6319:
38:
6442:. It allows you to glue affine varieties (along common open sets) without worrying whether the resulting object can be put into some projective space. This also leads to difficulties since one can introduce somewhat pathological objects, e.g. an affine line with zero doubled. Such objects are usually not considered varieties, and are eliminated by requiring the schemes underlying a variety to be
204:
is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way. The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. It is not obvious
4632:
which ensures a set of isomorphism classes has a (reducible) quasi-projective variety structure. Moduli such as the moduli of curves of fixed genus is typically not a projective variety; roughly the reason is that a degeneration (limit) of a smooth curve tends to be non-smooth or reducible. This
1055:
subset of a projective variety. Notice that every affine variety is quasi-projective. Notice also that the complement of an algebraic set in an affine variety is a quasi-projective variety; in the context of affine varieties, such a quasi-projective variety is usually not called a variety but a
1164:
over an algebraically closed field, although some authors drop the irreducibility or the reducedness or the separateness condition or allow the underlying field to be not algebraically closed. Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an
3747:
5169:
In general, in contrast to the case of moduli of curves, a compactification of a moduli need not be unique and, in some cases, different non-equivalent compactifications are constructed using different methods and by different authors. An example over
5582:). The non-uniqueness of compactifications is due to the lack of moduli interpretations of those compactifications; i.e., they do not represent (in the category-theory sense) any natural moduli problem or, in the precise language, there is no natural
463:
3009:
961:
4423:
679:
3972:
1825:
4954:
The moduli of curves exemplifies a typical situation: a moduli of nice objects tend not to be projective but only quasi-projective. Another case is a moduli of vector bundles on a curve. Here, there are the notions of
3593:
7028:
Mark
Goresky. Compactifications and cohomology of modular varieties. In Harmonic analysis, the trace formula, and Shimura varieties, volume 4 of Clay Math. Proc., pages 551–582. Amer. Math. Soc., Providence, RI,
6387:— some foundational changes are required. The modern notion of a variety is considerably more abstract than the one above, though equivalent in the case of varieties over algebraically closed fields. An
1177:(the analog of compactness), but soon afterwards he found an algebraic surface that was complete and non-projective. Since then other examples have been found: for example, it is straightforward to construct
6147:
A product of finitely many algebraic varieties (over an algebraically closed field) is an algebraic variety. A finite product of affine varieties is affine and a finite product of projective varieties is
5792:
4154:
2411:
2317:
1208:
says that closed subvarieties of an affine or projective variety are in one-to-one correspondence with the prime ideals or non-irrelevant homogeneous prime ideals of the coordinate ring of the variety.
6064:
6023:
1472:
5393:
1754:
1198:
is a subset of a variety that is itself a variety (with respect to the topological structure induced by the ambient variety). For example, every open subset of a variety is a variety. See also
4857:
4790:
4697:
5955:
5829:
2813:
2588:
2360:
2266:
1096:) refers to a more general object, which locally is a quasi-projective variety, but when viewed as a whole is not necessarily quasi-projective; i.e. it might not have an embedding into
6391:
is a particular kind of scheme; the generalization to schemes on the geometric side enables an extension of the correspondence described above to a wider class of rings. A scheme is a
3206:
1152:'s definition of a scheme is more general still and has received the most widespread acceptance. In Grothendieck's language, an abstract algebraic variety is usually defined to be an
4186:
4095:
4063:
4007:
3918:
3874:
6133:
Every nonempty affine algebraic set may be written uniquely as a finite union of algebraic varieties (where none of the varieties in the decomposition is a subvariety of any other).
5323:
1630:
1104:
on the variety. The disadvantage of such a definition is that not all varieties come with natural embeddings into projective space. For example, under this definition, the product
3524:
3318:
3288:
3258:
3232:
3104:
3039:
2216:
2160:
1939:
5452:
4923:
4892:
4821:
4754:
4626:
3139:
6383:
The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not
5705:
1100:. So classically the definition of an algebraic variety required an embedding into projective space, and this embedding was used to define the topology on the variety and the
3078:
3545:, which has genus zero. Using genus to distinguish curves is very basic: in fact, the genus is the first invariant one uses to classify curves (see also the construction of
3456:
2504:
6509:
Allowing nilpotent elements in rings is related to keeping track of "multiplicities" in algebraic geometry. For example, the closed subscheme of the affine line defined by
1334:
7133:
5727:
5190:
3791:
2920:
2767:
5062:
2853:
5104:
2549:
2124:
6348:
5872:
5564:
5532:
5500:
5421:
5286:
5218:
2008:
2447:
6090:
5347:
5258:
4949:
4723:
4661:
4551:
5977:. This may be explained as follows: the affine line has dimension one and so any subvariety of it other than itself must have strictly less dimension; namely, zero.
5638:), but an affine variety cannot contain a projective variety of positive dimension as a closed subvariety. It is not projective either, since there is a nonconstant
4282:
2089:
1969:
4453:
4236:
2059:
281:
370:
5918:
4517:
4487:
5975:
5892:
5472:
5238:
5164:
5144:
5124:
5017:
4997:
4977:
4595:
4571:
4256:
4206:
2675:
2180:
2036:
6972:
76:. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.
6483:
was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way.
2928:
872:
4025:
structure on it (the name "abelian" is however not because it is an abelian group). An abelian variety turns out to be projective (in short, algebraic
5474:(a principal polarization identifies an abelian variety with its dual). The theory of toric varieties (or torus embeddings) gives a way to compactify
4290:
3742:{\displaystyle {\begin{cases}G_{n}(V)\hookrightarrow \mathbf {P} \left(\wedge ^{n}V\right)\\\langle b_{1},\ldots ,b_{n}\rangle \mapsto \end{cases}}}
560:
3931:
1749:
733:
6613:
1504:
is not empty. It is irreducible, as it cannot be written as the union of two proper algebraic subsets. Thus it is an affine algebraic variety.
1118:. Furthermore, any variety that admits one embedding into projective space admits many others, for example by composing the embedding with the
158:
79:
Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be
4663:, a not-necessarily-smooth complete curve with no terribly bad singularities and not-so-large automorphism group. The moduli of stable curves
7416:
7388:
7228:
7154:
7097:
7057:
6895:
6476:
4030:
6446:. (Strictly speaking, there is also a third condition, namely, that one needs only finitely many affine patches in the definition above.)
3461:
in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation:
3364:
2591:
7333:
7307:
7194:
6901:, a remark describes a complete toric variety that has no non-trivial line bundle; thus, in particular, it has no ample line bundle.
6757:
6370:
6141:
2268:. It is an affine variety, since, in general, the complement of a hypersurface in an affine variety is affine. Explicitly, consider
1131:
162:
6648:
5732:
4628:. There are few ways to show this moduli has a structure of a possibly reducible algebraic variety; for example, one way is to use
4100:
7325:
7299:
7178:
6749:
146:
and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of
6218:
2365:
2271:
7245:
5349:
6879:
1161:
107:
65:
6028:
5987:
4860:
1400:
6591:
5355:
7466:
7044:
5326:
3050:
1205:
1057:
131:
6331:
1173:
One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata. Nagata's example was not
7461:
7343:
6623:
1383:
coming from the topology on the complex numbers, a complex line is a real manifold of dimension two.) This is the set
1034:
31:
6341:
6335:
6327:
4826:
4759:
4666:
7170:
6887:
6384:
5923:
5797:
4629:
3546:
1830:
The irreducibility of this algebraic set needs a proof. One approach in this case is to check that the projection (
201:
6633:
6541:
are reduced. Geometrically, this says that fibers of good mappings may have nontrivial "infinitesimal" structure.
1861:
computation to compute the dimension, followed by a random linear change of variables (not always needed); then a
7291:
3385:
A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. The
1857:
For more difficult examples, a similar proof may always be given, but may imply a difficult computation: first a
1157:
1153:
1077:
6352:
2779:
2554:
2326:
2232:
6607:
6518:
6468:
into it can be extended uniquely to the whole curve. Every projective variety is complete, but not vice versa.
5535:
5503:
5146:
as an open subset. Since a line bundle is stable, such a moduli is a generalization of the
Jacobian variety of
3155:
1878:
1085:
1047:
1003:
Projective varieties are also equipped with the
Zariski topology by declaring all algebraic sets to be closed.
717:
284:
6590:. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions.
4159:
4068:
4036:
3980:
3891:
3847:
1739:
is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane. It is the
1125:
The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by
3348:
2039:
1069:
781:
154:
7364:, published as Chapter VII of Arithmetic geometry (Storrs, Conn., 1984), 167–212, Springer, New York, 1986.
5291:
1568:
6981:
6503:
5579:
3467:
3321:
3297:
3267:
3237:
3211:
3083:
3018:
2923:
2879:
2863:
2185:
2129:
1908:
1680:
1149:
759:
5426:
4897:
4866:
4795:
4728:
4600:
3113:
7379:. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) . Vol. 34 (3rd ed.). Berlin, New York:
6158:
5666:
1122:; thus many notions that should be intrinsic, such as that of a regular function, are not obviously so.
80:
7445:
122:(an algebraic object) in one variable with complex number coefficients is determined by the set of its
5681:
6454:
6392:
6261:
4956:
3825:
3584:
3562:
3059:
2227:
1145:
177:
3602:
3412:
2455:
6986:
6618:
6408:
5575:
3921:
3821:
3261:
2867:
1683:
polynomial, this is an algebraic variety. The set of its real points (that is the points for which
1286:
1137:
181:
135:
7116:
5710:
5173:
3766:
2903:
2680:
1114:
is not a variety until it is embedded into a larger projective space; this is usually done by the
7268:
6999:
6860:
6638:
6561:
6428:
6396:
5022:
3817:
3344:
2818:
1851:
1119:
713:
185:
123:
84:
53:
6092:. (Over a different base field, a unitary group can however be given a structure of a variety.)
5067:
2516:
2094:
6453:
affine charts, and when speaking of a variety only require that the affine charts have trivial
5834:
5541:
5509:
5477:
5398:
5263:
5195:
1974:
7422:
7412:
7384:
7329:
7303:
7224:
7190:
7150:
7136:
7093:
7053:
6964:
6891:
6753:
6472:
6395:
such that every point has a neighbourhood that, as a locally ringed space, is isomorphic to a
6284:
6198:
5567:
4894:. Historically a paper of Mumford and Deligne introduced the notion of a stable curve to show
2416:
170:
88:
7210:
6502:. This is one of several generalizations of classical algebraic geometry that are built into
6069:
5332:
5243:
4928:
4702:
4640:
4530:
458:{\displaystyle Z(S)=\left\{x\in \mathbf {A} ^{n}\mid f(x)=0{\text{ for all }}f\in S\right\}.}
7260:
7216:
7182:
7142:
7085:
6991:
6842:
6826:
6798:
6782:
6741:
6628:
6480:
6461:
5639:
4574:
4261:
3975:
2064:
1944:
1870:
1862:
1858:
1199:
1174:
1141:
1101:
1097:
1073:
1052:
522:
206:
119:
96:
7398:
7067:
6856:
6812:
4431:
4214:
3392:
is an example of a projective curve; it can be viewed as the curve in the projective plane
2044:
259:
7394:
7380:
7063:
6852:
6808:
6575:
6549:
6545:
6465:
6450:
6420:
6404:
6126:
4018:
3386:
2856:
1376:
1115:
166:
161:
while a differentiable manifold cannot. Algebraic varieties can be characterized by their
139:
5897:
4496:
4466:
529:
to be precisely the affine algebraic sets. This topology is called the
Zariski topology.
4519:
turns out to be an isomorphism; in particular, an elliptic curve is an abelian variety.
3347:
is a closed subvariety of a projective space. That is, it is the zero locus of a set of
7372:
7317:
6956:
6595:
6570:-dimensional manifold, and hence every sufficiently small local patch is isomorphic to
6486:
One way that leads to generalizations is to allow reducible algebraic sets (and fields
6106:
5984:(over the complex numbers) is not an algebraic variety, while the special linear group
5960:
5877:
5457:
5223:
5149:
5129:
5109:
5002:
4982:
4962:
4580:
4556:
4241:
4191:
4026:
3798:
3534:
2596:
2165:
2021:
1866:
228:
218:
73:
6910:
Definition 1.1.12 in
Ginzburg, V., 1998. Lectures on D-modules. University of Chicago.
7455:
7368:
7040:
6960:
6864:
6587:
5981:
4022:
3813:
3538:
3004:{\displaystyle \operatorname {gr} A=\bigoplus _{i=-\infty }^{\infty }A_{i}/{A_{i-1}}}
1740:
1178:
1126:
701:
127:
42:
7003:
7406:
6643:
5626:
is an algebraic variety since it is a product of varieties. It is not affine since
5594:
An algebraic variety can be neither affine nor projective. To give an example, let
5583:
5571:
4634:
3877:
1874:
1708:
1704:
1237:
241:
61:
37:
30:
This article is about algebraic varieties. For the term "variety of algebras", see
5019:(degree of the determinant of the bundle) is then a projective variety denoted as
956:{\displaystyle Z(S)=\{x\in \mathbf {P} ^{n}\mid f(x)=0{\text{ for all }}f\in S\}.}
7347:
7437:
7146:
6432:
6117:
3829:
3352:
3107:
3054:
1692:
1359:
on which this function vanishes: it is the set of all pairs of complex numbers (
147:
69:
57:
7113:
Chai, Ching-Li (1986). "Siegel Moduli
Schemes and Their Compactifications over
4418:{\displaystyle C^{n}\to \operatorname {Jac} (C),\,(P_{1},\dots ,P_{r})\mapsto }
7441:
7220:
7186:
6689:
Harris, p.9; that it is irreducible is stated as an exercise in
Hartshorne p.7
3363:
1854:
on the set of the solutions and that its image is an irreducible plane curve.
674:{\displaystyle I(V)=\left\{f\in K\mid f(x)=0{\text{ for all }}x\in V\right\}.}
526:
92:
7426:
7361:
6847:
6835:
Memoirs of the
College of Science, University of Kyoto. Series A: Mathematics
6830:
6803:
6791:
Memoirs of the
College of Science, University of Kyoto. Series A: Mathematics
6786:
3583:. It is a projective variety: it is embedded into a projective space via the
6495:
4013:
is the kernel of this degree map; i.e., the group of the divisor classes on
3967:{\displaystyle \operatorname {deg} :\operatorname {Pic} (C)\to \mathbb {Z} }
1820:{\displaystyle {\begin{aligned}y-x^{2}&=0\\z-x^{3}&=0\end{aligned}}}
205:
that one can construct genuinely new examples of varieties in this way, but
17:
6471:
These varieties have been called "varieties in the sense of Serre", since
6494:
may not be integral domains. A more significant modification is to allow
3329:
510:
algebraic subsets. An irreducible affine algebraic set is also called an
115:
6787:"On the imbedding problem of abstract varieties in projective varieties"
6449:
Some modern researchers also remove the restriction on a variety having
5957:
is an algebraic (affine) variety, since the origin is the zero-locus of
2551:
has a solution. This is best seen algebraically: the coordinate ring of
99:. Under this definition, non-irreducible algebraic varieties are called
7272:
7089:
6995:
111:
184:
is an integral (irreducible and reduced) scheme over that field whose
1072:, meaning that they were open subvarieties of closed subvarieties of
1024:) be the ideal generated by all homogeneous polynomials vanishing on
507:
7264:
6464:
is a variety such that any map from an open subset of a nonsingular
4553:, the set of isomorphism classes of smooth complete curves of genus
5831:
is not an algebraic variety (nor even an algebraic set). Note that
150:. This correspondence is a defining feature of algebraic geometry.
3362:
1068:
In classical algebraic geometry, all varieties were by definition
36:
3880:
of it; i.e., the group of isomorphism classes of line bundles on
3381:. The corresponding projective curve is called an elliptic curve.
1148:, which served a similar purpose, but was more general. However,
6831:"On the imbeddings of abstract surfaces in projective varieties"
5454:
of principally polarized complex abelian varieties of dimension
3805:, and the bracket means the line spanned by the nonzero vector
3110:); more precisely, the coordinate ring of the dual vector space
7411:. Oxford science publications. Oxford University Press. 2006.
7215:. Encyclopaedia of Mathematical Sciences. Vol. 23. 1994.
6566:
An algebraic manifold is an algebraic variety that is also an
6312:
3541:); in particular, it is not isomorphic to the projective line
2870:
in such a way the group operations are morphism of varieties.
1743:
shown in the above figure. It may be defined by the equations
27:
Mathematical object studied in the field of algebraic geometry
5787:{\displaystyle \{z\in \mathbb {C} {\text{ with }}|z|^{2}=1\}}
4149:{\displaystyle \operatorname {H} ^{1}(C,{\mathcal {O}}_{C});}
3406:. For another example, first consider the affine cubic curve
1169:
Existence of non-quasiprojective abstract algebraic varieties
6140:
of a variety may be defined in various equivalent ways. See
4129:
1651:
on which this function vanishes, that is the set of points (
6594:
are an equivalent definition for projective varieties. The
4699:, the set of isomorphism classes of stable curves of genus
3735:
2406:{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}
2312:{\displaystyle \mathbb {A} ^{n^{2}}\times \mathbb {A} ^{1}}
550:) to be the ideal of all polynomial functions vanishing on
518:
to refer to any affine algebraic set, irreducible or not.)
6965:"The irreducibility of the space of curves of given genus"
5653:
Another example of a non-affine non-projective variety is
5586:
that would be an analog of moduli stack of stable curves.
4979:. The moduli of semistable vector bundles of a given rank
7322:
Commutative
Algebra with a View Toward Algebraic Geometry
4959:
and semistable vector bundles on a smooth complete curve
5106:
of isomorphism classes of stable vector bundles of rank
3041:
is the coordinate ring of an affine (reducible) variety
4017:
of degree zero. A Jacobian variety is an example of an
7046:
Smooth compactification of locally symmetric varieties
6124:
is a variety if and only if its coordinate ring is an
5920:). On the other hand, the complement of the origin in
3328:. The notion plays an important role in the theory of
2888:
be a not-necessarily-commutative algebra over a field
2815:
and thus is an affine variety. A finite product of it
1695:; this name is also often given to the whole variety.
1084:
over an algebraically closed field is defined to be a
1042:
is the quotient of the polynomial ring by this ideal.
996:. An irreducible projective algebraic set is called a
60:. Classically, an algebraic variety is defined as the
7119:
6072:
6059:{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}
6031:
6018:{\displaystyle \operatorname {SL} _{n}(\mathbb {C} )}
5990:
5963:
5926:
5900:
5894:(although it is a polynomial in the real cooridnates
5880:
5837:
5800:
5735:
5713:
5684:
5544:
5512:
5480:
5460:
5429:
5401:
5358:
5335:
5294:
5266:
5246:
5226:
5198:
5176:
5152:
5132:
5112:
5070:
5025:
5005:
4985:
4965:
4931:
4900:
4869:
4829:
4798:
4762:
4731:
4705:
4669:
4643:
4603:
4583:
4559:
4533:
4499:
4469:
4434:
4293:
4264:
4244:
4217:
4194:
4162:
4103:
4071:
4039:
3983:
3934:
3894:
3850:
3769:
3596:
3470:
3415:
3300:
3270:
3240:
3214:
3158:
3116:
3086:
3062:
3021:
2931:
2906:
2821:
2782:
2683:
2599:
2557:
2519:
2458:
2419:
2368:
2329:
2274:
2235:
2188:
2168:
2132:
2097:
2067:
2047:
2024:
1977:
1947:
1911:
1752:
1571:
1467:{\displaystyle Z(f)=\{(x,1-x)\in \mathbf {C} ^{2}\}.}
1403:
1289:
875:
852:
of homogeneous polynomials, define the zero-locus of
563:
373:
262:
6498:
in the sheaf of rings, that is, rings which are not
5388:{\displaystyle \operatorname {Sp} (2g,\mathbb {Z} )}
3011:
is commutative, reduced and finitely generated as a
209:
gave an example of such a new variety in the 1950s.
169:
and algebraic varieties of dimension two are called
4097:at the identity element is naturally isomorphic to
708:
Projective varieties and quasi-projective varieties
103:. Other conventions do not require irreducibility.
7127:
6680:, p. 55 Definition 2.3.47, and p. 88 Example 3.2.3
6610:— listing also several mathematical meanings
6084:
6058:
6017:
5969:
5949:
5912:
5886:
5866:
5823:
5786:
5721:
5699:
5558:
5526:
5494:
5466:
5446:
5415:
5387:
5341:
5317:
5280:
5252:
5232:
5212:
5184:
5158:
5138:
5118:
5098:
5056:
5011:
4991:
4971:
4943:
4917:
4886:
4851:
4815:
4784:
4748:
4717:
4691:
4655:
4620:
4589:
4565:
4545:
4511:
4481:
4447:
4417:
4276:
4250:
4230:
4200:
4180:
4148:
4089:
4057:
4001:
3966:
3912:
3868:
3785:
3741:
3518:
3450:
3312:
3282:
3252:
3226:
3200:
3133:
3098:
3072:
3033:
3003:
2914:
2847:
2807:
2761:
2669:
2582:
2543:
2498:
2441:
2405:
2354:
2311:
2260:
2210:
2174:
2154:
2118:
2083:
2053:
2030:
2002:
1963:
1933:
1869:to compute the projection and to prove that it is
1819:
1624:
1466:
1328:
955:
673:
457:
275:
165:. Algebraic varieties of dimension one are called
6427:-algebras, that is to say, they are quotients of
7446:Creative Commons Attribution/Share-Alike License
7177:. Graduate Texts in Mathematics. Vol. 133.
6490:that aren't algebraically closed), so the rings
6340:but its sources remain unclear because it lacks
6073:
5568:projective variety associated to the graded ring
2747:
2651:
2523:
2465:
2110:
2048:
7084:. Lecture Notes in Mathematics. Vol. 812.
6513:= 0 is different from the subscheme defined by
4852:{\displaystyle {\overline {\mathfrak {M}}}_{g}}
4785:{\displaystyle {\overline {\mathfrak {M}}}_{g}}
4692:{\displaystyle {\overline {\mathfrak {M}}}_{g}}
4029:give an embedding into a projective space. See
3759:are any set of linearly independent vectors in
5506:of it. But there are other ways to compactify
4725:, is then a projective variety which contains
4493:is an elliptic curve), the above morphism for
4065:is a projective variety. The tangent space to
3812:The Grassmannian variety comes with a natural
2866:, an affine variety that has a structure of a
134:provides a fundamental correspondence between
6736:
6734:
6732:
6730:
6728:
6726:
5950:{\displaystyle \mathbb {A} ^{1}=\mathbb {C} }
5824:{\displaystyle \mathbb {A} ^{1}=\mathbb {C} }
5240:by an action of an arithmetic discrete group
5220:, the quotient of a bounded symmetric domain
2896:is not commutative, it can still happen that
2772:The multiplicative group k of the base field
1534:can be viewed as complex valued functions on
1248:can be viewed as complex valued functions on
8:
6724:
6722:
6720:
6718:
6716:
6714:
6712:
6710:
6708:
6706:
5781:
5736:
3691:
3659:
1458:
1419:
1181:that are not quasi-projective but complete.
947:
891:
506:if it cannot be written as the union of two
6415:-algebras with the property that the rings
1715:be the three-dimensional affine space over
7408:Algebraic geometry and arithmetic curves /
7082:Toroidal Compactification of Siegel Spaces
6544:There are further generalizations called
3928:and thus there is the degree homomorphism
3561:be a finite-dimensional vector space. The
2862:A general linear group is an example of a
2808:{\displaystyle \operatorname {GL} _{1}(k)}
2583:{\displaystyle \operatorname {GL} _{n}(k)}
2355:{\displaystyle \operatorname {GL} _{n}(k)}
2319:where the affine line is given coordinate
2261:{\displaystyle \operatorname {GL} _{n}(k)}
1881:to prove the irreducibility of the image.
7121:
7120:
7118:
6985:
6846:
6802:
6371:Learn how and when to remove this message
6071:
6049:
6048:
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4878:
4872:
4871:
4868:
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4833:
4831:
4828:
4807:
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4792:is obtained by adding boundary points to
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3507:
3491:
3475:
3469:
3433:
3420:
3414:
3299:
3269:
3239:
3213:
3201:{\displaystyle A_{i}M_{j}\subset M_{i+j}}
3186:
3173:
3163:
3157:
3125:
3119:
3118:
3115:
3085:
3064:
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3061:
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2475:
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2424:
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2397:
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2392:
2380:
2375:
2371:
2370:
2367:
2334:
2328:
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2299:
2298:
2286:
2281:
2277:
2276:
2273:
2240:
2234:
2200:
2195:
2191:
2190:
2187:
2167:
2144:
2139:
2135:
2134:
2131:
2096:
2072:
2066:
2046:
2023:
1982:
1976:
1952:
1946:
1923:
1918:
1914:
1913:
1910:
1797:
1767:
1753:
1751:
1610:
1597:
1570:
1526:be the two-dimensional affine space over
1452:
1447:
1402:
1288:
933:
906:
901:
874:
726:be an algebraically closed field and let
649:
619:
600:
562:
433:
406:
401:
372:
267:
261:
7436:This article incorporates material from
6678:Algebraic Geometry and Arithmetic Curves
6702:
6660:
4181:{\displaystyle \operatorname {Jac} (C)}
4090:{\displaystyle \operatorname {Jac} (C)}
4058:{\displaystyle \operatorname {Jac} (C)}
4021:, a complete variety with a compatible
4002:{\displaystyle \operatorname {Jac} (C)}
3913:{\displaystyle \operatorname {Pic} (C)}
3869:{\displaystyle \operatorname {Pic} (C)}
1088:, but from Chapter 2 onwards, the term
7052:, Brookline, Mass.: Math. Sci. Press,
6614:Function field of an algebraic variety
6517:= 0 (the origin). More generally, the
766:. It is not well-defined to evaluate
704:of the polynomial ring by this ideal.
364:simultaneously vanish, that is to say
6931:
6919:
6438:This definition works over any field
5667:Morphism of varieties § Examples
5590:Non-affine and non-projective example
5318:{\displaystyle D={\mathfrak {H}}_{g}}
3824:, which is important in the study of
1625:{\displaystyle g(x,y)=x^{2}+y^{2}-1.}
1497:
7:
6973:Publications Mathématiques de l'IHÉS
5423:has an interpretation as the moduli
4031:equations defining abelian varieties
3836:Jacobian variety and abelian variety
3519:{\displaystyle y^{2}z=x^{3}-xz^{2},}
3313:{\displaystyle \operatorname {gr} M}
3283:{\displaystyle \operatorname {gr} M}
3253:{\displaystyle \operatorname {gr} A}
3227:{\displaystyle \operatorname {gr} M}
3099:{\displaystyle \operatorname {gr} A}
3034:{\displaystyle \operatorname {gr} A}
2859:, which is again an affine variety.
2218:that consists of all the invertible
2211:{\displaystyle \mathbb {A} ^{n^{2}}}
2155:{\displaystyle \mathbb {A} ^{n^{2}}}
1934:{\displaystyle \mathbb {A} ^{n^{2}}}
848:vanishes at a point . For each set
180:theory, an algebraic variety over a
157:, but an algebraic variety may have
52:are the central objects of study in
7175:Algebraic Geometry - A first course
7016:
6943:
5447:{\displaystyle {\mathfrak {A}}_{g}}
5433:
5304:
4918:{\displaystyle {\mathfrak {M}}_{g}}
4904:
4887:{\displaystyle {\mathfrak {M}}_{g}}
4873:
4834:
4816:{\displaystyle {\mathfrak {M}}_{g}}
4802:
4767:
4749:{\displaystyle {\mathfrak {M}}_{g}}
4735:
4674:
4621:{\displaystyle {\mathfrak {M}}_{g}}
4607:
3134:{\displaystyle {\mathfrak {g}}^{*}}
3120:
3065:
1703:The following example is neither a
1691:are real numbers), is known as the
1028:. For any projective algebraic set
7294:; John Little; Don O'Shea (1997).
6578:(free from singular points). When
6153:Isomorphism of algebraic varieties
5566:due to Baily and Borel: it is the
5553:
5521:
5489:
5410:
5336:
5275:
5247:
5207:
4105:
2963:
2958:
2091:and thus defines the hypersurface
1901:can be identified with the affine
1873:injective and that its image is a
498:. A nonempty affine algebraic set
223:For an algebraically closed field
45:is a projective algebraic variety.
25:
7296:Ideals, Varieties, and Algorithms
7246:"Faisceaux Algebriques Coherents"
6586:, algebraic manifolds are called
6142:Dimension of an algebraic variety
3820:in other terminology) called the
1132:Foundations of Algebraic Geometry
188:is separated and of finite type.
83:, which means that it is not the
7043:; Rapoport, M.; Tai, Y. (1975),
6772:Hartshorne, Exercise I.2.9, p.12
6317:
5700:{\displaystyle \mathbb {A} ^{1}}
5192:is the problem of compactifying
3628:
1448:
902:
521:Affine varieties can be given a
402:
6884:Introduction to toric varieties
6574:. Equivalently, the variety is
6423:and are all finitely generated
6179:be algebraic varieties. We say
5729:. The complement of the circle
3844:be a smooth complete curve and
3073:{\displaystyle {\mathfrak {g}}}
2677:, which can be identified with
1538:by evaluating at the points in
1252:by evaluating at the points in
1076:. For example, in Chapter 1 of
514:. (Some authors use the phrase
7444:, which is licensed under the
6592:Projective algebraic manifolds
6309:Discussion and generalizations
6053:
6045:
6012:
6004:
5848:
5839:
5765:
5756:
5382:
5365:
5093:
5081:
5051:
5039:
4412:
4364:
4361:
4358:
4326:
4319:
4313:
4304:
4284:, there is a natural morphism
4175:
4169:
4140:
4117:
4084:
4078:
4052:
4046:
3996:
3990:
3956:
3953:
3947:
3907:
3901:
3863:
3857:
3729:
3697:
3694:
3624:
3621:
3615:
3451:{\displaystyle y^{2}=x^{3}-x.}
3320:does not vanish is called the
2836:
2822:
2802:
2796:
2756:
2741:
2733:
2687:
2664:
2646:
2643:
2603:
2577:
2571:
2532:
2526:
2499:{\displaystyle t\cdot \det-1,}
2484:
2468:
2349:
2343:
2255:
2249:
2113:
2107:
1997:
1991:
1587:
1575:
1440:
1422:
1413:
1407:
1305:
1293:
924:
918:
885:
879:
640:
634:
625:
593:
573:
567:
424:
418:
383:
377:
142:and algebraic sets. Using the
108:fundamental theorem of algebra
66:system of polynomial equations
1:
6667:Hartshorne, p.xv, Harris, p.3
4859:is colloquially said to be a
2362:amounts to the zero-locus in
1897:matrices over the base field
1379:in the affine plane. (In the
1329:{\displaystyle f(x,y)=x+y-1.}
792:is homogeneous, meaning that
684:For any affine algebraic set
354:) to be the set of points in
325:, i.e. by choosing values in
153:Many algebraic varieties are
7128:{\displaystyle \mathbb {C} }
6533:may be non-reduced, even if
6399:. Basically, a variety over
5722:{\displaystyle \mathbb {C} }
5534:; for example, there is the
5185:{\displaystyle \mathbb {C} }
4838:
4771:
4678:
3786:{\displaystyle \wedge ^{n}V}
3051:universal enveloping algebra
2915:{\displaystyle \mathbb {Z} }
2762:{\displaystyle k/(t\det -1)}
1165:algebraically closed field.
130:. Generalizing this result,
126:(a geometric object) in the
7244:Serre, Jean-Pierre (1955).
7147:10.1007/978-1-4613-8655-1_9
7080:Namikawa, Yukihiko (1980).
6649:Mnëv's universality theorem
6624:Motive (algebraic geometry)
5057:{\displaystyle SU_{C}(n,d)}
3537:. The curve has genus one (
3234:is fintiely generated as a
2848:{\displaystyle (k^{*})^{r}}
856:to be the set of points in
110:establishes a link between
32:Variety (universal algebra)
7483:
7377:Geometric invariant theory
6888:Princeton University Press
6559:
6389:abstract algebraic variety
6156:
6025:is a closed subvariety of
5630:is a closed subvariety of
5099:{\displaystyle U_{C}(n,d)}
4630:geometric invariant theory
3579:-dimensional subspaces of
3547:moduli of algebraic curves
3148:be a filtered module over
3106:is a polynomial ring (the
2877:
2544:{\displaystyle t\det(A)=1}
2509:i.e., the set of matrices
2182:is then an open subset of
2119:{\displaystyle H=V(\det )}
1711:, nor a single point. Let
1647:) is the set of points in
1530:. Polynomials in the ring
1244:. Polynomials in the ring
1070:quasi-projective varieties
862:on which the functions in
840:make sense to ask whether
711:
360:on which the functions in
216:
202:algebraically closed field
29:
7221:10.1007/978-3-642-57878-6
7187:10.1007/978-1-4757-2189-8
6521:of a morphism of schemes
6419:that occur above are all
5867:{\displaystyle |z|^{2}-1}
5678:Consider the affine line
5559:{\displaystyle D/\Gamma }
5527:{\displaystyle D/\Gamma }
5504:toroidal compactification
5495:{\displaystyle D/\Gamma }
5416:{\displaystyle D/\Gamma }
5327:Siegel's upper half-space
5281:{\displaystyle D/\Gamma }
5213:{\displaystyle D/\Gamma }
5064:, which contains the set
4756:as an open subset. Since
4633:leads to the notion of a
3920:can be identified as the
3529:which defines a curve in
2018:)-th entry of the matrix
2003:{\displaystyle x_{ij}(A)}
1550:contain a single element
1264:contain a single element
1206:Hilbert's Nullstellensatz
283:through the choice of an
176:In the context of modern
132:Hilbert's Nullstellensatz
7438:Isomorphism of varieties
6608:Variety (disambiguation)
6326:This section includes a
6101:An affine algebraic set
5536:minimal compactification
4156:hence, the dimension of
3294:; i.e., the locus where
3053:of a finite-dimensional
2922:-filtration so that the
2442:{\displaystyle x_{ij},t}
2061:is then a polynomial in
1879:polynomial factorization
1865:computation for another
1355:is the set of points in
1086:quasi-projective variety
1048:quasi-projective variety
978:projective algebraic set
718:Quasi-projective variety
346:, define the zero-locus
285:affine coordinate system
192:Overview and definitions
155:differentiable manifolds
6934:, The beginning of § 5.
6355:more precise citations.
6085:{\displaystyle \det -1}
5980:For similar reasons, a
5874:is not a polynomial in
5342:{\displaystyle \Gamma }
5253:{\displaystyle \Gamma }
4944:{\displaystyle g\geq 2}
4718:{\displaystyle g\geq 2}
4656:{\displaystyle g\geq 2}
4546:{\displaystyle g\geq 0}
3553:Example 2: Grassmannian
3367:The affine plane curve
3349:homogeneous polynomials
1236:be the two-dimensional
1144:made a definition of a
782:homogeneous coordinates
7129:
7019:, Appendix C to Ch. 5.
6848:10.1215/kjm/1250777007
6804:10.1215/kjm/1250777138
6506:'s theory of schemes.
6475:'s foundational paper
6086:
6060:
6019:
5971:
5951:
5914:
5888:
5868:
5825:
5788:
5723:
5701:
5634:(as the zero locus of
5580:Siegel modular variety
5560:
5528:
5496:
5468:
5448:
5417:
5389:
5343:
5319:
5282:
5254:
5234:
5214:
5186:
5160:
5140:
5120:
5100:
5058:
5013:
4993:
4973:
4945:
4919:
4888:
4853:
4817:
4786:
4750:
4719:
4693:
4657:
4622:
4591:
4567:
4547:
4513:
4483:
4449:
4419:
4278:
4277:{\displaystyle n>0}
4252:
4232:
4202:
4182:
4150:
4091:
4059:
4003:
3968:
3914:
3870:
3826:characteristic classes
3787:
3743:
3520:
3452:
3382:
3322:characteristic variety
3314:
3284:
3254:
3228:
3202:
3135:
3100:
3074:
3035:
3005:
2967:
2916:
2880:Characteristic variety
2874:Characteristic variety
2864:linear algebraic group
2849:
2809:
2763:
2671:
2584:
2545:
2500:
2443:
2407:
2356:
2313:
2262:
2212:
2176:
2156:
2120:
2085:
2084:{\displaystyle x_{ij}}
2055:
2032:
2004:
1965:
1964:{\displaystyle x_{ij}}
1935:
1821:
1681:absolutely irreducible
1626:
1468:
1330:
1150:Alexander Grothendieck
957:
760:homogeneous polynomial
675:
459:
277:
46:
7253:Annals of Mathematics
7130:
6634:Zariski–Riemann space
6582:is the real numbers,
6159:Morphism of varieties
6087:
6061:
6020:
5972:
5952:
5915:
5889:
5869:
5826:
5789:
5724:
5702:
5622:the projection. Here
5574:(in the Siegel case,
5561:
5529:
5497:
5469:
5449:
5418:
5390:
5344:
5320:
5283:
5260:. A basic example of
5255:
5235:
5215:
5187:
5161:
5141:
5121:
5101:
5059:
5014:
4994:
4974:
4946:
4920:
4889:
4854:
4818:
4787:
4751:
4720:
4694:
4658:
4623:
4592:
4568:
4548:
4514:
4484:
4450:
4448:{\displaystyle C^{n}}
4420:
4279:
4253:
4233:
4231:{\displaystyle P_{0}}
4203:
4183:
4151:
4092:
4060:
4004:
3969:
3915:
3871:
3788:
3744:
3521:
3453:
3366:
3315:
3285:
3255:
3229:
3203:
3136:
3101:
3075:
3036:
3006:
2944:
2917:
2850:
2810:
2764:
2672:
2585:
2546:
2501:
2444:
2413:of the polynomial in
2408:
2357:
2314:
2263:
2213:
2177:
2157:
2121:
2086:
2056:
2054:{\displaystyle \det }
2033:
2005:
1966:
1936:
1822:
1719:. The set of points (
1627:
1469:
1331:
958:
676:
460:
305:-valued functions on
278:
276:{\displaystyle K^{n}}
40:
7348:"Algebraic Geometry"
7212:Algebraic Geometry I
7141:. pp. 231–251.
7117:
6393:locally ringed space
6385:algebraically closed
6070:
6066:, the zero-locus of
6029:
5988:
5961:
5924:
5898:
5878:
5835:
5798:
5733:
5711:
5682:
5576:Siegel modular forms
5542:
5510:
5478:
5458:
5427:
5399:
5356:
5333:
5292:
5264:
5244:
5224:
5196:
5174:
5150:
5130:
5110:
5068:
5023:
5003:
4983:
4963:
4929:
4925:is irreducible when
4898:
4867:
4827:
4796:
4760:
4729:
4703:
4667:
4641:
4601:
4581:
4557:
4531:
4497:
4467:
4432:
4291:
4262:
4242:
4215:
4192:
4160:
4101:
4069:
4037:
3981:
3932:
3892:
3848:
3767:
3594:
3575:) is the set of all
3563:Grassmannian variety
3468:
3413:
3298:
3268:
3238:
3212:
3156:
3114:
3084:
3060:
3019:
2929:
2904:
2819:
2780:
2681:
2597:
2555:
2517:
2456:
2417:
2366:
2327:
2272:
2233:
2228:general linear group
2186:
2166:
2162:. The complement of
2130:
2095:
2065:
2045:
2022:
1975:
1945:
1909:
1885:General linear group
1750:
1569:
1401:
1287:
873:
784:. However, because
561:
480:affine algebraic set
371:
260:
7467:Algebraic varieties
7298:(second ed.).
7138:Arithmetic Geometry
6619:Birational geometry
6556:Algebraic manifolds
6429:polynomial algebras
5913:{\displaystyle x,y}
4999:and a given degree
4512:{\displaystyle n=1}
4482:{\displaystyle g=1}
4258:. For each integer
3922:divisor class group
3822:tautological bundle
3260:-algebra, then the
1375:. This is called a
935: for all
651: for all
435: for all
50:Algebraic varieties
7462:Algebraic geometry
7362:Jacobian Varieties
7125:
7090:10.1007/BFb0091051
6996:10.1007/bf02684599
6922:, Proposition 2.1.
6746:Algebraic Geometry
6639:Semi-algebraic set
6562:Algebraic manifold
6403:is a scheme whose
6397:spectrum of a ring
6328:list of references
6082:
6056:
6015:
5967:
5947:
5910:
5884:
5864:
5821:
5784:
5719:
5697:
5556:
5524:
5492:
5464:
5444:
5413:
5385:
5339:
5315:
5278:
5250:
5230:
5210:
5182:
5156:
5136:
5116:
5096:
5054:
5009:
4989:
4969:
4941:
4915:
4884:
4849:
4813:
4782:
4746:
4715:
4689:
4653:
4618:
4597:and is denoted as
4587:
4563:
4543:
4509:
4479:
4455:is the product of
4445:
4415:
4274:
4248:
4228:
4198:
4178:
4146:
4087:
4055:
3999:
3964:
3910:
3866:
3818:locally free sheaf
3783:
3739:
3734:
3516:
3448:
3383:
3345:projective variety
3339:Projective variety
3310:
3280:
3250:
3224:
3198:
3131:
3096:
3070:
3045:. For example, if
3031:
3001:
2912:
2845:
2805:
2759:
2667:
2580:
2541:
2496:
2439:
2403:
2352:
2309:
2258:
2208:
2172:
2152:
2116:
2081:
2051:
2028:
2000:
1961:
1931:
1817:
1815:
1635:The zero-locus of
1622:
1464:
1381:classical topology
1339:The zero-locus of
1326:
1120:Veronese embedding
1064:Abstract varieties
998:projective variety
953:
714:Projective variety
671:
455:
340:of polynomials in
287:. The polynomials
273:
186:structure morphism
171:algebraic surfaces
118:by showing that a
54:algebraic geometry
47:
7418:978-0-19-154780-5
7390:978-3-540-56963-3
7371:; Fogarty, John;
7230:978-3-540-63705-9
7156:978-1-4613-8657-5
7099:978-3-540-10021-8
7059:978-0-521-73955-9
6897:978-0-691-00049-7
6827:Nagata, Masayoshi
6783:Nagata, Masayoshi
6742:Hartshorne, Robin
6381:
6380:
6373:
5970:{\displaystyle z}
5887:{\displaystyle z}
5753:
5467:{\displaystyle g}
5233:{\displaystyle D}
5159:{\displaystyle C}
5139:{\displaystyle d}
5119:{\displaystyle n}
5012:{\displaystyle d}
4992:{\displaystyle n}
4972:{\displaystyle C}
4841:
4774:
4681:
4590:{\displaystyle g}
4566:{\displaystyle g}
4527:Given an integer
4251:{\displaystyle C}
4201:{\displaystyle C}
3585:Plücker embedding
2670:{\displaystyle k}
2175:{\displaystyle H}
2031:{\displaystyle A}
1941:with coordinates
1867:monomial ordering
1102:regular functions
1058:constructible set
936:
652:
525:by declaring the
436:
319:at the points in
301:can be viewed as
56:, a sub-field of
16:(Redirected from
7474:
7430:
7402:
7357:
7355:
7354:
7339:
7313:
7277:
7276:
7250:
7241:
7235:
7234:
7207:
7201:
7200:
7167:
7161:
7160:
7134:
7132:
7131:
7126:
7124:
7110:
7104:
7103:
7077:
7071:
7070:
7051:
7036:
7030:
7026:
7020:
7014:
7008:
7007:
6989:
6969:
6953:
6947:
6941:
6935:
6929:
6923:
6917:
6911:
6908:
6902:
6900:
6875:
6869:
6868:
6850:
6823:
6817:
6816:
6806:
6779:
6773:
6770:
6764:
6763:
6738:
6690:
6687:
6681:
6674:
6668:
6665:
6629:Analytic variety
6598:is one example.
6581:
6546:algebraic spaces
6489:
6481:sheaf cohomology
6462:complete variety
6441:
6426:
6421:integral domains
6414:
6402:
6376:
6369:
6365:
6362:
6356:
6351:this section by
6342:inline citations
6321:
6320:
6313:
6304:
6295:
6282:
6272:
6259:
6239:
6216:
6196:
6187:
6178:
6130:
6120:; equivalently,
6091:
6089:
6088:
6083:
6065:
6063:
6062:
6057:
6052:
6041:
6040:
6024:
6022:
6021:
6016:
6011:
6000:
5999:
5976:
5974:
5973:
5968:
5956:
5954:
5953:
5948:
5946:
5938:
5937:
5932:
5919:
5917:
5916:
5911:
5893:
5891:
5890:
5885:
5873:
5871:
5870:
5865:
5857:
5856:
5851:
5842:
5830:
5828:
5827:
5822:
5820:
5812:
5811:
5806:
5793:
5791:
5790:
5785:
5774:
5773:
5768:
5759:
5754:
5752: with
5751:
5749:
5728:
5726:
5725:
5720:
5718:
5706:
5704:
5703:
5698:
5696:
5695:
5690:
5663:
5640:regular function
5621:
5607:
5565:
5563:
5562:
5557:
5552:
5533:
5531:
5530:
5525:
5520:
5501:
5499:
5498:
5493:
5488:
5473:
5471:
5470:
5465:
5453:
5451:
5450:
5445:
5443:
5442:
5437:
5436:
5422:
5420:
5419:
5414:
5409:
5395:; in that case,
5394:
5392:
5391:
5386:
5381:
5348:
5346:
5345:
5340:
5324:
5322:
5321:
5316:
5314:
5313:
5308:
5307:
5287:
5285:
5284:
5279:
5274:
5259:
5257:
5256:
5251:
5239:
5237:
5236:
5231:
5219:
5217:
5216:
5211:
5206:
5191:
5189:
5188:
5183:
5181:
5165:
5163:
5162:
5157:
5145:
5143:
5142:
5137:
5125:
5123:
5122:
5117:
5105:
5103:
5102:
5097:
5080:
5079:
5063:
5061:
5060:
5055:
5038:
5037:
5018:
5016:
5015:
5010:
4998:
4996:
4995:
4990:
4978:
4976:
4975:
4970:
4950:
4948:
4947:
4942:
4924:
4922:
4921:
4916:
4914:
4913:
4908:
4907:
4893:
4891:
4890:
4885:
4883:
4882:
4877:
4876:
4861:compactification
4858:
4856:
4855:
4850:
4848:
4847:
4842:
4837:
4832:
4822:
4820:
4819:
4814:
4812:
4811:
4806:
4805:
4791:
4789:
4788:
4783:
4781:
4780:
4775:
4770:
4765:
4755:
4753:
4752:
4747:
4745:
4744:
4739:
4738:
4724:
4722:
4721:
4716:
4698:
4696:
4695:
4690:
4688:
4687:
4682:
4677:
4672:
4662:
4660:
4659:
4654:
4627:
4625:
4624:
4619:
4617:
4616:
4611:
4610:
4596:
4594:
4593:
4588:
4575:moduli of curves
4572:
4570:
4569:
4564:
4552:
4550:
4549:
4544:
4523:Moduli varieties
4518:
4516:
4515:
4510:
4488:
4486:
4485:
4480:
4454:
4452:
4451:
4446:
4444:
4443:
4424:
4422:
4421:
4416:
4411:
4410:
4395:
4394:
4376:
4375:
4357:
4356:
4338:
4337:
4303:
4302:
4283:
4281:
4280:
4275:
4257:
4255:
4254:
4249:
4237:
4235:
4234:
4229:
4227:
4226:
4207:
4205:
4204:
4199:
4188:is the genus of
4187:
4185:
4184:
4179:
4155:
4153:
4152:
4147:
4139:
4138:
4133:
4132:
4113:
4112:
4096:
4094:
4093:
4088:
4064:
4062:
4061:
4056:
4008:
4006:
4005:
4000:
3976:Jacobian variety
3973:
3971:
3970:
3965:
3963:
3919:
3917:
3916:
3911:
3875:
3873:
3872:
3867:
3792:
3790:
3789:
3784:
3779:
3778:
3748:
3746:
3745:
3740:
3738:
3737:
3728:
3727:
3709:
3708:
3690:
3689:
3671:
3670:
3654:
3650:
3646:
3645:
3631:
3614:
3613:
3525:
3523:
3522:
3517:
3512:
3511:
3496:
3495:
3480:
3479:
3457:
3455:
3454:
3449:
3438:
3437:
3425:
3424:
3405:
3398:
3380:
3351:that generate a
3319:
3317:
3316:
3311:
3289:
3287:
3286:
3281:
3259:
3257:
3256:
3251:
3233:
3231:
3230:
3225:
3207:
3205:
3204:
3199:
3197:
3196:
3178:
3177:
3168:
3167:
3140:
3138:
3137:
3132:
3130:
3129:
3124:
3123:
3105:
3103:
3102:
3097:
3079:
3077:
3076:
3071:
3069:
3068:
3040:
3038:
3037:
3032:
3015:-algebra; i.e.,
3010:
3008:
3007:
3002:
3000:
2999:
2998:
2982:
2977:
2976:
2966:
2961:
2921:
2919:
2918:
2913:
2911:
2854:
2852:
2851:
2846:
2844:
2843:
2834:
2833:
2814:
2812:
2811:
2806:
2792:
2791:
2768:
2766:
2765:
2760:
2740:
2702:
2701:
2676:
2674:
2673:
2668:
2663:
2662:
2654:
2618:
2617:
2589:
2587:
2586:
2581:
2567:
2566:
2550:
2548:
2547:
2542:
2505:
2503:
2502:
2497:
2483:
2482:
2448:
2446:
2445:
2440:
2432:
2431:
2412:
2410:
2409:
2404:
2402:
2401:
2396:
2387:
2386:
2385:
2384:
2374:
2361:
2359:
2358:
2353:
2339:
2338:
2318:
2316:
2315:
2310:
2308:
2307:
2302:
2293:
2292:
2291:
2290:
2280:
2267:
2265:
2264:
2259:
2245:
2244:
2217:
2215:
2214:
2209:
2207:
2206:
2205:
2204:
2194:
2181:
2179:
2178:
2173:
2161:
2159:
2158:
2153:
2151:
2150:
2149:
2148:
2138:
2125:
2123:
2122:
2117:
2090:
2088:
2087:
2082:
2080:
2079:
2060:
2058:
2057:
2052:
2037:
2035:
2034:
2029:
2009:
2007:
2006:
2001:
1990:
1989:
1970:
1968:
1967:
1962:
1960:
1959:
1940:
1938:
1937:
1932:
1930:
1929:
1928:
1927:
1917:
1877:, and finally a
1826:
1824:
1823:
1818:
1816:
1802:
1801:
1772:
1771:
1631:
1629:
1628:
1623:
1615:
1614:
1602:
1601:
1521:
1491:
1477:Thus the subset
1473:
1471:
1470:
1465:
1457:
1456:
1451:
1393:
1354:
1335:
1333:
1332:
1327:
1279:
1231:
1200:closed immersion
1142:Claude Chevalley
1113:
1098:projective space
1094:abstract variety
1092:(also called an
1074:projective space
1015:
975:
962:
960:
959:
954:
937:
934:
911:
910:
905:
861:
847:
835:
791:
779:
773:
757:
751:
743:
731:
725:
680:
678:
677:
672:
667:
663:
653:
650:
624:
623:
605:
604:
541:
523:natural topology
477:
464:
462:
461:
456:
451:
447:
437:
434:
411:
410:
405:
359:
345:
324:
318:
310:
300:
294:
282:
280:
279:
274:
272:
271:
256:, identified to
255:
247:
239:
233:
226:
213:Affine varieties
167:algebraic curves
140:polynomial rings
120:monic polynomial
97:Zariski topology
62:set of solutions
21:
7482:
7481:
7477:
7476:
7475:
7473:
7472:
7471:
7452:
7451:
7433:
7419:
7405:
7391:
7381:Springer-Verlag
7373:Kirwan, Frances
7367:
7352:
7350:
7344:Milne, James S.
7342:
7336:
7326:Springer-Verlag
7318:Eisenbud, David
7316:
7310:
7300:Springer-Verlag
7290:
7286:
7281:
7280:
7265:10.2307/1969915
7248:
7243:
7242:
7238:
7231:
7209:
7208:
7204:
7197:
7179:Springer-Verlag
7169:
7168:
7164:
7157:
7115:
7114:
7112:
7111:
7107:
7100:
7079:
7078:
7074:
7060:
7049:
7038:
7037:
7033:
7027:
7023:
7015:
7011:
6967:
6957:Deligne, Pierre
6955:
6954:
6950:
6946:, Theorem 5.11.
6942:
6938:
6930:
6926:
6918:
6914:
6909:
6905:
6898:
6880:Fulton, William
6878:
6876:
6872:
6825:
6824:
6820:
6781:
6780:
6776:
6771:
6767:
6760:
6750:Springer-Verlag
6740:
6739:
6704:
6699:
6694:
6693:
6688:
6684:
6675:
6671:
6666:
6662:
6657:
6604:
6579:
6564:
6558:
6487:
6451:integral domain
6439:
6424:
6412:
6405:structure sheaf
6400:
6377:
6366:
6360:
6357:
6346:
6332:related reading
6322:
6318:
6311:
6303:
6297:
6294:
6288:
6274:
6264:
6258:
6251:
6241:
6238:
6231:
6221:
6217:, if there are
6215:
6208:
6202:
6195:
6189:
6186:
6180:
6177:
6170:
6164:
6161:
6155:
6127:integral domain
6125:
6098:
6068:
6067:
6032:
6027:
6026:
5991:
5986:
5985:
5959:
5958:
5927:
5922:
5921:
5896:
5895:
5876:
5875:
5846:
5833:
5832:
5801:
5796:
5795:
5763:
5731:
5730:
5709:
5708:
5685:
5680:
5679:
5676:
5654:
5609:
5595:
5592:
5540:
5539:
5508:
5507:
5476:
5475:
5456:
5455:
5430:
5425:
5424:
5397:
5396:
5354:
5353:
5331:
5330:
5301:
5290:
5289:
5262:
5261:
5242:
5241:
5222:
5221:
5194:
5193:
5172:
5171:
5148:
5147:
5128:
5127:
5108:
5107:
5071:
5066:
5065:
5029:
5021:
5020:
5001:
5000:
4981:
4980:
4961:
4960:
4927:
4926:
4901:
4896:
4895:
4870:
4865:
4864:
4830:
4825:
4824:
4799:
4794:
4793:
4763:
4758:
4757:
4732:
4727:
4726:
4701:
4700:
4670:
4665:
4664:
4639:
4638:
4604:
4599:
4598:
4579:
4578:
4555:
4554:
4529:
4528:
4525:
4495:
4494:
4465:
4464:
4435:
4430:
4429:
4402:
4386:
4367:
4348:
4329:
4294:
4289:
4288:
4260:
4259:
4240:
4239:
4218:
4213:
4212:
4190:
4189:
4158:
4157:
4126:
4104:
4099:
4098:
4067:
4066:
4035:
4034:
4027:theta functions
4019:abelian variety
3979:
3978:
3930:
3929:
3890:
3889:
3846:
3845:
3838:
3770:
3765:
3764:
3757:
3733:
3732:
3719:
3700:
3681:
3662:
3656:
3655:
3637:
3636:
3632:
3605:
3598:
3592:
3591:
3569:
3555:
3503:
3487:
3471:
3466:
3465:
3429:
3416:
3411:
3410:
3400:
3393:
3387:projective line
3368:
3361:
3341:
3296:
3295:
3266:
3265:
3236:
3235:
3210:
3209:
3182:
3169:
3159:
3154:
3153:
3117:
3112:
3111:
3082:
3081:
3058:
3057:
3017:
3016:
2984:
2968:
2927:
2926:
2924:associated ring
2902:
2901:
2882:
2876:
2857:algebraic torus
2835:
2825:
2817:
2816:
2783:
2778:
2777:
2776:is the same as
2690:
2679:
2678:
2649:
2606:
2595:
2594:
2558:
2553:
2552:
2515:
2514:
2471:
2454:
2453:
2420:
2415:
2414:
2391:
2376:
2369:
2364:
2363:
2330:
2325:
2324:
2297:
2282:
2275:
2270:
2269:
2236:
2231:
2230:
2196:
2189:
2184:
2183:
2164:
2163:
2140:
2133:
2128:
2127:
2093:
2092:
2068:
2063:
2062:
2043:
2042:
2020:
2019:
1978:
1973:
1972:
1948:
1943:
1942:
1919:
1912:
1907:
1906:
1887:
1814:
1813:
1803:
1793:
1784:
1783:
1773:
1763:
1748:
1747:
1701:
1606:
1593:
1567:
1566:
1513:
1510:
1478:
1446:
1399:
1398:
1384:
1340:
1285:
1284:
1265:
1223:
1220:
1215:
1192:
1187:
1179:toric varieties
1171:
1116:Segre embedding
1105:
1066:
1035:coordinate ring
1011:
1006:Given a subset
971:
900:
871:
870:
857:
841:
832:
826:
810:
804:
793:
785:
775:
767:
753:
745:
741:
727:
723:
720:
712:Main articles:
710:
690:coordinate ring
615:
596:
583:
579:
559:
558:
537:
532:Given a subset
473:
400:
393:
389:
369:
368:
355:
341:
336:. For each set
334:
320:
312:
306:
296:
288:
263:
258:
257:
251:
243:
235:
231:
224:
221:
215:
194:
159:singular points
144:Nullstellensatz
87:of two smaller
74:complex numbers
35:
28:
23:
22:
15:
12:
11:
5:
7480:
7478:
7470:
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7454:
7453:
7432:
7431:
7417:
7403:
7389:
7369:Mumford, David
7365:
7358:
7340:
7334:
7314:
7308:
7287:
7285:
7282:
7279:
7278:
7259:(2): 197–278.
7236:
7229:
7202:
7195:
7162:
7155:
7123:
7105:
7098:
7072:
7058:
7041:Mumford, David
7031:
7021:
7009:
6987:10.1.1.589.288
6961:Mumford, David
6948:
6936:
6924:
6912:
6903:
6896:
6877:In page 65 of
6870:
6841:(3): 231–235.
6818:
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6626:
6621:
6616:
6611:
6603:
6600:
6596:Riemann sphere
6588:Nash manifolds
6560:Main article:
6557:
6554:
6529:at a point of
6379:
6378:
6336:external links
6325:
6323:
6316:
6310:
6307:
6305:respectively.
6301:
6292:
6260:such that the
6256:
6249:
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6229:
6213:
6206:
6193:
6184:
6175:
6168:
6154:
6151:
6150:
6149:
6145:
6134:
6131:
6107:if and only if
6097:
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6081:
6078:
6075:
6055:
6051:
6047:
6044:
6039:
6035:
6014:
6010:
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5855:
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5783:
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5777:
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4714:
4711:
4708:
4686:
4680:
4676:
4652:
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4615:
4609:
4586:
4573:is called the
4562:
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3799:exterior power
3782:
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3535:elliptic curve
3527:
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2878:Main article:
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2279:
2257:
2254:
2251:
2248:
2243:
2239:
2226:matrices, the
2203:
2199:
2193:
2171:
2147:
2143:
2137:
2115:
2112:
2109:
2106:
2103:
2100:
2078:
2075:
2071:
2050:
2027:
1999:
1996:
1993:
1988:
1985:
1981:
1958:
1955:
1951:
1926:
1922:
1916:
1886:
1883:
1828:
1827:
1812:
1809:
1806:
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1796:
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1427:
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1406:
1337:
1336:
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1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1219:
1216:
1214:
1213:Affine variety
1211:
1191:
1188:
1186:
1183:
1170:
1167:
1065:
1062:
964:
963:
952:
949:
946:
943:
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932:
929:
926:
923:
920:
917:
914:
909:
904:
899:
896:
893:
890:
887:
884:
881:
878:
830:
824:
808:
802:
709:
706:
694:structure ring
682:
681:
670:
666:
662:
659:
656:
648:
645:
642:
639:
636:
633:
630:
627:
622:
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611:
608:
603:
599:
595:
592:
589:
586:
582:
578:
575:
572:
569:
566:
516:affine variety
512:affine variety
466:
465:
454:
450:
446:
443:
440:
432:
429:
426:
423:
420:
417:
414:
409:
404:
399:
396:
392:
388:
385:
382:
379:
376:
332:
311:by evaluating
270:
266:
229:natural number
219:Affine variety
217:Main article:
214:
211:
198:affine variety
193:
190:
101:algebraic sets
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7479:
7468:
7465:
7463:
7460:
7459:
7457:
7450:
7449:
7447:
7443:
7439:
7428:
7424:
7420:
7414:
7410:
7409:
7404:
7400:
7396:
7392:
7386:
7382:
7378:
7374:
7370:
7366:
7363:
7359:
7349:
7345:
7341:
7337:
7335:0-387-94269-6
7331:
7327:
7323:
7319:
7315:
7311:
7309:0-387-94680-2
7305:
7301:
7297:
7293:
7289:
7288:
7283:
7274:
7270:
7266:
7262:
7258:
7254:
7247:
7240:
7237:
7232:
7226:
7222:
7218:
7214:
7213:
7206:
7203:
7198:
7196:0-387-97716-3
7192:
7188:
7184:
7180:
7176:
7172:
7166:
7163:
7158:
7152:
7148:
7144:
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7018:
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7005:
7001:
6997:
6993:
6988:
6983:
6979:
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6974:
6966:
6962:
6958:
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6937:
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6928:
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6916:
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6907:
6904:
6899:
6893:
6889:
6885:
6881:
6874:
6871:
6866:
6862:
6858:
6854:
6849:
6844:
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6836:
6832:
6828:
6822:
6819:
6814:
6810:
6805:
6800:
6796:
6792:
6788:
6784:
6778:
6775:
6769:
6766:
6761:
6759:0-387-90244-9
6755:
6751:
6747:
6743:
6737:
6735:
6733:
6731:
6729:
6727:
6725:
6723:
6721:
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6709:
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6703:
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6686:
6683:
6679:
6673:
6670:
6664:
6661:
6654:
6650:
6647:
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6640:
6637:
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6627:
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6622:
6620:
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6615:
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6609:
6606:
6605:
6601:
6599:
6597:
6593:
6589:
6585:
6577:
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6569:
6563:
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6536:
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6516:
6512:
6507:
6505:
6501:
6497:
6493:
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6469:
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6406:
6398:
6394:
6390:
6386:
6375:
6372:
6364:
6354:
6350:
6344:
6343:
6337:
6333:
6329:
6324:
6315:
6314:
6308:
6306:
6300:
6291:
6286:
6285:identity maps
6281:
6277:
6271:
6267:
6263:
6255:
6248:
6244:
6235:
6228:
6224:
6220:
6212:
6205:
6200:
6192:
6183:
6174:
6167:
6160:
6152:
6146:
6143:
6139:
6135:
6132:
6128:
6123:
6119:
6115:
6111:
6108:
6105:is a variety
6104:
6100:
6099:
6096:Basic results
6095:
6093:
6079:
6076:
6042:
6037:
6033:
6001:
5996:
5992:
5983:
5982:unitary group
5978:
5964:
5939:
5934:
5907:
5904:
5901:
5881:
5861:
5858:
5853:
5843:
5813:
5808:
5778:
5775:
5770:
5760:
5742:
5739:
5692:
5673:
5671:
5669:
5668:
5661:
5657:
5651:
5649:
5645:
5641:
5637:
5633:
5629:
5625:
5620:
5616:
5612:
5606:
5602:
5598:
5589:
5587:
5585:
5581:
5577:
5573:
5572:modular forms
5569:
5549:
5545:
5537:
5517:
5513:
5505:
5485:
5481:
5461:
5439:
5406:
5402:
5374:
5371:
5368:
5362:
5359:
5351:
5350:commensurable
5328:
5310:
5298:
5295:
5271:
5267:
5227:
5203:
5199:
5167:
5153:
5133:
5113:
5090:
5087:
5084:
5076:
5072:
5048:
5045:
5042:
5034:
5030:
5026:
5006:
4986:
4966:
4958:
4952:
4938:
4935:
4932:
4910:
4879:
4862:
4844:
4808:
4777:
4741:
4712:
4709:
4706:
4684:
4650:
4647:
4644:
4636:
4631:
4613:
4584:
4576:
4560:
4540:
4537:
4534:
4522:
4520:
4506:
4503:
4500:
4492:
4476:
4473:
4470:
4462:
4458:
4440:
4436:
4407:
4403:
4399:
4396:
4391:
4387:
4383:
4380:
4377:
4372:
4368:
4353:
4349:
4345:
4342:
4339:
4334:
4330:
4322:
4316:
4310:
4307:
4299:
4295:
4287:
4286:
4285:
4271:
4268:
4265:
4245:
4223:
4219:
4209:
4195:
4172:
4166:
4163:
4143:
4135:
4123:
4120:
4114:
4109:
4081:
4075:
4072:
4049:
4043:
4040:
4032:
4028:
4024:
4023:abelian group
4020:
4016:
4012:
3993:
3987:
3984:
3977:
3950:
3944:
3941:
3938:
3935:
3927:
3923:
3904:
3898:
3895:
3887:
3883:
3879:
3860:
3854:
3851:
3843:
3835:
3833:
3831:
3830:Chern classes
3827:
3823:
3819:
3815:
3814:vector bundle
3810:
3808:
3804:
3800:
3796:
3780:
3775:
3771:
3762:
3758:
3724:
3720:
3716:
3713:
3710:
3705:
3701:
3686:
3682:
3678:
3675:
3672:
3667:
3663:
3651:
3647:
3642:
3638:
3633:
3618:
3610:
3606:
3599:
3590:
3589:
3588:
3586:
3582:
3578:
3574:
3570:
3564:
3560:
3552:
3550:
3548:
3544:
3540:
3539:genus formula
3536:
3532:
3513:
3508:
3504:
3500:
3497:
3492:
3488:
3484:
3481:
3476:
3472:
3464:
3463:
3462:
3445:
3442:
3439:
3434:
3430:
3426:
3421:
3417:
3409:
3408:
3407:
3403:
3399:} defined by
3396:
3391:
3388:
3379:
3375:
3371:
3365:
3358:
3356:
3354:
3350:
3346:
3338:
3336:
3334:
3332:
3327:
3323:
3307:
3304:
3301:
3293:
3277:
3274:
3271:
3263:
3247:
3244:
3241:
3221:
3218:
3215:
3193:
3190:
3187:
3183:
3179:
3174:
3170:
3164:
3160:
3151:
3147:
3142:
3126:
3109:
3093:
3090:
3087:
3056:
3052:
3048:
3044:
3028:
3025:
3022:
3014:
2995:
2992:
2989:
2985:
2979:
2973:
2969:
2955:
2952:
2949:
2945:
2941:
2938:
2935:
2932:
2925:
2899:
2895:
2891:
2887:
2881:
2873:
2871:
2869:
2865:
2860:
2858:
2840:
2830:
2826:
2799:
2793:
2788:
2784:
2775:
2770:
2753:
2750:
2744:
2737:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2698:
2695:
2691:
2684:
2659:
2656:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2614:
2611:
2607:
2600:
2593:
2574:
2568:
2563:
2559:
2538:
2535:
2529:
2520:
2512:
2493:
2490:
2487:
2479:
2476:
2472:
2462:
2459:
2452:
2451:
2450:
2436:
2433:
2428:
2425:
2421:
2398:
2388:
2381:
2377:
2346:
2340:
2335:
2331:
2322:
2304:
2294:
2287:
2283:
2252:
2246:
2241:
2237:
2229:
2225:
2221:
2201:
2197:
2169:
2145:
2141:
2104:
2101:
2098:
2076:
2073:
2069:
2041:
2025:
2017:
2013:
1994:
1986:
1983:
1979:
1956:
1953:
1949:
1924:
1920:
1904:
1900:
1896:
1892:
1884:
1882:
1880:
1876:
1872:
1868:
1864:
1863:Gröbner basis
1860:
1859:Gröbner basis
1855:
1853:
1849:
1845:
1841:
1837:
1833:
1810:
1807:
1805:
1798:
1794:
1790:
1787:
1780:
1777:
1775:
1768:
1764:
1760:
1757:
1746:
1745:
1744:
1742:
1741:twisted cubic
1738:
1734:
1730:
1726:
1722:
1718:
1714:
1710:
1706:
1698:
1696:
1694:
1690:
1686:
1682:
1678:
1674:
1670:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1638:
1619:
1616:
1611:
1607:
1603:
1598:
1594:
1590:
1584:
1581:
1578:
1572:
1565:
1564:
1563:
1561:
1557:
1553:
1549:
1545:
1542:. Let subset
1541:
1537:
1533:
1529:
1525:
1520:
1516:
1507:
1505:
1503:
1499:
1498:algebraic set
1495:
1489:
1485:
1481:
1461:
1453:
1443:
1437:
1434:
1431:
1428:
1425:
1416:
1410:
1404:
1397:
1396:
1395:
1391:
1387:
1382:
1378:
1374:
1370:
1366:
1362:
1358:
1352:
1348:
1344:
1323:
1320:
1317:
1314:
1311:
1308:
1302:
1299:
1296:
1290:
1283:
1282:
1281:
1277:
1273:
1269:
1263:
1259:
1256:. Let subset
1255:
1251:
1247:
1243:
1239:
1235:
1230:
1226:
1217:
1212:
1210:
1207:
1203:
1201:
1197:
1189:
1184:
1182:
1180:
1176:
1168:
1166:
1163:
1159:
1155:
1151:
1147:
1143:
1139:
1135:
1133:
1128:
1123:
1121:
1117:
1112:
1108:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1063:
1061:
1059:
1054:
1050:
1049:
1043:
1041:
1037:
1036:
1031:
1027:
1023:
1019:
1014:
1009:
1004:
1001:
999:
995:
991:
987:
983:
979:
974:
969:
950:
944:
941:
938:
930:
927:
921:
915:
912:
907:
897:
894:
888:
882:
876:
869:
868:
867:
865:
860:
855:
851:
845:
839:
833:
823:
819:
815:
811:
801:
797:
789:
783:
778:
774:on points in
771:
765:
761:
756:
749:
739:
737:
730:
719:
715:
707:
705:
703:
699:
695:
691:
687:
668:
664:
660:
657:
654:
646:
643:
637:
631:
628:
620:
616:
612:
609:
606:
601:
597:
590:
587:
584:
580:
576:
570:
564:
557:
556:
555:
553:
549:
545:
540:
535:
530:
528:
524:
519:
517:
513:
509:
505:
501:
497:
493:
489:
485:
481:
478:is called an
476:
471:
452:
448:
444:
441:
438:
430:
427:
421:
415:
412:
407:
397:
394:
390:
386:
380:
374:
367:
366:
365:
363:
358:
353:
349:
344:
339:
335:
328:
323:
316:
309:
304:
299:
292:
286:
268:
264:
254:
249:
246:
238:
230:
220:
212:
210:
208:
203:
199:
191:
189:
187:
183:
179:
174:
172:
168:
164:
160:
156:
151:
149:
145:
141:
137:
133:
129:
128:complex plane
125:
121:
117:
113:
109:
104:
102:
98:
94:
90:
86:
82:
77:
75:
71:
67:
63:
59:
55:
51:
44:
43:twisted cubic
39:
33:
19:
7435:
7434:
7407:
7376:
7351:. Retrieved
7321:
7295:
7256:
7252:
7239:
7211:
7205:
7174:
7165:
7137:
7108:
7081:
7075:
7045:
7034:
7024:
7012:
6977:
6971:
6951:
6939:
6927:
6915:
6906:
6883:
6873:
6838:
6834:
6821:
6794:
6790:
6777:
6768:
6745:
6685:
6677:
6672:
6663:
6644:Fano variety
6583:
6571:
6567:
6565:
6543:
6538:
6534:
6530:
6526:
6522:
6514:
6510:
6508:
6504:Grothendieck
6499:
6491:
6485:
6470:
6459:
6448:
6443:
6437:
6433:prime ideals
6416:
6388:
6382:
6367:
6358:
6347:Please help
6339:
6298:
6289:
6279:
6275:
6269:
6265:
6262:compositions
6253:
6246:
6242:
6233:
6226:
6222:
6219:regular maps
6210:
6203:
6201:, and write
6190:
6181:
6172:
6165:
6162:
6144:for details.
6137:
6121:
6113:
6109:
6102:
5979:
5677:
5674:Non-examples
5665:
5659:
5655:
5652:
5647:
5643:
5635:
5631:
5627:
5623:
5618:
5614:
5610:
5604:
5600:
5596:
5593:
5584:moduli stack
5168:
4953:
4635:stable curve
4526:
4490:
4460:
4456:
4427:
4211:Fix a point
4210:
4014:
4010:
3925:
3885:
3881:
3878:Picard group
3841:
3839:
3811:
3806:
3802:
3794:
3760:
3753:
3751:
3580:
3576:
3572:
3565:
3558:
3556:
3542:
3530:
3528:
3460:
3401:
3394:
3389:
3384:
3377:
3373:
3369:
3342:
3330:
3325:
3291:
3149:
3145:
3143:
3046:
3042:
3012:
2897:
2893:
2889:
2885:
2883:
2861:
2773:
2771:
2592:localization
2510:
2508:
2320:
2223:
2219:
2015:
2011:
1902:
1898:
1894:
1890:
1888:
1875:hypersurface
1856:
1847:
1843:
1839:
1835:
1831:
1829:
1736:
1732:
1728:
1724:
1720:
1716:
1712:
1709:linear space
1705:hypersurface
1702:
1688:
1684:
1676:
1672:
1668:
1664:
1660:
1659:) such that
1656:
1652:
1648:
1644:
1640:
1636:
1634:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1531:
1527:
1523:
1518:
1514:
1511:
1501:
1493:
1487:
1483:
1479:
1476:
1389:
1385:
1380:
1372:
1368:
1367:) such that
1364:
1360:
1356:
1350:
1346:
1342:
1338:
1275:
1271:
1267:
1261:
1257:
1253:
1249:
1245:
1241:
1238:affine space
1233:
1228:
1224:
1221:
1204:
1195:
1193:
1172:
1130:
1124:
1110:
1106:
1093:
1089:
1081:
1067:
1053:Zariski open
1046:
1044:
1039:
1033:
1029:
1025:
1021:
1017:
1012:
1007:
1005:
1002:
997:
993:
989:
985:
981:
977:
976:is called a
972:
967:
965:
863:
858:
853:
849:
843:
837:
828:
821:
817:
813:
806:
799:
795:
787:
776:
769:
763:
754:
747:
735:
728:
721:
697:
693:
689:
685:
683:
551:
547:
543:
542:, we define
538:
533:
531:
520:
515:
511:
503:
499:
495:
491:
487:
483:
479:
474:
469:
467:
361:
356:
351:
347:
342:
337:
330:
326:
321:
314:
307:
302:
297:
295:in the ring
290:
252:
244:
236:
222:
197:
195:
175:
152:
143:
105:
100:
78:
49:
48:
18:Affine curve
7171:Harris, Joe
6676:Liu, Qing.
6353:introducing
6148:projective.
6118:prime ideal
5578:; see also
5126:and degree
3888:is smooth,
3353:prime ideal
3108:PBW theorem
3055:Lie algebra
2040:determinant
1889:The set of
1871:generically
1693:unit circle
1162:finite type
992:) for some
734:projective
527:closed sets
504:irreducible
494:) for some
148:ring theory
81:irreducible
58:mathematics
7456:Categories
7442:PlanetMath
7360:Milne J.,
7353:2009-09-01
7292:Cox, David
6980:: 75–109.
6932:Milne 2008
6920:Milne 2008
6697:References
6496:nilpotents
6455:nilradical
6361:March 2013
6199:isomorphic
6157:See also:
5646:; namely,
5570:formed by
4459:copies of
3533:called an
2892:. Even if
2513:such that
1971:such that
1500:. The set
1196:subvariety
1190:Subvariety
1160:scheme of
1138:valuations
1127:André Weil
1078:Hartshorne
762:of degree
502:is called
7427:646747871
7039:Ash, A.;
6982:CiteSeerX
6865:118328992
6797:: 71–82.
6444:separated
6138:dimension
6077:−
6043:
6002:
5859:−
5743:∈
5554:Γ
5522:Γ
5490:Γ
5411:Γ
5363:
5337:Γ
5276:Γ
5248:Γ
5208:Γ
4936:≥
4839:¯
4772:¯
4710:≥
4679:¯
4648:≥
4637:of genus
4577:of genus
4538:≥
4397:−
4381:⋯
4362:↦
4343:…
4311:
4305:→
4167:
4115:
4076:
4044:
4033:); thus,
3988:
3957:→
3945:
3899:
3855:
3772:∧
3717:∧
3714:⋯
3711:∧
3695:↦
3692:⟩
3676:…
3660:⟨
3639:∧
3625:↪
3498:−
3440:−
3359:Example 1
3305:
3275:
3245:
3219:
3180:⊂
3127:∗
3091:
3026:
2993:−
2964:∞
2959:∞
2956:−
2946:⨁
2936:
2831:∗
2794:
2751:−
2728:≤
2716:≤
2710:∣
2657:−
2638:≤
2626:≤
2620:∣
2569:
2488:−
2463:⋅
2389:×
2341:
2295:×
2247:
1852:injective
1791:−
1761:−
1699:Example 3
1617:−
1508:Example 2
1444:∈
1435:−
1345: (
1321:−
1270: (
1218:Example 1
1158:separated
1129:. In his
966:A subset
942:∈
913:∣
898:∈
820: (
798: (
658:∈
629:∣
610:…
588:∈
468:A subset
442:∈
413:∣
398:∈
329:for each
163:dimension
91:that are
68:over the
7375:(1994).
7346:(2008).
7320:(1999).
7173:(1992).
7017:MFK 1994
7004:16482150
6963:(1969).
6944:MFK 1994
6882:(1993),
6829:(1957).
6785:(1956).
6744:(1977).
6602:See also
6283:are the
6245: :
6225: :
5662:− (0, 0)
5288:is when
3884:. Since
3828:such as
3333:-modules
2010:is the (
1707:, nor a
1679:) is an
1667:= 1. As
1490: )
1486:(
1392: )
1388:(
1185:Examples
1175:complete
1154:integral
866:vanish:
702:quotient
200:over an
116:geometry
7399:1304906
7284:Sources
7273:1969915
7068:0457437
6857:0094358
6813:0088035
6500:reduced
6349:improve
6116:) is a
4489:(i.e.,
3793:is the
3262:support
3152:(i.e.,
3080:, then
3049:is the
2590:is the
2323:. Then
1905:-space
1341:
1266:
1090:variety
1082:variety
846:
842:
827:, ...,
816:
805:, ...,
794:
790:
786:
772:
768:
750:
746:
744:. Let
732:be the
700:is the
317:
313:
293:
289:
242:affine
112:algebra
95:in the
7425:
7415:
7397:
7387:
7332:
7306:
7271:
7227:
7193:
7153:
7096:
7066:
7056:
7002:
6984:
6894:
6863:
6855:
6811:
6756:
6576:smooth
6550:stacks
4957:stable
4463:. For
4428:where
3974:. The
3752:where
3208:). If
2900:has a
2855:is an
2038:. The
1731:) for
1522:, and
1496:is an
1371:= 1 −
1232:, and
1146:scheme
1136:using
1032:, the
1016:, let
738:-space
688:, the
508:proper
248:-space
240:be an
234:, let
227:and a
207:Nagata
178:scheme
136:ideals
93:closed
7269:JSTOR
7249:(PDF)
7050:(PDF)
7029:2005.
7000:S2CID
6968:(PDF)
6861:S2CID
6655:Notes
6519:fiber
6473:Serre
6466:curve
6409:sheaf
6407:is a
6334:, or
5707:over
5664:(cf.
5352:with
2868:group
1850:) is
1842:) → (
1240:over
1051:is a
836:, it
758:be a
740:over
250:over
182:field
124:roots
85:union
64:of a
7423:OCLC
7413:ISBN
7385:ISBN
7330:ISBN
7304:ISBN
7225:ISBN
7191:ISBN
7151:ISBN
7094:ISBN
7054:ISBN
6892:ISBN
6754:ISBN
6548:and
6537:and
6296:and
6273:and
6240:and
6197:are
6188:and
6163:Let
6136:The
5608:and
5502:, a
5329:and
4269:>
3876:the
3840:Let
3816:(or
3797:-th
3557:Let
3144:Let
2884:Let
2222:-by-
1893:-by-
1687:and
1512:Let
1377:line
1222:Let
838:does
812:) =
722:Let
716:and
114:and
106:The
89:sets
70:real
41:The
7440:on
7261:doi
7217:doi
7183:doi
7143:doi
7135:".
7086:doi
6992:doi
6843:doi
6799:doi
6479:on
6477:FAC
6431:by
6411:of
6287:on
6074:det
5794:in
5670:.)
5642:on
5538:of
4863:of
4308:Jac
4238:on
4164:Jac
4073:Jac
4041:Jac
4009:of
3985:Jac
3942:Pic
3936:deg
3924:of
3896:Pic
3852:Pic
3801:of
3549:).
3404:= 0
3397:= {
3324:of
3290:in
3264:of
2748:det
2652:det
2524:det
2466:det
2126:in
2111:det
2049:det
1735:in
1562:):
1546:of
1492:of
1260:of
1038:of
1010:of
980:if
970:of
780:in
752:in
696:of
692:or
536:of
482:if
472:of
196:An
138:of
72:or
7458::
7421:.
7395:MR
7393:.
7383:.
7328:.
7324:.
7302:.
7267:.
7257:61
7255:.
7251:.
7223:.
7189:.
7181:.
7149:.
7092:.
7064:MR
7062:,
6998:.
6990:.
6978:36
6976:.
6970:.
6959:;
6890:,
6886:,
6859:.
6853:MR
6851:.
6839:30
6837:.
6833:.
6809:MR
6807:.
6795:30
6793:.
6789:.
6752:.
6748:.
6705:^
6552:.
6525:→
6460:A
6457:.
6435:.
6338:,
6330:,
6278:∘
6268:∘
6252:→
6232:→
6209:≅
6171:,
6034:GL
5993:SL
5658:=
5650:.
5617:→
5613::
5603:×
5599:=
5360:Sp
5325:,
5166:.
4951:.
4823:,
4208:.
3832:.
3809:.
3763:,
3587::
3376:−
3372:=
3355:.
3343:A
3335:.
3302:gr
3272:gr
3242:gr
3216:gr
3141:.
3088:gr
3023:gr
2933:gr
2785:GL
2769:.
2560:GL
2449::
2332:GL
2238:GL
2014:,
1846:,
1838:,
1834:,
1727:,
1723:,
1675:,
1663:+
1643:,
1620:1.
1558:,
1517:=
1482:=
1394::
1363:,
1349:,
1324:1.
1280::
1274:,
1227:=
1202:.
1194:A
1156:,
1140:.
1109:×
1080:a
1060:.
1045:A
1000:.
984:=
807:λx
800:λx
554::
486:=
173:.
7448:.
7429:.
7401:.
7356:.
7338:.
7312:.
7275:.
7263::
7233:.
7219::
7199:.
7185::
7159:.
7145::
7122:C
7102:.
7088::
7006:.
6994::
6867:.
6845::
6815:.
6801::
6762:.
6584:R
6580:k
6572:k
6568:m
6539:Y
6535:X
6531:Y
6527:Y
6523:X
6515:x
6511:x
6492:R
6488:k
6440:k
6425:k
6417:R
6413:k
6401:k
6374:)
6368:(
6363:)
6359:(
6345:.
6302:2
6299:V
6293:1
6290:V
6280:ψ
6276:φ
6270:φ
6266:ψ
6257:1
6254:V
6250:2
6247:V
6243:ψ
6237:2
6234:V
6230:1
6227:V
6223:φ
6214:2
6211:V
6207:1
6204:V
6194:2
6191:V
6185:1
6182:V
6176:2
6173:V
6169:1
6166:V
6129:.
6122:V
6114:V
6112:(
6110:I
6103:V
6080:1
6054:)
6050:C
6046:(
6038:n
6013:)
6009:C
6005:(
5997:n
5965:z
5944:C
5940:=
5935:1
5930:A
5908:y
5905:,
5902:x
5882:z
5862:1
5854:2
5849:|
5844:z
5840:|
5818:C
5814:=
5809:1
5804:A
5782:}
5779:1
5776:=
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34:.
20:)
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