419:
12236:locally free
12119:polarization
12011:
11999:
11967:
11654:
11502:
11494:
11447:Hodge number
11382:
11346:
11342:
11339:Picard group
11333:Picard group
11316:
11307:
11288:Nowadays an
11234:
11230:
11229:of a scheme
11226:
11219:
11215:
11207:
11203:
11198:
11194:
11190:
11094:
11086:
11082:
11066:
11062:
11050:
10973:
10969:
10883:
10864:
10860:
10733:
10730:normal sheaf
10725:
10721:
10717:
10705:
10697:
10694:affine cover
10651:
10647:
10589:
10585:
10537:
10408:
10320:
10315:
10310:moduli space
10202:
10198:
10187:
10021:
10016:
10012:
10008:
9970:by means of
9950:. See also:
9935:
9927:
9920:
9897:
9893:
9875:
9872:localization
9867:
9860:
9845:
9841:
9829:
9825:
9818:
9811:
9788:
9781:
9777:
9773:
9769:
9765:
9687:
9680:
9676:
9590:
9561:
9557:
9554:
9547:Affine space
9526:
9519:
9515:
9509:
9505:
9500:irreducible
9496:≠ 0) is not.
9493:
9489:
9485:
9481:
9474:
9470:
9466:
9454:
9389:
9385:
9381:
9377:
9305:
9301:
9297:
9222:
9218:
9214:
9208:
9204:
9199:
9195:
9191:
9187:
9176:
9172:
9168:
9164:
9160:
9156:
9152:
9148:
9144:
9140:
9136:
9132:
9128:
9124:
9120:
9116:
9112:
9108:
9104:
9100:
9093:
9089:
9085:
9081:
9077:
9072:
9068:
9064:
9056:
9052:
9041:
9037:
9033:
9028:
9024:
9020:
9015:
9011:
9007:
8929:
8928:if it has a
8871:
8864:Hodge bundle
8858:Hodge bundle
8802:
8768:group scheme
8762:group scheme
8682:
8678:
8602:extends the
8416:
8412:
8411:of a scheme
8222:GIT quotient
8216:GIT quotient
8193:
8189:
8188:of a scheme
8081:
8077:
7952:
7948:
7932:
7806:
7798:
7778:
7774:
7770:
7766:
7762:
7662:
7661:, a divisor
7658:
7564:
7560:
7556:
7512:flag variety
7506:flag variety
7499:
7495:
7490:
7486:
7485:
7481:
7476:
7472:
7468:
7463:
7459:
7455:
7451:
7447:
7359:
7355:
7351:
7347:
7343:
7339:
7335:
7330:
7328:
7323:
7321:
7317:
7313:
7309:
7297:quasi-finite
7295:
7265:
7260:
7256:
7252:
7085:
7080:
7076:
7072:
6913:
6908:
6904:
6900:
6729:
6724:
6720:
6716:
6669:
6624:final object
6619:
6603:
6475:
6471:
6467:
6463:
6459:
6455:
6400:
6396:
6392:
6388:
6384:
6380:
6376:
6372:
6368:
6266:
6262:
6193:
6105:
6101:
6022:
6018:is a smooth
6016:Fano variety
5995:
5962:
5695:
5691:
5687:
5665:is a smooth
5648:
5633:
5615:
5611:
5607:
5507:
5479:
5451:
5447:
5443:
5439:
5435:
5427:
5423:
5417:
5412:
5408:
5404:
5384:
5382:Weil divisor
5377:
5372:
5368:
5345:ring spectra
5249:
5207:
5105:
5101:
5084:schemes, 1:
5076:. Examples:
5000:
4952:
4924:to a scheme
4917:
4912:degeneration
4812:
4808:
4760:
4756:
4752:
4748:
4744:
4740:
4736:
4732:
4557:
4553:
4549:
4542:affine space
4537:
4529:
4521:
4504:
4479:
4475:
4471:
4449:
4445:
4428:
4420:
4347:
4343:
4254:
4250:
4249:of a scheme
4246:
4170:for details.
4149:base changed
4145:number field
4141:plane curves
4124:
4120:
4091:group scheme
3920:
3916:
3912:
3869:
3848:A scheme is
3837:
3768:
3764:
3714:
3698:
3694:
3690:
3686:
3685:on a scheme
3682:
3647:
3542:
3505:
3501:
3497:
3461:
3459:smooth locus
3454:
3395:
3391:
3356:
3210:
3206:
3176:
3061:
3057:
3053:
3038:
3029:generalizes
3015:
3003:
2889:
2885:
2714:
2678:
2614:
2532:
2527:
2520:
2516:
2490:
2485:
2477:
2453:Affine space
2440:
2436:
2324:
2320:
2182:
2178:
2042:
2034:
2026:
2018:
2014:
2010:
2006:
2001:
1997:
1985:
1980:
1971:
1963:
1955:
1873:
1869:
1865:
1861:
1857:
1853:
1747:
1743:
1731:
1727:
1673:
1619:
1615:
1611:
1603:
1514:
1475:
1397:
1300:
1285:
1283:Weil divisor
1235:
1231:
1195:
1148:
1144:
1136:
864:
860:
819:
815:
811:
804:
797:
790:
789:of a scheme
787:GIT quotient
742:
738:
723:
715:
711:
707:
703:
699:
613:
609:
605:
603:Weil divisor
590:
582:
578:
574:
570:
566:
562:
476:
441:
437:
318:
271:
263:
259:
255:
251:
58:
54:
52:
33:
28:
26:
19569:Proof: Let
18866:universally
18612:split torus
18500:; that is,
18119:. The term
17443:is said to
17383:split torus
16383:. See also
15311:, then the
14730:A morphism
14720:affine line
14635:, that is,
14506:is trivial.
13719:. See also
13490:Quot scheme
13484:Quot scheme
13432:quasi-split
13383:quasi-split
13314:A morphism
13183:A morphism
12834:global Proj
12372:(in short,
11445:. See also
10880:normal cone
10728:, then the
10520:nonsingular
10138:where each
9539:irreducible
9535:prime ideal
9511:irreducible
8924:A curve is
7777:. One says
7773:has degree
7526:A morphism
7202:where each
7030:where each
6002:Related to
5685:A morphism
5402:A morphism
5380:) for some
5347:instead of
5341:commutative
4920:is said to
4696:deformation
4534:idempotents
4513:refine the
2967:Artin stack
2513:affine cone
2228:, then the
1880:Q-factorial
616:; that is,
61:-morphism.
20431:Categories
19944:References
19685:1503.02309
19643:1509.05576
19593:such that
19481:very ample
19467:very ample
19232:unramified
19078:, and let
18882:unramified
18696:-point of
17083:. Usually
17004:-variety (
16921:A divisor
16913:is smooth.
16525:there are
15673:Todd class
15435:Chow group
14870:such that
13919:such that
13874:A variety
13605:-morphism
13367:is called
13328:is called
13197:is called
12985:property P
12652:such that
12486:projective
12483:is called
12403:projective
11886:André Weil
11495:a fortiori
11387:plurigenus
11377:plurigenus
10291:ind-scheme
10287:loop group
10281:loop group
10204:noetherian
10195:Noetherian
10122:Spec
10094:Spec
10053:Spec
9402:ind-scheme
9396:ind-scheme
9188:Immersions
9080:such that
9031:subscheme
8528:Gorenstein
8451:such that
7350:such that
7186:Spec
7158:Spec
7117:Spec
7014:Spec
6986:Spec
6945:Spec
6830:Spec
6802:Spec
6761:Spec
6608:2-commutes
6541:such that
6317:given for
5510:such that
5416:is called
5359:divisorial
5108:such that
4922:degenerate
4028:birational
3945:invariants
3919:(group of
3874:Chow group
3864:Chow group
3771:such that
3500:of degree
3370:Calabi–Yau
2807:is not an
1856:) and its
867:copies of
205:References
27:This is a
19880:1302.0385
19784:1410.1716
19739:1409.1169
19706:0704.2030
19605:and then
19389:→
19101:#
18976:→
18944::
18939:#
18836:→
18816:π
18672:, then a
18657:universal
18525:−
18388:subscheme
18382:subscheme
18363:~
18354:↪
18348:~
18310:−
18278:→
18272:~
18232:→
18196:→
18190:~
18178:π
18024:⊆
17775:⊃
17772:…
17769:⊃
17756:⊃
17680:connected
17632:⊆
17472:¯
17464:⊆
17458:⊆
17340:¯
17302:¯
17274:¯
17116:¯
16967:−
16899:¯
16885:×
16819:¯
16774:¯
16736:×
16722:¯
16682:¯
16643:¯
16401:separated
16329:Γ
16324:∞
16315:⊕
16229:⊂
16179:∈
16168:for some
16053:
15897:
15843:−
15831:
15800:∗
15779:−
15739:∗
15735:π
15688:∗
15684:π
15613:∗
15609:π
15582:−
15570:∑
15557:π
15524:
15518:⋅
15506:
15495:∗
15491:π
15475:
15469:⋅
15454:π
15447:
15402:→
15393:π
15349:−
15331:χ
15289:different
15261:Ω
15220:Ω
15213:
15184:∈
15177:∑
15146:∼
15108:∗
15104:π
15100:∼
15064:−
15043:∈
15036:∑
15023:−
14999:π
14993:
14981:−
14926:π
14903:→
14894:π
14855:→
14846:π
14798:×
14767:→
14741:→
14588:↪
14516:reflexive
14492:¯
14444:¯
14409:¯
14331:connected
14325:reductive
14246:⊂
14174:
14054:≥
14011:∗
13932:∗
13904:→
13830:⋯
13816:−
13789:↦
13755:→
13687:
13682:→
13628:→
13616:
13445:⊆
13273:∈
13164:sheaf of
13119:connected
13115:reductive
13083:→
13071:×
13040:→
13010:→
12914:∨
12903:
12628:-Algebra
12533:×
12451:→
12385:∗
12357:⊗
12346:∗
12324:∗
12316:⊗
12305:∗
12164:→
12090:¯
12066:×
12060:′
12032:→
12027:′
11932:with the
11852:that are
11807:¯
11757:¯
11503:point of
11469:A scheme
11425:⊗
11416:ω
11403:Γ
11400:
11153:⊗
11136:Γ
11133:→
11125:⊗
11105:Γ
10994:
10920:∞
10911:⊕
10904:
10770:∗
10668:⊂
10646:if it is
10560:⊂
10429:∘
10392:→
10324:Chapter 1
10235:∪
10227:∞
10178:-algebra.
10082:−
9930:denote a
9908:∪
9890:nilpotent
9799:∪
9607:
9504:A scheme
9353:
9319:
9268:∗
9260:→
9238:♯
9183:immersion
8895:ω
8882:Γ
8813:χ
8754:concerns
8723:×
8702:⋅
8659:−
8467:∗
8436:→
8384:
8339:
8313:
8268:
8132:
8120:
8113:×
8086:geometric
8018:
8005:
7982:Ω
7969:Γ
7966:
7919:δ
7916:−
7899:−
7884:−
7676:⊂
7601:−
7279:∈
7242:-algebra.
7146:−
6974:−
6790:−
6590:ψ
6587:∘
6581:β
6566:α
6557:∘
6550:ψ
6521:→
6512:β
6498:→
6489:α
6470:, ψ) to (
6428:∼
6425:→
6352:→
6334:→
6296:×
6208:×
6147:∈
6121:−
6085:→
6060:is ample.
6043:−
6034:ω
5941:→
5923:→
5903:⊕
5866:→
5840:→
5777:→
5592:∗
5562:≃
5556:ω
5553:⊗
5548:∨
5526:−
5494:ω
5298:
5279:∈
5265:∗
5233:
5227:→
5183:−
5127:⊗
5116:χ
5070:dimension
5064:dimension
5048:π
4972:→
4963:π
4890:≤
4869:
4855:∈
4792:→
4711:→
4659:ω
4646:ω
4640:∗
4612:→
4528:the ring
4506:connected
4499:connected
4261:sheaf of
4104:≥
3973:by their
3927:) modulo
3810:−
3796:∗
3748:→
3587:ω
3473:Ω
3430:Ω
3424:∗
3407:ω
3382:canonical
3316:π
3293:~
3284:↪
3267:−
3263:π
3239:→
3233:~
3221:π
3190:⊂
3095:Γ
3092:
3084:∞
3081:→
3035:Frobenius
2945:−
2922:χ
2903:−
2782:⊗
2751:
2737:×
2728:
2692:
2628:
2572:0909.0069
2266:⊗
2257:ω
2241:ω
2210:ω
2197:ω
2005:-algebra
1950:-Cartier.
1825:∗
1814:
1805:
1778:
1705:
1685:Ω
1654:
1634:Ω
1581:
1561:Ω
1526:Ω
1490:Ω
1447:Ω
1412:Ω
1294:reflexive
1209:
1071:−
926:−
648:Γ
506:⊗
475:-module (
379:⊗
228:η
34:See also
20386:See also
20332:(1977),
20294:(1967).
20252:(1966).
20210:(1965).
20168:(1964).
20126:(1963).
20084:(1961).
20042:(1961).
20000:(1960).
19953:(1998),
19860:Q. Liu,
17683:solvable
16575:∈
16571:for any
16558: :
16475:diagonal
16473:has its
16030:Schubert
15963:rigidify
15647:means a
15158:means a
14954:and the
14716:morphism
14073:reduced
14032:′
13953:′
13900:′
13521:rational
13515:rational
13319: :
13249: :
13207:by some
13188: :
13173:-modules
12474: :
12268:-module
12210:-module
11710:, where
11657:, where
11527:a point
11290:orbifold
11284:orbifold
11193: :
10642:. It is
10353:morphism
10011: :
9817:, every
9698:See the
9455:integral
9449:integral
9194: :
9151:). When
9127:, so if
9067: :
9036: :
9010: :
7824:G-bundle
7559: :
7552:flat map
7471: :
7255: :
7075: :
6903: :
6719: :
6620:relative
6604:does not
6553:′
6528:′
6505:′
6279:pullback
6235:, where
5998:-regular
5690: :
5419:dominant
5407: :
5398:dominant
5082:Artinian
5060:is flat.
4718:′
4582:Cox ring
4576:Cox ring
4483:-module.
4160:surfaces
4024:rational
3965:over an
3957:Complete
3850:catenary
3844:catenary
3836:for all
2979:artinian
1750:; i.e.,
200:See also
20358:0463157
20322:0238860
20280:0217086
20238:0199181
20196:0173675
20154:0163911
20112:0217085
20070:0217084
20028:0217083
19987:1644323
19920:, 1.2.1
19851:, 4.2.5
19839:, §II.3
19824:, 4.2.1
19520:divisor
19459:variety
19235:(resp.
18547:theorem
18402:surface
18257:blow-up
16931:special
16917:special
16856:regular
16613:affine
15425:and if
14560:is not.
14522:regular
14099:reduced
12951:#normal
12832:is the
12616:global
11882:Zariski
11369:of the
11365:is the
10869:regular
10260:is not.
9886:reduced
9787:, i.e.
9570:is not.
9390:reduced
9088:, then
8866:on the
7781:has a g
7769:+1 and
4588:crepant
4522:Spec(R)
4458:is not.
4300:of the
4298:support
4232:torsors
3840:> 0.
3674:Cartier
3666:is the
3173:blow-up
3167:blow-up
2484:global
2056:abelian
1996:of the
1849:of the
1672:is the
1513:is the
317:and if
20356:
20346:
20320:
20278:
20236:
20194:
20152:
20110:
20068:
20026:
19985:
19975:
19896:, II.4
19767:, §1.6
19755:, §1.4
19435:finite
19008:. Let
18418:. See
18247:, the
18127:stable
18097:resp.
16617:-space
16537:smooth
16503:simple
16109:degree
16095:scroll
16080:where
16012:scheme
16006:scheme
15971:resp.
15871:Every
15702:is an
15548:where
15193:length
15138:where
14876:smooth
14540:Spec k
13601:is an
13232:) are
13216:Spec B
13122:smooth
12964:proper
12957:proper
12817:, the
12422:is an
12234:and a
11821:is an
11794:where
11325:pencil
11208:closed
10712:normal
10539:normal
10532:normal
10457:where
10304:moduli
9998:above.
9921:Spec A
9846:Spec A
9760:local
9482:Spec A
9467:Spec k
9414:torsor
9029:closed
8677:where
7931:where
7757:is a g
7631:formal
6730:finite
6710:finite
6379:, ψ),
6068:Given
5961:where
5645:scheme
5482:, the
5440:Spec B
5436:Spec A
5351:; see
5096:degree
4747:is an
4446:Spec k
4296:. The
4294:ideals
4242:closed
4133:degree
4127:. See
4059:. (b)
3996:. (a)
3963:curves
3960:smooth
3925:cycles
3453:where
3209:along
2868:. See
2504:, and
2479:affine
2447:affine
2391:where
2232:says:
1614:along
1288:(on a
1139:is an
793:by an
444:is an
321:is an
42:, and
19875:arXiv
19779:arXiv
19734:arXiv
19701:arXiv
19680:arXiv
19659:arXiv
19638:arXiv
19622:) if
19577:. If
19558:Notes
19433:is a
18606:torus
18330:. If
18160:stack
18154:stack
18121:split
18008:split
17943:over
17727:split
17445:split
17381:is a
17255:torus
17253:is a
17170:then
17016:split
16858:over
16619:over
16481:is a
16130:in a
16103:is a
16038:is a
16022:of a
16014:is a
15867:rigid
14678:when
14623:. If
14603:is a
14375:is a
13858:. If
13162:is a
13055:with
12828:) of
12000:fiber
11732:field
11730:is a
11493:, so
11489:is a
11465:point
10968:. if
10863:into
10700:is a
10696:over
10341:is a
9932:field
9123:) in
9001:image
8772:group
8367:when
8296:when
8210:stack
8206:gerbe
8200:gerbe
7840:genus
7458:. If
7406:over
7402:is a
6614:final
6479:'
6104:over
6064:fiber
5745:field
5701:étale
5681:étale
5432:dense
5430:) is
5088:, 2:
4996:with
4743:' of
4681:curve
4487:conic
4257:be a
3975:genus
3689:over
2833:ample
2827:ample
2591:group
2567:arXiv
2025:Spec(
1954:Spec(
1845:(the
1303:is a
822:. If
718:is a
601:of a
20379:book
20344:ISBN
19973:ISBN
19601:) =
19530:See
18750:and
18708:. A
18551:See
18449:The
18437:See
16583:and
16529:and
16394:See
16379:See
16294:The
16214:The
16189:>
15718:and
15651:and
15378:- 2.
15374:is 2
15162:and
14171:Spec
13988:and
13613:Spec
13590:, a
13527:and
13334:or (
13258:has
12989:Let
12949:See
12618:Proj
12143:The
12131:See
12127:Proj
12077:Spec
11968:e.g.
11744:Spec
11687:Spec
11457:See
11385:-th
11381:The
11361:The
11337:The
11314:See
11212:open
11204:open
11184:open
10985:Spec
10895:Spec
10512:See
10337:The
10285:See
10273:See
10193:are
10186:The
9994:Cf.
9962:The
9855:and
9741:see
9729:See
9710:See
9643:The
9585:The
9564:) =
9555:Spec
9549:and
9384:but
9350:Spec
9342:and
9316:Spec
9135:and
9071:′ →
8862:The
8797:The
8617:The
8598:The
8407:The
8381:Proj
8336:Proj
8328:and
8310:Spec
8265:Spec
8220:The
8184:The
8129:Spec
8117:Spec
8084:is "
7943:The
7860:The
7844:See
7793:The
7635:See
7548:flat
7522:flat
7516:flag
7510:The
7484:(or
7362:and
7340:f(y)
7327:(or
7264:has
6681:Spec
6670:over
6010:Fano
5723:and
5639:The
5462:See
5295:Spec
5230:Spec
5068:The
5003:and
4765:flat
4700:Let
4550:Spec
4350:and
4230:for
4135:for
3872:-th
3868:The
3709:The
3668:Proj
3662:The
3464:and
3374:The
3135:>
2998:The
2880:The
2748:Spec
2725:Spec
2689:Spec
2625:Spec
2563:and
2486:Spec
2439:and
2431:are
2323:and
2011:Spec
1992:The
1978:Spec
1847:Proj
1802:Proj
1775:Proj
947:The
785:The
733:The
597:The
215:!$ @
75:!$ @
20310:doi
20268:doi
20226:doi
20184:doi
20142:doi
20100:doi
20058:doi
20016:doi
19963:doi
19906:EGA
19597:+ (
19589:on
19483:if
19479:is
19444:in
19259:in
19229:is
19172:in
18906:in
18293:of
18251:of
18006:is
17725:is
17617:is
17513:to
16933:if
16929:is
16700:to
16658:of
16607:to
16587:of
16433:of
16257:in
15828:deg
15319:of
14990:deg
14938:is
14874:is
14631:of
14459:of
14097:is
13679:lim
13597:of
13430:is
13371:if
12883:Sym
12809:If
12592:EGA
12570:to
11825:of
11774:to
11610:to
11397:dim
11341:of
11067:snc
10886:is
10882:to
10867:is
10736:is
10732:to
10508:nef
10024:if
10020:is
9651:klt
9604:Pic
9524:iff
9400:An
9062:if
9023:of
9005:If
8257:is
8002:dim
7963:dim
7955:is
7805:on
7761:if
7546:is
7480:is
7450:is
7346:of
7338:of
7320:is
7308:If
7088:if
7084:is
6916:if
6912:is
6732:if
6728:is
6668:is
6454:in
6395:in
6391:),
6383:in
6192:of
5699:is
5661:An
5641:EGA
5610:on
5506:on
4735:an
4526:iff
4491:An
4346:of
4189:or
4166:or
3398:is
3089:dim
3056:on
3045:big
3037:on
3014:of
2892:is
2673:An
2657:An
2609:An
2585:An
2435:on
2041:of
1970:of
1811:Sym
1702:log
1651:log
1610:on
1578:log
1474:on
1365:is
1230:of
1206:Pic
1143:on
585:).)
561:If
258:),
195:XYZ
20433::
20377:A
20354:MR
20352:,
20338:,
20318:MR
20316:.
20308:.
20306:32
20304:.
20298:.
20290:;
20276:MR
20274:.
20266:.
20264:28
20262:.
20256:.
20248:;
20234:MR
20232:.
20224:.
20222:24
20220:.
20214:.
20206:;
20192:MR
20190:.
20182:.
20180:20
20178:.
20172:.
20164:;
20150:MR
20148:.
20140:.
20138:17
20136:.
20130:.
20122:;
20108:MR
20106:.
20098:.
20096:11
20094:.
20088:.
20080:;
20066:MR
20064:.
20056:.
20052:.
20046:.
20038:;
20024:MR
20022:.
20014:.
20010:.
20004:.
19996:;
19983:MR
19981:,
19971:,
19961:,
19829:^
19624:D'
19603:D'
19581:~
19579:D'
19549:A
19279:,
18874:,
18688:→
18610:A
18586:A
18563:,
18559:,
18555:,
18386:A
18158:A
17361:.
16996:A
16729::=
16591:,
16567::
16562:→
16405:A
16099:A
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