Divisor (algebraic geometry)

3309: 7777: 5233: 7628: 2623:

it is a fractional ideal sheaf (see below). Conversely, every rank one reflexive sheaf corresponds to a Weil divisor: The sheaf can be restricted to the regular locus, where it becomes free and so corresponds to a Cartier divisor (again, see below), and because the singular locus has codimension at

4324: 97:

can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties and the corresponding

372:

On a compact Riemann surface, the degree of a principal divisor is zero; that is, the number of zeros of a meromorphic function is equal to the number of poles, counted with multiplicity. As a result, the degree is well-defined on linear equivalence classes of divisors.

1651: 5369: 6123:

is then a Cartier divisor. This is well-defined because the only choices involved were of the covering and of the isomorphism, neither of which change the Cartier divisor. This Cartier divisor may be used to produce a sheaf, which for distinction we will notate

5047: 7772:{\displaystyle {\begin{array}{ccc}\operatorname {Pic} (X)&\longrightarrow &H^{2}(X,\mathbf {Z} )\\\downarrow &&\downarrow \\\operatorname {Cl} (X)&\longrightarrow &H_{2n-2}^{\operatorname {BM} }(X,\mathbf {Z} )\end{array}}} 7964:

One reason to study the space of global sections of a line bundle is to understand the possible maps from a given variety to projective space. This is essential for the classification of algebraic varieties. Explicitly, a morphism from a variety

105:

On singular varieties, this property can also fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors.

5503: 3594: 1882: 2233: 5989: 5395:

with the group of Cartier divisors modulo linear equivalence. This holds more generally for reduced Noetherian schemes, or for quasi-projective schemes over a Noetherian ring, but it can fail in general (even for proper schemes over

7522: 3897: 1446: 6437:

Effective Cartier divisors are those which correspond to ideal sheaves. In fact, the theory of effective Cartier divisors can be developed without any reference to sheaves of rational functions or fractional ideal sheaves.

1368: 4213: 4398: 7312: 7053: 5884: 8407: 7609: 4044: 2385: 340: 7182: 869: 6746: 1768: 4898: 1519: 5245: 8515: 2446: 3211: 2305: 5228:{\displaystyle H^{0}(X,{\mathcal {M}}_{X}^{\times })\to H^{0}(X,{\mathcal {M}}_{X}^{\times }/{\mathcal {O}}_{X}^{\times })\to H^{1}(X,{\mathcal {O}}_{X}^{\times })=\operatorname {Pic} (X).} 3945: 3255: 2894: 4704: 8577:

generalized Lefschetz's theorem in several directions, involving arbitrary base fields, singular varieties, and results on local rings rather than projective varieties. In particular, if

4555: 4140: 4847: 6928: 6875: 1052: 968: 1920: 8332: 3721: 3669: 1698: 1507: 1175: 6630: 6346: 1269: 921: 4932: 4775: 4205: 2621: 2141: 4987: 4741: 4453: 2844: 2042: 2011: 8473: 8436: 8020: 7919: 7859: 7329:

is factorial (as defined above). In particular, Cartier divisors can be identified with Weil divisors on any regular scheme, and so the first Chern class is an isomorphism for

7105: 6427: 6318: 6211: 6163: 6121: 6062: 6022: 5798: 5758: 5718: 5685: 5588: 5555: 5024: 2813: 2780: 2587: 2550: 2490: 2083: 1980: 779: 730: 682: 4665: 6992: 6585: 6264: 6791: 4625: 8140:

Because the canonical divisor is intrinsically associated to a variety, a key role in the classification of varieties is played by the maps to projective space given by

4486: 5628: 4519: 5413: 4585: 1389:

is a regular function, then its principal Weil divisor is effective, but in general this is not true. The additivity of the order of vanishing function implies that

3518: 1806: 2149: 5889: 1943: 7448: 499:

of meromorphic functions, any two nonzero meromorphic 1-forms yield linearly equivalent divisors. Any divisor in this linear equivalence class is called the

6820:

from one scheme to the other. Whether this is possible depends on whether the divisor is a Weil or Cartier divisor, whether the divisor is to be moved from

3809: 7230: 1395: 6429:

is always a line bundle. In general, however, a Weil divisor on a normal scheme need not be locally principal; see the examples of quadric cones above.

7206:

is undefined because the corresponding local sections would be everywhere zero. (The pullback of the corresponding line bundle, however, is defined.)

4319:{\displaystyle 0\to {\mathcal {O}}_{X}^{\times }\to {\mathcal {M}}_{X}^{\times }\to {\mathcal {M}}_{X}^{\times }/{\mathcal {O}}_{X}^{\times }\to 0.} 1306: 6369:

is normal, a Cartier divisor is determined by the associated Weil divisor, and a Weil divisor is Cartier if and only if it is locally principal.

4340: 7251: 8977: 7811:

are regular (not just rational functions). In that case, the Cartier divisor can be identified with a closed subscheme of codimension 1 in

6997: 6954:

is not quasi-compact, then the pushforward may fail to be a locally finite sum.) This is a special case of the pushforward on Chow groups.

5810: 1373:

It can be shown that this sum is locally finite and hence that it indeed defines a Weil divisor. The principal Weil divisor associated to

8351: 7549: 3982: 2324: 272: 8114: 7110: 365:), the set of principal divisors is a subgroup of the group of divisors. Two divisors that differ by a principal divisor are called 8110: 6930:

if that subscheme is a prime divisor and is defined to be the zero divisor otherwise. Extending this by linearity will, assuming

4934:

Each Cartier divisor defines an invertible fractional ideal sheaf using the description of the Cartier divisor as a collection

2648:

if their difference is principal, so the divisor class group is the group of divisors modulo linear equivalence. For a variety

1097: 9018: 834: 9168: 9044: 8797: 6674: 6064:

is defined to be a subsheaf of the sheaf of rational functions, the image of 1 may be identified with some rational function

1646:{\displaystyle \Gamma (U,{\mathcal {O}}_{X}(D))=\{f\in k(X):f=0{\text{ or }}\operatorname {div} (f)+D\geq 0{\text{ on }}U\}.} 5364:{\displaystyle H^{0}(X,{\mathcal {M}}_{X}^{\times })\to H^{0}(X,{\mathcal {M}}_{X}^{\times }/{\mathcal {O}}_{X}^{\times }),} 3627:: it is the Weil divisor (up to linear equivalence) corresponding to the line bundle of differential forms of top degree on 1718: 428:

has negative degree, then this vector space is zero (because a meromorphic function cannot have more zeros than poles). If

6881:. Depending on φ, it may or may not be a prime Weil divisor. For example, if φ is the blow up of a point in the plane and 4852: 6816:

be a morphism of integral locally Noetherian schemes. It is often—but not always—possible to use φ to transfer a divisor

8478: 603: 8955: 6529: 2393: 827:

is a formal sum of prime numbers with integer coefficients and therefore corresponds to a non-zero fractional ideal in

820:

For example, a divisor on an algebraic curve over a field is a formal sum of finitely many closed points. A divisor on

8597:) is Cartier). The dimension bound here is optimal, as shown by the example of the 3-dimensional quadric cone, above. 548:

has genus zero), zero degree (genus one), or positive degree (genus at least 2). For example, this determines whether

9205: 9086: 8929: 7055:, then this pullback can be used to define pullback of Cartier divisors. In terms of local sections, the pullback of 3173: 2245: 3909: 3220: 2853: 4670: 3156: 6388:. (Some authors say "locally factorial".) In particular, every regular scheme is factorial. On a factorial scheme 4524: 4055: 4812: 159:

is a higher codimension generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1.

8973:"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie" 6895: 6842: 4989:

and conversely, invertible fractional ideal sheaves define Cartier divisors. If the Cartier divisor is denoted

1003: 930: 47: 1890: 9248: 9111: 17: 7245:, the natural homomorphism from the group of Cartier divisors to that of Weil divisors gives a homomorphism 8525: 8284: 3969:. Switching to a different affine chart changes only the sign of ω and so we see ω has a simple pole along 3681: 3634: 1663: 1472: 1140: 6331: 1237: 889: 6385: 4903: 4746: 4176: 3018: 2592: 2107: 4937: 4715: 4403: 2818: 2016: 1985: 9152: 8582: 8445: 8415: 7992: 7891: 7831: 7058: 6399: 6271: 6183: 6135: 6074: 6034: 5994: 5770: 5730: 5690: 5657: 5560: 5527: 4996: 2785: 2752: 2559: 2522: 2462: 2055: 1952: 739: 690: 495:

along the lines above. Since the space of meromorphic 1-forms is a 1-dimensional vector space over the

7229:

continues to have codimension one. This can fail for morphisms which are not flat, for example, for a

644: 7958: 6594: 4630: 587: 7184:. Pullback is always defined if φ is dominant, but it cannot be defined in general. For example, if 6632:

and this pullback is an effective Cartier divisor. In particular, this is true for the fibers of φ.

3160: 1815: 8102: 7439: 6967: 457: 6224: 2459:. While the canonical section is the image of a nowhere vanishing rational function, its image in 560:, zero curvature, or negative curvature. The canonical divisor has negative degree if and only if 8585:

local ring which is factorial in codimension at most 3 (for example, if the non-regular locus of

6755: 4590: 3299: 1272: 46:. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for 1455:

is a homomorphism, and in particular its image is a subgroup of the group of all Weil divisors.

9006: 8964: 8574: 5498:{\displaystyle 0\to {\mathcal {O}}_{X}\to {\mathcal {O}}_{X}(D)\to {\mathcal {O}}_{D}(D)\to 0.} 4458: 110: 74: 63: 9015:

Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2)

6556: 5597: 3589:{\displaystyle j^{*}:\operatorname {Cl} (X)\to \operatorname {Cl} (U)=\operatorname {Pic} (U)} 1877:{\displaystyle {\begin{cases}{\mathcal {O}}(D)\to {\mathcal {O}}_{X}\\f\mapsto fg\end{cases}}} 8556: 4491: 2228:{\displaystyle 0\to {\mathcal {O}}_{X}(-D)\to {\mathcal {O}}_{X}\to {\mathcal {O}}_{D}\to 0.} 90: 8227:-divisor is a finite formal linear combination of irreducible codimension-1 subvarieties of 6513:. The sum of two effective Cartier divisors corresponds to multiplication of ideal sheaves. 5984:{\displaystyle 1\in \Gamma (U_{i},{\mathcal {O}}_{U_{i}})=\Gamma (U_{i},{\mathcal {O}}_{X})} 4560: 9215: 9178: 9137: 9125: 9096: 9054: 9032: 8998: 8939: 5387:

is the class of some Cartier divisor. As a result, the exact sequence above identifies the

484:. The distinctive features of a compact Riemann surface are reflected in these dimensions. 460:

is a more precise statement along these lines. On the other hand, the precise dimension of

218: 6640:

As a basic result of the (big) Cartier divisor, there is a result called Kodaira's lemma:

8: 8158: 7828:

is linearly equivalent to an effective divisor if and only if its associated line bundle

7633: 7615: 7540: 7517:{\displaystyle \operatorname {Cl} (X)\to H_{2n-2}^{\operatorname {BM} }(X,\mathbf {Z} ).} 5508:

This sequence is derived from the short exact sequence relating the structure sheaves of

557: 517: 496: 43: 9129: 9036: 2504:

is a smooth Cartier divisor, the cokernel of the above inclusion may be identified; see

1925: 122: 9115: 9022: 8914: 7881: 7209:

If φ is flat, then pullback of Weil divisors is defined. In this case, the pullback of

6649: 3676: 3042: 997: 195: 145: 31: 8972: 8559:

variety of dimension at least 3 in complex projective space, then the Picard group of

6221:) are compatible, and this amounts to the fact that these functions all have the form 3298:. It is a finite abelian group. Understanding ideal class groups is a central goal of 3102:, the degree homomorphism is surjective, and the kernel is isomorphic to the group of 9201: 9185: 9164: 9144: 9082: 9040: 8951: 8925: 8793: 8150: 3892:{\displaystyle \omega ={dy_{1} \over y_{1}}\wedge \dots \wedge {dy_{n} \over y_{n}}.} 3612: 3291: 2731: 488: 175:, and so its codimension-1 submanifolds have dimension 0. The group of divisors on a 133: 8968: 8909: 4207:

All regular functions are rational functions, which leads to a short exact sequence

2100:

can be a reduced divisor or a prime divisor), then the ideal sheaf of the subscheme

9193: 9156: 9074: 9062: 8986: 8789: 8785: 8211:+2 classes, which (very roughly) go from positive curvature to negative curvature. 5038: 3275: 3107: 2934: 2239: 1441:{\displaystyle \operatorname {div} fg=\operatorname {div} f+\operatorname {div} g.} 1054:

This length is finite, and it is additive with respect to multiplication, that is,

172: 149: 129: 70: 7925:

does not change its zero locus. As a result, the projective space of lines in the

7368:

by analogy with the divisor of a rational function. Then the first Chern class of

553: 9211: 9174: 9133: 9092: 9070: 9050: 8994: 8935: 8921: 7986: 3115: 2850:

form a monoid with product given as the reflexive hull of a tensor product. Then

2553: 924: 584: 168: 156: 141: 137: 7435: 9103: 7945:)) can be identified with the set of effective divisors linearly equivalent to 7336:

Explicitly, the first Chern class can be defined as follows. For a line bundle

4154: 3258: 3123: 3099: 1467: 565: 118: 9197: 9078: 7539:, with its classical (Euclidean) topology. Likewise, the Picard group maps to 6885:

is the exceptional divisor, then its image is not a Weil divisor. Therefore, φ

2644:) by the subgroup of all principal Weil divisors. Two divisors are said to be 1513:. Concretely it may be defined as subsheaf of the sheaf of rational functions 9242: 9160: 8533: 8059: 3489: 3485: 1231: 176: 94: 55: 8105:

is a fundamental tool for computing the dimension of this vector space when

8038:(the intersection of their zero sets) is empty. Conversely, any line bundle 7618:, where the right vertical map is cap product with the fundamental class of 1363:{\displaystyle \operatorname {div} f=\sum _{Z}\operatorname {ord} _{Z}(f)Z.} 51: 8764: 8703: 8654: 8638: 8622: 5388: 3279: 3168: 385: 89:

is greater than 1. (That is, not every subvariety of projective space is a

8046:+1 global sections whose common base locus is empty determines a morphism 4393:{\displaystyle {\mathcal {M}}_{X}^{\times }/{\mathcal {O}}_{X}^{\times }.} 3308: 2925:

is a unique factorization domain, the divisor class group of affine space

9010: 8834: 8063: 7318: 5727:) is the line bundle associated to a Cartier divisor. More precisely, if 3127: 611: 99: 39: 7307:{\displaystyle c_{1}:\operatorname {Pic} (X)\to \operatorname {Cl} (X),} 480:

of low degree is subtle, and not completely determined by the degree of

8990: 8593:

is a unique factorization domain (and hence every Weil divisor on Spec(

8035: 2657: 2515:

is a normal integral separated scheme of finite type over a field. Let

627: 491:. To define it, one first defines the divisor of a nonzero meromorphic 114: 6516:

There is a good theory of families of effective Cartier divisors. Let

5407:

is an effective Cartier divisor. Then there is a short exact sequence

5400:), which lessens the interest of Cartier divisors in full generality. 2900:

to the monoid of isomorphism classes of rank-one reflexive sheaves on

9120: 9069:, Graduate Texts in Mathematics, vol. 52, New York, Heidelberg: 9027: 7793: 7048:{\displaystyle \varphi ^{-1}{\mathcal {M}}_{Y}\to {\mathcal {M}}_{X}} 5879:{\displaystyle {\mathcal {O}}_{U_{i}}\to {\mathcal {O}}(D)|_{U_{i}}.} 3135:

Generalizing the previous example: for any smooth projective variety

3122:. It follows, for example, that the divisor class group of a complex 424:

says a lot about the dimension of this vector space. For example, if

8745:

Grothendieck, EGA IV, Part 4, Proposition 21.3.4, Corollaire 21.3.5.

7957:. A projective linear subspace of this projective space is called a 4400:

An equivalent description is that a Cartier divisor is a collection

27:

Generalizations of codimension-1 subvarieties of algebraic varieties

9230: 5641: 3500:

is an open subset whose complement has codimension at least 2. Let

8402:{\displaystyle \lfloor D\rfloor =\sum \lfloor a_{j}\rfloor Z_{j},} 6961:

is a Cartier divisor, then under mild hypotheses on φ, there is a

6180:}. The key fact to check here is that the transition functions of 2896:

defines a monoid isomorphism from the Weil divisor class group of

7604:{\displaystyle \operatorname {Pic} (X)\to H^{2}(X,\mathbf {Z} ).} 4039:{\displaystyle \operatorname {div} (\omega )=-Z_{0}-\dots -Z_{n}} 3328: 2624:

least two, the closure of the Cartier divisor is a Weil divisor.

2380:{\displaystyle 0\to {\mathcal {O}}_{X}\to {\mathcal {O}}_{X}(D).} 1284:

is the corresponding valuation. For a non-zero rational function

203: 59: 8519: 3443:

by one equation near the origin, even as a set. It follows that

2972:. Concretely, this means that every codimension-1 subvariety of 335:{\displaystyle (f):=\sum _{p\in X}\operatorname {ord} _{p}(f)p,} 8784:. Mathematical Society of Japan Memoirs. 2017. pp. 16–47. 2976:

is defined by the vanishing of a single homogeneous polynomial.

492: 7177:{\displaystyle \{(\varphi ^{-1}(U_{i}),f_{i}\circ \varphi )\}} 2448:

namely, the image of the global section 1. This is called the

148:

generated by all divisors) is closely related to the group of

1205:. With this definition, the order of vanishing is a function 6828:

or vice versa, and what additional properties φ might have.

4708:

Cartier divisors also have a sheaf-theoretic description. A

3415:

be the quadric cone of dimension 3, defined by the equation

6472:

is principal. It is equivalent to require that around each

5037:

By the exact sequence above, there is an exact sequence of

4993:, then the corresponding fractional ideal sheaf is denoted 2730:) is an isomorphism. (These facts are special cases of the 2092:

is an effective divisor that corresponds to a subscheme of

1870: 1462:

be a normal integral Noetherian scheme. Every Weil divisor

58:). Both are derived from the notion of divisibility in the 9110:, Math. Surveys Monogr., vol. 123, Providence, R.I.: 6797:

Kodaira's lemma gives some results about the big divisor.

3611:. For example, one can use this isomorphism to define the 5800:

restricts to a trivial bundle on each open set. For each

5590:

yields another short exact sequence, the one above. When

3512:

be the inclusion map, then the restriction homomorphism:

544:

is whether the canonical divisor has negative degree (so

8948:

3264 and All That: A Second Course in Algebraic Geometry

8829:

over a field, the Chern classes of any vector bundle on

5371:

that is, if it is the divisor of a rational function on

864:{\displaystyle \operatorname {Spec} {\mathcal {O}}_{K},} 781:

being finite. The group of all Weil divisors is denoted

8109:

is a projective curve. Successive generalizations, the

6741:{\displaystyle H^{0}(X,{\mathcal {O}}_{X}(mD-H))\neq 0} 540:− 2. The key trichotomy among compact Riemann surfaces 9228: 7543:, by the first Chern class in the topological sense: 7225:. The flatness of φ ensures that the inverse image of 5557:

is locally free, and hence tensoring that sequence by

2846:-modules. Isomorphism classes of reflexive sheaves on 1763:{\displaystyle \operatorname {ord} _{Z}(f)\geq -n_{Z}} 8963: 8528:

implies that for a smooth complex projective variety

8481: 8448: 8418: 8354: 8287: 8161:

invariant, measuring the growth of the vector spaces

7995: 7894: 7834: 7631: 7552: 7451: 7254: 7113: 7061: 7000: 6970: 6898: 6845: 6758: 6677: 6597: 6559: 6402: 6334: 6274: 6227: 6186: 6138: 6077: 6037: 5997: 5892: 5813: 5773: 5733: 5693: 5660: 5600: 5563: 5530: 5416: 5248: 5050: 4999: 4940: 4906: 4855: 4815: 4749: 4718: 4673: 4633: 4593: 4563: 4527: 4494: 4461: 4406: 4343: 4216: 4179: 4058: 3985: 3912: 3812: 3684: 3637: 3521: 3223: 3176: 3057:

has degree zero. As a result, for a projective curve

2960:. From there, it is straightforward to check that Cl( 2856: 2821: 2788: 2755: 2595: 2562: 2525: 2465: 2396: 2327: 2307:

contains information on whether regular functions on

2248: 2152: 2110: 2058: 2019: 1988: 1955: 1928: 1893: 1809: 1721: 1666: 1522: 1475: 1398: 1309: 1240: 1143: 1006: 933: 892: 837: 831:. A similar characterization is true for divisors on 797:

if all the coefficients are non-negative. One writes

742: 693: 647: 275: 190:

Equivalently, a divisor on a compact Riemann surface

7352:(that is, a section on some nonempty open subset of 6994:. Sheaf-theoretically, when there is a pullback map 5379:

if their difference is principal. Every line bundle

4893:{\displaystyle f\in {\mathcal {M}}_{X}^{\times }(U)} 736:

is quasi-compact, local finiteness is equivalent to

487:

One key divisor on a compact Riemann surface is the

9005: 7380:

changes this divisor by linear equivalence, since (

8913: 8718:Kleiman (2005), Theorems 2.5 and 5.4, Remark 6.19. 8567:, generated by the restriction of the line bundle 8510:{\displaystyle {\mathcal {O}}(\lfloor D\rfloor ).} 8509: 8467: 8430: 8401: 8326: 8014: 7913: 7853: 7794:Global sections of line bundles and linear systems 7771: 7603: 7516: 7306: 7176: 7099: 7047: 6986: 6922: 6869: 6785: 6740: 6624: 6579: 6457:which is invertible and such that for every point 6421: 6340: 6312: 6258: 6205: 6157: 6115: 6056: 6016: 5983: 5878: 5792: 5752: 5712: 5679: 5622: 5582: 5549: 5497: 5363: 5227: 5018: 4981: 4926: 4892: 4841: 4769: 4735: 4698: 4659: 4619: 4579: 4549: 4513: 4480: 4447: 4392: 4318: 4199: 4134: 4038: 3939: 3891: 3715: 3663: 3588: 3249: 3205: 3049:, then the divisor of a nonzero rational function 3029:. Extending this by linearity gives the notion of 2888: 2838: 2807: 2774: 2615: 2581: 2544: 2484: 2440: 2379: 2299: 2227: 2143:This leads to an often used short exact sequence, 2135: 2077: 2036: 2005: 1974: 1937: 1914: 1876: 1762: 1692: 1645: 1501: 1440: 1362: 1263: 1169: 1046: 962: 915: 863: 773: 724: 676: 334: 125:says that Weil and Cartier divisors are the same. 117:classes. On a smooth variety (or more generally a 5391:of line bundles on an integral Noetherian scheme 2921:be a positive integer. Since the polynomial ring 2441:{\displaystyle \Gamma (X,{\mathcal {O}}_{X}(D)),} 9240: 8841:, and the homomorphism here can be described as 8058:for Cartier divisors (or line bundles), such as 8054:. These observations lead to several notions of 6553:. Because of the flatness assumption, for every 5760:is invertible, then there exists an open cover { 5642:Comparison of Weil divisors and Cartier divisors 3459:is Cartier. In fact, the divisor class group Cl( 527:can be read from the canonical divisor: namely, 258:. The divisor of a nonzero meromorphic function 162: 93:.) Locally, every codimension-1 subvariety of a 9017:, Documents Mathématiques, vol. 4, Paris: 8259:is a Cartier divisor for some positive integer 8117:, give some information about the dimension of 3423:in affine 4-space over a field. Then the plane 3206:{\displaystyle \operatorname {Pic} _{X/k}^{0}.} 2300:{\displaystyle H^{1}(X,{\mathcal {O}}_{X}(-D))} 8945: 8924:, Belmont, CA: Wadsworth International Group, 8920:, Wadsworth Mathematics Series, translated by 8862: 8438:is the greatest integer less than or equal to 3940:{\displaystyle \Omega _{\mathbf {P} ^{n}}^{n}} 3339:in affine 3-space over a field. Then the line 3250:{\displaystyle \operatorname {Pic} _{X/k}^{0}} 2889:{\displaystyle D\mapsto {\mathcal {O}}_{X}(D)} 2496:because the transition functions vanish along 109:Topologically, Weil divisors play the role of 8520:The Grothendieck–Lefschetz hyperplane theorem 8203:increases. The Kodaira dimension divides all 6664:be an arbitrary effective Cartier divisor on 6432: 6268:In the opposite direction, a Cartier divisor 4699:{\displaystyle {\mathcal {O}}_{X}^{\times }.} 3331:cone of dimension 2, defined by the equation 2952:, it follows that the divisor class group of 2311:are the restrictions of regular functions on 345:which is a finite sum. Divisors of the form ( 8498: 8492: 8425: 8419: 8383: 8370: 8361: 8355: 7869:is linearly equivalent to the zero locus of 7527:The latter group is defined using the space 7171: 7114: 7094: 7062: 6307: 6275: 6110: 6078: 4973: 4941: 4550:{\displaystyle {\mathcal {M}}_{X}^{\times }} 4475: 4462: 4439: 4407: 4135:{\displaystyle K_{\mathbf {P} ^{n}}==-(n+1)} 3151:-rational point, the divisor class group Cl( 2702:) is isomorphic to the quotient group of Cl( 1637: 1564: 768: 743: 719: 694: 128:The name "divisor" goes back to the work of 69:Globally, every codimension-1 subvariety of 2694:is irreducible of codimension one, then Cl( 2656:over a field, the divisor class group is a 432:has positive degree, then the dimension of 228:, one can define the order of vanishing of 183:is the free abelian group on the points of 9184: 9149:Singularities of the Minimal Model Program 9061: 8884: 8813: 5242:if it is in the image of the homomorphism 4842:{\displaystyle {\mathcal {O}}_{U}\cdot f,} 3902:Then ω is a rational differential form on 3061:, the degree gives a homomorphism deg: Cl( 113:classes, while Cartier divisors represent 9119: 9026: 8908: 6923:{\displaystyle {\overline {\varphi (Z)}}} 6870:{\displaystyle {\overline {\varphi (Z)}}} 1047:{\displaystyle {\mathcal {O}}_{X,Z}/(f).} 963:{\displaystyle f\in {\mathcal {O}}_{X,Z}} 81:subvariety need not be definable by only 9192:, vol. 1, Berlin: Springer-Verlag, 8896:Grothendieck, SGA 2, Corollaire XI.3.14. 8782:Foundations of the minimal model program 7430:, not necessarily smooth or proper over 7321:. The first Chern class is injective if 7236: 6934:is quasi-compact, define a homomorphism 6173:) defined by working on the open cover { 3749:-space with the homogeneous coordinates 3475: 3307: 2964:) is in fact isomorphic to the integers 1915:{\displaystyle \operatorname {div} (fg)} 886:is a prime divisor, then the local ring 144:. The group of divisors on a curve (the 9102: 8243:if the coefficients are nonnegative. A 7888:. Then multiplying a global section of 7790:, both vertical maps are isomorphisms. 7614:The two homomorphisms are related by a 7434:, there is a natural homomorphism, the 7356:), which exists by local triviality of 7325:is normal, and it is an isomorphism if 4169:be an integral Noetherian scheme. Then 3366:be defined as a set by one equation on 2749:are linearly equivalent if and only if 2737:On a normal integral Noetherian scheme 2085:is invertible; that is, a line bundle. 384:, it is important to study the complex 14: 9241: 9143: 8693:Hartshorne (1977), Proposition II.6.2. 8689: 8687: 8681:Hartshorne (1977), Proposition II.6.5. 8668: 8666: 3394:. In fact, the divisor class group Cl( 2627: 2390:This furnishes a canonical element of 2318:There is also an inclusion of sheaves 1945:is regular thanks to the normality of 1796:is the divisor of a rational function 8946:Eisenbud, David; Harris, Joe (2016), 8551:) is an isomorphism. For example, if 8532:of dimension at least 4 and a smooth 8327:{\displaystyle D=\sum _{j}a_{j}Z_{j}} 6877:is a closed irreducible subscheme of 6476:, there exists an open affine subset 4667:up to multiplication by a section of 3716:{\displaystyle j_{*}\Omega _{U}^{n},} 3664:{\displaystyle {\mathcal {O}}(K_{X})} 3082:, the degree gives an isomorphism Cl( 2505: 1704:if and only if for any prime divisor 1693:{\displaystyle {\mathcal {O}}_{X}(D)} 1656:That is, a nonzero rational function 1502:{\displaystyle {\mathcal {O}}_{X}(D)} 1170:{\displaystyle {\mathcal {O}}_{X,Z},} 8978:Publications Mathématiques de l'IHÉS 8727:Hartshorne (1977), Example II.6.5.2. 6341:{\displaystyle \operatorname {div} } 4793:, there exists an open neighborhood 3906:; thus, it is a rational section of 3398:) is isomorphic to the cyclic group 2052:is locally principal if and only if 1264:{\displaystyle {\mathcal {O}}_{X,Z}} 916:{\displaystyle {\mathcal {O}}_{X,Z}} 414:space of sections of the line bundle 8684: 8663: 7815:, the subscheme defined locally by 4927:{\displaystyle {\mathcal {M}}_{X}.} 4770:{\displaystyle {\mathcal {M}}_{X}.} 4200:{\displaystyle {\mathcal {M}}_{X}.} 4160: 3976:as well. Thus, the divisor of ω is 3455:; that is, no positive multiple of 3118:of dimension equal to the genus of 2985:be an algebraic curve over a field 2616:{\displaystyle {\mathcal {M}}_{X},} 2136:{\displaystyle {\mathcal {O}}(-D).} 73:is defined by the vanishing of one 24: 8874:Hartshorne (1977), Theorem II.7.1. 8816:, p. 141, Proposition 2.2.6.) 8736:Hartshorne(1977), Exercise II.6.5. 8589:has codimension at least 4), then 8484: 8451: 8235:-divisor is defined similarly.) A 7998: 7897: 7837: 7438:, from the divisor class group to 7392:) for a nonzero rational function 7372:can be defined to be the divisor ( 7241:For an integral Noetherian scheme 7034: 7017: 6700: 6635: 6530:relative effective Cartier divisor 6405: 6189: 6141: 6040: 6000: 5967: 5945: 5921: 5899: 5840: 5817: 5776: 5736: 5696: 5687:is invertible. When this happens, 5663: 5566: 5533: 5469: 5443: 5426: 5339: 5315: 5271: 5185: 5141: 5117: 5073: 5002: 4982:{\displaystyle \{(U_{i},f_{i})\},} 4910: 4865: 4819: 4753: 4736:{\displaystyle {\mathcal {O}}_{X}} 4722: 4677: 4531: 4448:{\displaystyle \{(U_{i},f_{i})\},} 4371: 4347: 4294: 4270: 4248: 4226: 4183: 4173:has a sheaf of rational functions 3914: 3696: 3640: 3090:. For any smooth projective curve 2866: 2839:{\displaystyle {\mathcal {O}}_{X}} 2825: 2791: 2758: 2599: 2565: 2528: 2468: 2412: 2397: 2354: 2337: 2271: 2208: 2191: 2162: 2113: 2061: 2037:{\displaystyle {\mathcal {O}}_{X}} 2023: 2006:{\displaystyle {\mathcal {O}}_{X}} 1992: 1958: 1840: 1820: 1670: 1538: 1523: 1479: 1300:is defined to be the Weil divisor 1244: 1147: 1010: 943: 896: 847: 25: 9260: 9222: 8754:Lazarsfeld (2004), Example 1.1.6. 8468:{\displaystyle {\mathcal {O}}(D)} 8431:{\displaystyle \lfloor a\rfloor } 8115:Grothendieck–Riemann–Roch theorem 8015:{\displaystyle {\mathcal {O}}(1)} 7929:-vector space of global sections 7914:{\displaystyle {\mathcal {O}}(D)} 7854:{\displaystyle {\mathcal {O}}(D)} 7376:). Changing the rational section 7348:be a nonzero rational section of 7340:on an integral Noetherian scheme 7100:{\displaystyle \{(U_{i},f_{i})\}} 6422:{\displaystyle {\mathcal {O}}(D)} 6320:on an integral Noetherian scheme 6313:{\displaystyle \{(U_{i},f_{i})\}} 6206:{\displaystyle {\mathcal {O}}(D)} 6158:{\displaystyle {\mathcal {O}}(D)} 6116:{\displaystyle \{(U_{i},f_{i})\}} 6057:{\displaystyle {\mathcal {O}}(D)} 6017:{\displaystyle {\mathcal {O}}(D)} 5793:{\displaystyle {\mathcal {O}}(D)} 5753:{\displaystyle {\mathcal {O}}(D)} 5713:{\displaystyle {\mathcal {O}}(D)} 5680:{\displaystyle {\mathcal {O}}(D)} 5583:{\displaystyle {\mathcal {O}}(D)} 5550:{\displaystyle {\mathcal {O}}(D)} 5383:on an integral Noetherian scheme 5019:{\displaystyle {\mathcal {O}}(D)} 3390:is Cartier (as defined below) on 2808:{\displaystyle {\mathcal {O}}(E)} 2775:{\displaystyle {\mathcal {O}}(D)} 2582:{\displaystyle {\mathcal {O}}(D)} 2545:{\displaystyle {\mathcal {O}}(D)} 2485:{\displaystyle {\mathcal {O}}(D)} 2078:{\displaystyle {\mathcal {O}}(D)} 1975:{\displaystyle {\mathcal {O}}(D)} 774:{\displaystyle \{Z:n_{Z}\neq 0\}} 725:{\displaystyle \{Z:n_{Z}\neq 0\}} 250:). It is an integer, negative if 9190:Positivity in Algebraic Geometry 8231:with rational coefficients. (An 8149:and its positive multiples. The 7802:if its local defining functions 7758: 7678: 7591: 7504: 6800: 6541:is an effective Cartier divisor 5238:A Cartier divisor is said to be 4066: 3920: 3463:) is isomorphic to the integers 3157:finitely generated abelian group 3001:for some finite extension field 1800:, then there is an isomorphism 677:{\displaystyle \sum _{Z}n_{Z}Z,} 574: 214:is the sum of its coefficients. 8890: 8877: 8868: 8856: 8819: 8806: 8773: 8757: 8748: 8739: 8730: 8721: 8612:Dieudonné (1985), section VI.6. 8137:of any dimension over a field. 8111:Hirzebruch–Riemann–Roch theorem 7396:and a nonzero rational section 6625:{\displaystyle X\times _{S}S',} 5991:under this map is a section of 4660:{\displaystyle U_{i}\cap U_{j}} 1922:is an effective divisor and so 262:on the compact Riemann surface 9108:Fundamental Algebraic Geometry 9019:Société Mathématique de France 8712: 8696: 8675: 8647: 8631: 8615: 8606: 8501: 8489: 8462: 8456: 8223:be a normal variety. A (Weil) 8009: 8003: 7908: 7902: 7848: 7842: 7762: 7748: 7719: 7714: 7708: 7695: 7689: 7682: 7668: 7653: 7648: 7642: 7595: 7581: 7568: 7565: 7559: 7508: 7494: 7467: 7464: 7458: 7298: 7292: 7283: 7280: 7274: 7168: 7146: 7133: 7117: 7091: 7065: 7028: 6911: 6905: 6858: 6852: 6780: 6768: 6729: 6726: 6711: 6688: 6568: 6416: 6410: 6328:in a natural way, by applying 6304: 6278: 6200: 6194: 6152: 6146: 6132:). There is an isomorphism of 6107: 6081: 6051: 6045: 6011: 6005: 5978: 5948: 5939: 5902: 5856: 5851: 5845: 5835: 5787: 5781: 5747: 5741: 5707: 5701: 5674: 5668: 5617: 5611: 5577: 5571: 5544: 5538: 5489: 5486: 5480: 5463: 5460: 5454: 5437: 5420: 5355: 5303: 5290: 5287: 5259: 5219: 5213: 5201: 5173: 5160: 5157: 5105: 5092: 5089: 5061: 5013: 5007: 4970: 4944: 4887: 4881: 4436: 4410: 4310: 4264: 4242: 4220: 4129: 4123: 4120: 4108: 4099: 4096: 4090: 4081: 3998: 3992: 3658: 3645: 3607:has codimension at least 2 in 3583: 3577: 3565: 3559: 3550: 3547: 3541: 3402:/2, generated by the class of 2883: 2877: 2860: 2802: 2796: 2769: 2763: 2714:has codimension at least 2 in 2576: 2570: 2539: 2533: 2479: 2473: 2432: 2429: 2423: 2400: 2371: 2365: 2348: 2331: 2294: 2291: 2282: 2259: 2219: 2202: 2185: 2182: 2173: 2156: 2127: 2118: 2072: 2066: 2048:is principal. It follows that 1969: 1963: 1909: 1900: 1858: 1834: 1831: 1825: 1741: 1735: 1687: 1681: 1614: 1608: 1582: 1576: 1558: 1555: 1549: 1526: 1496: 1490: 1351: 1345: 1177:and the order of vanishing of 1038: 1032: 323: 317: 282: 276: 136:, who showed the relevance of 13: 1: 9112:American Mathematical Society 9106:(2005), "The Picard scheme", 8916:History of Algebraic Geometry 8902: 8214: 8101:)) has finite dimension. The 7861:has a nonzero global section 6987:{\displaystyle \varphi ^{*}Z} 6396:is locally principal, and so 6324:determines a Weil divisor on 3947:which has simple poles along 3282:, the divisor class group Cl( 2956:is generated by the class of 1119:may be written as a quotient 380:on a compact Riemann surface 163:Divisors on a Riemann surface 77:; by contrast, a codimension- 9229:The Stacks Project Authors, 9009:; Raynaud, Michèle (2005) , 8790:10.2969/msjmemoirs/03501C020 8780:"Chapter 2. Preliminaries". 8672:Kollár (2013), Notation 1.2. 8526:Lefschetz hyperplane theorem 8207:-dimensional varieties into 8133:)) for a projective variety 7989:of the standard line bundle 6915: 6862: 6656:be a big Cartier divisor on 6386:unique factorization domains 6259:{\displaystyle f_{i}/f_{j}.} 4900:and the product is taken in 4801:on which the restriction of 3782:-space with the coordinates 3778:is isomorphic to the affine 3467:, generated by the class of 3386:, and so we only find that 2 2589:is defined as a subsheaf of 2242:of this sequence shows that 392:with poles at most given by 388:of meromorphic functions on 7: 7360:. Define the Weil divisor ( 6835:is a prime Weil divisor on 6786:{\displaystyle m\in N(X,D)} 6752:for all sufficiently large 5375:. Two Cartier divisors are 4620:{\displaystyle f_{i}=f_{j}} 3484:be a normal variety over a 3359:near the origin. Note that 3257:is an abelian variety, the 2907: 1098:field of rational functions 10: 9265: 9153:Cambridge University Press 8863:Eisenbud & Harris 2016 7194:and φ is the inclusion of 6433:Effective Cartier divisors 3631:. Equivalently, the sheaf 3382:vanishes to order 2 along 2718:, then the restriction Cl( 9198:10.1007/978-3-642-18808-4 9079:10.1007/978-1-4757-3849-0 8571:(1) on projective space. 7977:determines a line bundle 7959:linear system of divisors 7622:in Borel–Moore homology: 6509:is a non-zero divisor in 6447:effective Cartier divisor 5654:if and only if the sheaf 4777:A fractional ideal sheaf 4481:{\displaystyle \{U_{i}\}} 4049:and its divisor class is 3599:is an isomorphism, since 3439:= 0 cannot be defined in 2640:) is the quotient of Div( 588:locally Noetherian scheme 121:), a result analogous to 9161:10.1017/CBO9781139547895 8766:Stacks Project, Tag 0AFW 8705:Stacks Project, Tag 02RS 8656:Stacks Project, Tag 02MD 8640:Stacks Project, Tag 02MC 8624:Stacks Project, Tag 00PF 8600: 8073:on a projective variety 6580:{\displaystyle S'\to S,} 5807:, choose an isomorphism 5630:is the normal bundle of 5623:{\displaystyle O_{D}(D)} 3355:= 0 is not principal on 3312:The affine quadric cone 3217:of characteristic zero, 3074:For the projective line 2933:is equal to zero. Since 2679:−1)-dimensional cycles. 2634:Weil divisor class group 2519:be a Weil divisor. Then 630:over the prime divisors 456:sufficiently large. The 38:are a generalization of 9007:Grothendieck, Alexander 8965:Grothendieck, Alexandre 7921:by a nonzero scalar in 7824:= 0. A Cartier divisor 7535:) of complex points of 6587:there is a pullback of 5720:(with its embedding in 5516:and the ideal sheaf of 4514:{\displaystyle X,f_{i}} 4337:is a global section of 3300:algebraic number theory 3167:-points of a connected 3155:) is an extension of a 1273:discrete valuation ring 152:for a Dedekind domain. 64:algebraic number fields 8837:on the Chow groups of 8543:, the restriction Pic( 8511: 8475:is then defined to be 8469: 8432: 8403: 8328: 8267:is smooth, then every 8016: 7951:complete linear system 7915: 7855: 7773: 7605: 7518: 7422:For a complex variety 7308: 7178: 7101: 7049: 6988: 6924: 6871: 6795: 6787: 6742: 6626: 6581: 6423: 6380:if all local rings of 6342: 6314: 6260: 6207: 6159: 6117: 6058: 6018: 5985: 5880: 5794: 5754: 5714: 5681: 5624: 5584: 5551: 5524:is a Cartier divisor, 5499: 5365: 5229: 5020: 4983: 4928: 4894: 4843: 4771: 4737: 4710:fractional ideal sheaf 4700: 4661: 4621: 4581: 4580:{\displaystyle U_{i},} 4551: 4515: 4482: 4449: 4394: 4320: 4201: 4136: 4040: 3941: 3893: 3717: 3665: 3590: 3374:= 0; but the function 3321: 3251: 3207: 2890: 2840: 2809: 2776: 2686:be a closed subset of 2664:) is the Chow group CH 2617: 2583: 2546: 2486: 2442: 2381: 2301: 2229: 2137: 2079: 2038: 2007: 1976: 1939: 1916: 1878: 1780:is the coefficient of 1764: 1694: 1647: 1503: 1442: 1364: 1294:principal Weil divisor 1265: 1234:, then the local ring 1171: 1048: 970:is non-zero, then the 964: 917: 865: 775: 732:is locally finite. If 726: 678: 336: 75:homogeneous polynomial 8583:complete intersection 8557:complete intersection 8512: 8470: 8433: 8404: 8329: 8017: 7916: 7856: 7798:A Cartier divisor is 7774: 7606: 7519: 7309: 7237:The first Chern class 7179: 7102: 7050: 6989: 6925: 6872: 6788: 6743: 6642: 6627: 6582: 6424: 6392:, every Weil divisor 6343: 6315: 6261: 6208: 6160: 6118: 6059: 6019: 5986: 5881: 5795: 5755: 5715: 5682: 5625: 5585: 5552: 5500: 5366: 5230: 5021: 4984: 4929: 4895: 4844: 4772: 4738: 4701: 4662: 4622: 4582: 4552: 4516: 4483: 4450: 4395: 4321: 4202: 4137: 4041: 3942: 3894: 3718: 3666: 3591: 3476:The canonical divisor 3311: 3290:) is also called the 3252: 3208: 3017:is defined to be the 2989:. Every closed point 2891: 2841: 2810: 2777: 2732:localization sequence 2618: 2584: 2547: 2487: 2443: 2382: 2302: 2230: 2138: 2080: 2039: 2008: 1977: 1940: 1917: 1879: 1765: 1695: 1648: 1504: 1443: 1365: 1266: 1172: 1049: 965: 918: 866: 776: 727: 687:where the collection 679: 564:is isomorphic to the 448:)) grows linearly in 337: 91:complete intersection 9249:Geometry of divisors 9114:, pp. 235–321, 8479: 8446: 8416: 8352: 8285: 8103:Riemann–Roch theorem 7993: 7969:to projective space 7892: 7832: 7629: 7550: 7449: 7440:Borel–Moore homology 7252: 7111: 7059: 6998: 6968: 6896: 6843: 6756: 6675: 6595: 6557: 6400: 6372:A Noetherian scheme 6332: 6272: 6225: 6184: 6136: 6075: 6035: 5995: 5890: 5811: 5771: 5731: 5691: 5658: 5598: 5561: 5528: 5414: 5246: 5048: 4997: 4938: 4904: 4853: 4813: 4747: 4716: 4671: 4631: 4591: 4561: 4525: 4492: 4488:is an open cover of 4459: 4404: 4341: 4214: 4177: 4056: 3983: 3910: 3810: 3727:is the dimension of 3682: 3635: 3519: 3221: 3174: 2917:be a field, and let 2854: 2819: 2786: 2753: 2741:, two Weil divisors 2593: 2560: 2523: 2463: 2394: 2325: 2246: 2150: 2108: 2056: 2017: 1986: 1953: 1926: 1891: 1807: 1719: 1664: 1520: 1473: 1396: 1307: 1238: 1141: 1104:, then any non-zero 1004: 931: 890: 835: 740: 691: 645: 458:Riemann–Roch theorem 273: 219:meromorphic function 9130:2005math......4020K 9037:2005math.....11279G 8341:-divisor, then its 7747: 7616:commutative diagram 7541:integral cohomology 7493: 7419:) is well-defined. 7317:known as the first 6549:which is flat over 5377:linearly equivalent 5354: 5330: 5286: 5200: 5156: 5132: 5088: 4880: 4692: 4546: 4386: 4362: 4309: 4285: 4263: 4241: 3936: 3709: 3246: 3199: 2944:minus a hyperplane 2646:linearly equivalent 2628:Divisor class group 2452:and may be denoted 1275:, and the function 875:is a number field. 596:irreducible divisor 367:linearly equivalent 171:is a 1-dimensional 44:algebraic varieties 42:-1 subvarieties of 9232:The Stacks Project 9186:Lazarsfeld, Robert 9067:Algebraic Geometry 8991:10.1007/bf02732123 8507: 8465: 8428: 8399: 8324: 8303: 8034:+1 sections whose 8012: 7911: 7882:projective variety 7851: 7769: 7767: 7724: 7601: 7514: 7470: 7304: 7174: 7097: 7045: 6984: 6920: 6867: 6783: 6738: 6650:projective variety 6622: 6577: 6453:is an ideal sheaf 6419: 6338: 6310: 6256: 6203: 6155: 6113: 6054: 6014: 5981: 5876: 5790: 5750: 5710: 5677: 5620: 5580: 5547: 5495: 5361: 5336: 5312: 5268: 5225: 5182: 5138: 5114: 5070: 5016: 4979: 4924: 4890: 4862: 4839: 4767: 4733: 4696: 4674: 4657: 4617: 4577: 4547: 4528: 4511: 4478: 4445: 4390: 4368: 4344: 4316: 4291: 4267: 4245: 4223: 4197: 4132: 4036: 3937: 3913: 3889: 3745:be the projective 3713: 3695: 3677:direct image sheaf 3661: 3586: 3322: 3286:) := Cl(Spec 3247: 3224: 3203: 3177: 3163:, by the group of 3161:Néron–Severi group 2997:has the form Spec 2886: 2836: 2815:are isomorphic as 2805: 2772: 2734:for Chow groups.) 2706:) by the class of 2613: 2579: 2542: 2482: 2438: 2377: 2297: 2225: 2133: 2075: 2034: 2003: 1972: 1938:{\displaystyle fg} 1935: 1912: 1874: 1869: 1760: 1690: 1643: 1499: 1438: 1360: 1331: 1261: 1167: 1044: 972:order of vanishing 960: 913: 861: 807:if the difference 771: 722: 674: 657: 351:principal divisors 349:) are also called 332: 303: 206:coefficients. The 196:linear combination 146:free abelian group 32:algebraic geometry 9170:978-1-107-03534-8 9063:Hartshorne, Robin 9046:978-2-85629-169-6 8799:978-4-86497-045-7 8563:is isomorphic to 8294: 8151:Kodaira dimension 7404:. So the element 7231:small contraction 7107:is defined to be 6918: 6892:is defined to be 6865: 6648:be a irreducible 6527:be a morphism. A 6355:on the open sets 6348:to the functions 6071:. The collection 3884: 3846: 3613:canonical divisor 3292:ideal class group 3033:for a divisor on 2948:is isomorphic to 2506:#Cartier divisors 2450:canonical section 1982:is isomorphic to 1949:. Conversely, if 1792:is principal, so 1632: 1600: 1322: 1181:is defined to be 789:. A Weil divisor 648: 606:closed subscheme 501:canonical divisor 489:canonical divisor 288: 150:fractional ideals 16:(Redirected from 9256: 9235: 9218: 9181: 9140: 9123: 9099: 9060:Section II.6 of 9057: 9030: 9002: 8960: 8942: 8919: 8897: 8894: 8888: 8881: 8875: 8872: 8866: 8860: 8854: 8823: 8817: 8810: 8804: 8803: 8777: 8771: 8769: 8761: 8755: 8752: 8746: 8743: 8737: 8734: 8728: 8725: 8719: 8716: 8710: 8708: 8700: 8694: 8691: 8682: 8679: 8673: 8670: 8661: 8659: 8651: 8645: 8643: 8635: 8629: 8627: 8619: 8613: 8610: 8516: 8514: 8513: 8508: 8488: 8487: 8474: 8472: 8471: 8466: 8455: 8454: 8437: 8435: 8434: 8429: 8408: 8406: 8405: 8400: 8395: 8394: 8382: 8381: 8333: 8331: 8330: 8325: 8323: 8322: 8313: 8312: 8302: 8021: 8019: 8018: 8013: 8002: 8001: 7920: 7918: 7917: 7912: 7901: 7900: 7860: 7858: 7857: 7852: 7841: 7840: 7778: 7776: 7775: 7770: 7768: 7761: 7746: 7741: 7693: 7681: 7667: 7666: 7610: 7608: 7607: 7602: 7594: 7580: 7579: 7523: 7521: 7520: 7515: 7507: 7492: 7487: 7313: 7311: 7310: 7305: 7264: 7263: 7224: 7193: 7183: 7181: 7180: 7175: 7161: 7160: 7145: 7144: 7132: 7131: 7106: 7104: 7103: 7098: 7090: 7089: 7077: 7076: 7054: 7052: 7051: 7046: 7044: 7043: 7038: 7037: 7027: 7026: 7021: 7020: 7013: 7012: 6993: 6991: 6990: 6985: 6980: 6979: 6945: 6929: 6927: 6926: 6921: 6919: 6914: 6900: 6876: 6874: 6873: 6868: 6866: 6861: 6847: 6815: 6792: 6790: 6789: 6784: 6747: 6745: 6744: 6739: 6710: 6709: 6704: 6703: 6687: 6686: 6667: 6663: 6659: 6655: 6647: 6631: 6629: 6628: 6623: 6618: 6610: 6609: 6586: 6584: 6583: 6578: 6567: 6526: 6504: 6485: 6445:be a scheme. An 6428: 6426: 6425: 6420: 6409: 6408: 6347: 6345: 6344: 6339: 6319: 6317: 6316: 6311: 6303: 6302: 6290: 6289: 6265: 6263: 6262: 6257: 6252: 6251: 6242: 6237: 6236: 6212: 6210: 6209: 6204: 6193: 6192: 6164: 6162: 6161: 6156: 6145: 6144: 6122: 6120: 6119: 6114: 6106: 6105: 6093: 6092: 6063: 6061: 6060: 6055: 6044: 6043: 6023: 6021: 6020: 6015: 6004: 6003: 5990: 5988: 5987: 5982: 5977: 5976: 5971: 5970: 5960: 5959: 5938: 5937: 5936: 5935: 5925: 5924: 5914: 5913: 5885: 5883: 5882: 5877: 5872: 5871: 5870: 5869: 5859: 5844: 5843: 5834: 5833: 5832: 5831: 5821: 5820: 5799: 5797: 5796: 5791: 5780: 5779: 5759: 5757: 5756: 5751: 5740: 5739: 5719: 5717: 5716: 5711: 5700: 5699: 5686: 5684: 5683: 5678: 5667: 5666: 5629: 5627: 5626: 5621: 5610: 5609: 5589: 5587: 5586: 5581: 5570: 5569: 5556: 5554: 5553: 5548: 5537: 5536: 5504: 5502: 5501: 5496: 5479: 5478: 5473: 5472: 5453: 5452: 5447: 5446: 5436: 5435: 5430: 5429: 5370: 5368: 5367: 5362: 5353: 5348: 5343: 5342: 5335: 5329: 5324: 5319: 5318: 5302: 5301: 5285: 5280: 5275: 5274: 5258: 5257: 5234: 5232: 5231: 5226: 5199: 5194: 5189: 5188: 5172: 5171: 5155: 5150: 5145: 5144: 5137: 5131: 5126: 5121: 5120: 5104: 5103: 5087: 5082: 5077: 5076: 5060: 5059: 5039:sheaf cohomology 5025: 5023: 5022: 5017: 5006: 5005: 4988: 4986: 4985: 4980: 4969: 4968: 4956: 4955: 4933: 4931: 4930: 4925: 4920: 4919: 4914: 4913: 4899: 4897: 4896: 4891: 4879: 4874: 4869: 4868: 4848: 4846: 4845: 4840: 4829: 4828: 4823: 4822: 4776: 4774: 4773: 4768: 4763: 4762: 4757: 4756: 4742: 4740: 4739: 4734: 4732: 4731: 4726: 4725: 4705: 4703: 4702: 4697: 4691: 4686: 4681: 4680: 4666: 4664: 4663: 4658: 4656: 4655: 4643: 4642: 4626: 4624: 4623: 4618: 4616: 4615: 4603: 4602: 4586: 4584: 4583: 4578: 4573: 4572: 4556: 4554: 4553: 4548: 4545: 4540: 4535: 4534: 4521:is a section of 4520: 4518: 4517: 4512: 4510: 4509: 4487: 4485: 4484: 4479: 4474: 4473: 4454: 4452: 4451: 4446: 4435: 4434: 4422: 4421: 4399: 4397: 4396: 4391: 4385: 4380: 4375: 4374: 4367: 4361: 4356: 4351: 4350: 4325: 4323: 4322: 4317: 4308: 4303: 4298: 4297: 4290: 4284: 4279: 4274: 4273: 4262: 4257: 4252: 4251: 4240: 4235: 4230: 4229: 4206: 4204: 4203: 4198: 4193: 4192: 4187: 4186: 4161:Cartier divisors 4153:. (See also the 4141: 4139: 4138: 4133: 4077: 4076: 4075: 4074: 4069: 4045: 4043: 4042: 4037: 4035: 4034: 4016: 4015: 3946: 3944: 3943: 3938: 3935: 3930: 3929: 3928: 3923: 3898: 3896: 3895: 3890: 3885: 3883: 3882: 3873: 3872: 3871: 3858: 3847: 3845: 3844: 3835: 3834: 3833: 3820: 3722: 3720: 3719: 3714: 3708: 3703: 3694: 3693: 3670: 3668: 3667: 3662: 3657: 3656: 3644: 3643: 3595: 3593: 3592: 3587: 3531: 3530: 3276:ring of integers 3256: 3254: 3253: 3248: 3245: 3240: 3236: 3212: 3210: 3209: 3204: 3198: 3193: 3189: 3108:Jacobian variety 2935:projective space 2895: 2893: 2892: 2887: 2876: 2875: 2870: 2869: 2845: 2843: 2842: 2837: 2835: 2834: 2829: 2828: 2814: 2812: 2811: 2806: 2795: 2794: 2781: 2779: 2778: 2773: 2762: 2761: 2622: 2620: 2619: 2614: 2609: 2608: 2603: 2602: 2588: 2586: 2585: 2580: 2569: 2568: 2551: 2549: 2548: 2543: 2532: 2531: 2491: 2489: 2488: 2483: 2472: 2471: 2447: 2445: 2444: 2439: 2422: 2421: 2416: 2415: 2386: 2384: 2383: 2378: 2364: 2363: 2358: 2357: 2347: 2346: 2341: 2340: 2306: 2304: 2303: 2298: 2281: 2280: 2275: 2274: 2258: 2257: 2240:sheaf cohomology 2234: 2232: 2231: 2226: 2218: 2217: 2212: 2211: 2201: 2200: 2195: 2194: 2172: 2171: 2166: 2165: 2142: 2140: 2139: 2134: 2117: 2116: 2084: 2082: 2081: 2076: 2065: 2064: 2043: 2041: 2040: 2035: 2033: 2032: 2027: 2026: 2012: 2010: 2009: 2004: 2002: 2001: 1996: 1995: 1981: 1979: 1978: 1973: 1962: 1961: 1944: 1942: 1941: 1936: 1921: 1919: 1918: 1913: 1883: 1881: 1880: 1875: 1873: 1872: 1850: 1849: 1844: 1843: 1824: 1823: 1769: 1767: 1766: 1761: 1759: 1758: 1731: 1730: 1699: 1697: 1696: 1691: 1680: 1679: 1674: 1673: 1660:is a section of 1652: 1650: 1649: 1644: 1633: 1630: 1601: 1598: 1548: 1547: 1542: 1541: 1508: 1506: 1505: 1500: 1489: 1488: 1483: 1482: 1454: 1447: 1445: 1444: 1439: 1384: 1377:is also notated 1369: 1367: 1366: 1361: 1341: 1340: 1330: 1283: 1270: 1268: 1267: 1262: 1260: 1259: 1248: 1247: 1225: 1204: 1176: 1174: 1173: 1168: 1163: 1162: 1151: 1150: 1128: 1118: 1087: 1053: 1051: 1050: 1045: 1031: 1026: 1025: 1014: 1013: 995: 969: 967: 966: 961: 959: 958: 947: 946: 922: 920: 919: 914: 912: 911: 900: 899: 870: 868: 867: 862: 857: 856: 851: 850: 826: 816: 806: 788: 780: 778: 777: 772: 761: 760: 731: 729: 728: 723: 712: 711: 683: 681: 680: 675: 667: 666: 656: 476:)) for divisors 420:. The degree of 376:Given a divisor 341: 339: 338: 333: 313: 312: 302: 217:For any nonzero 210:of a divisor on 179:Riemann surface 173:complex manifold 142:algebraic curves 140:to the study of 138:Dedekind domains 123:Poincaré duality 71:projective space 21: 9264: 9263: 9259: 9258: 9257: 9255: 9254: 9253: 9239: 9238: 9225: 9208: 9171: 9104:Kleiman, Steven 9089: 9071:Springer-Verlag 9047: 8969:Dieudonné, Jean 8958: 8932: 8922:Judith D. Sally 8910:Dieudonné, Jean 8905: 8900: 8895: 8891: 8885:Lazarsfeld 2004 8882: 8878: 8873: 8869: 8861: 8857: 8848: 8824: 8820: 8814:Lazarsfeld 2004 8811: 8807: 8800: 8779: 8778: 8774: 8763: 8762: 8758: 8753: 8749: 8744: 8740: 8735: 8731: 8726: 8722: 8717: 8713: 8702: 8701: 8697: 8692: 8685: 8680: 8676: 8671: 8664: 8653: 8652: 8648: 8637: 8636: 8632: 8621: 8620: 8616: 8611: 8607: 8603: 8522: 8483: 8482: 8480: 8477: 8476: 8450: 8449: 8447: 8444: 8443: 8417: 8414: 8413: 8390: 8386: 8377: 8373: 8353: 8350: 8349: 8345:is the divisor 8318: 8314: 8308: 8304: 8298: 8286: 8283: 8282: 8217: 8198: 8177: 8148: 7997: 7996: 7994: 7991: 7990: 7896: 7895: 7893: 7890: 7889: 7836: 7835: 7833: 7830: 7829: 7823: 7810: 7796: 7766: 7765: 7757: 7742: 7728: 7722: 7717: 7699: 7698: 7692: 7686: 7685: 7677: 7662: 7658: 7656: 7651: 7632: 7630: 7627: 7626: 7590: 7575: 7571: 7551: 7548: 7547: 7503: 7488: 7474: 7450: 7447: 7446: 7410: 7259: 7255: 7253: 7250: 7249: 7239: 7214: 7185: 7156: 7152: 7140: 7136: 7124: 7120: 7112: 7109: 7108: 7085: 7081: 7072: 7068: 7060: 7057: 7056: 7039: 7033: 7032: 7031: 7022: 7016: 7015: 7014: 7005: 7001: 6999: 6996: 6995: 6975: 6971: 6969: 6966: 6965: 6935: 6901: 6899: 6897: 6894: 6893: 6888: 6848: 6846: 6844: 6841: 6840: 6806: 6803: 6757: 6754: 6753: 6705: 6699: 6698: 6697: 6682: 6678: 6676: 6673: 6672: 6665: 6661: 6657: 6653: 6645: 6638: 6636:Kodaira's lemma 6611: 6605: 6601: 6596: 6593: 6592: 6560: 6558: 6555: 6554: 6517: 6487: 6477: 6470: 6435: 6404: 6403: 6401: 6398: 6397: 6360: 6353: 6333: 6330: 6329: 6298: 6294: 6285: 6281: 6273: 6270: 6269: 6247: 6243: 6238: 6232: 6228: 6226: 6223: 6222: 6188: 6187: 6185: 6182: 6181: 6178: 6140: 6139: 6137: 6134: 6133: 6101: 6097: 6088: 6084: 6076: 6073: 6072: 6069: 6039: 6038: 6036: 6033: 6032: 6029: 5999: 5998: 5996: 5993: 5992: 5972: 5966: 5965: 5964: 5955: 5951: 5931: 5927: 5926: 5920: 5919: 5918: 5909: 5905: 5891: 5888: 5887: 5865: 5861: 5860: 5855: 5854: 5839: 5838: 5827: 5823: 5822: 5816: 5815: 5814: 5812: 5809: 5808: 5805: 5775: 5774: 5772: 5769: 5768: 5765: 5735: 5734: 5732: 5729: 5728: 5725: 5695: 5694: 5692: 5689: 5688: 5662: 5661: 5659: 5656: 5655: 5646:A Weil divisor 5644: 5605: 5601: 5599: 5596: 5595: 5565: 5564: 5562: 5559: 5558: 5532: 5531: 5529: 5526: 5525: 5474: 5468: 5467: 5466: 5448: 5442: 5441: 5440: 5431: 5425: 5424: 5423: 5415: 5412: 5411: 5349: 5344: 5338: 5337: 5331: 5325: 5320: 5314: 5313: 5297: 5293: 5281: 5276: 5270: 5269: 5253: 5249: 5247: 5244: 5243: 5195: 5190: 5184: 5183: 5167: 5163: 5151: 5146: 5140: 5139: 5133: 5127: 5122: 5116: 5115: 5099: 5095: 5083: 5078: 5072: 5071: 5055: 5051: 5049: 5046: 5045: 5001: 5000: 4998: 4995: 4994: 4964: 4960: 4951: 4947: 4939: 4936: 4935: 4915: 4909: 4908: 4907: 4905: 4902: 4901: 4875: 4870: 4864: 4863: 4854: 4851: 4850: 4824: 4818: 4817: 4816: 4814: 4811: 4810: 4758: 4752: 4751: 4750: 4748: 4745: 4744: 4727: 4721: 4720: 4719: 4717: 4714: 4713: 4687: 4682: 4676: 4675: 4672: 4669: 4668: 4651: 4647: 4638: 4634: 4632: 4629: 4628: 4611: 4607: 4598: 4594: 4592: 4589: 4588: 4568: 4564: 4562: 4559: 4558: 4541: 4536: 4530: 4529: 4526: 4523: 4522: 4505: 4501: 4493: 4490: 4489: 4469: 4465: 4460: 4457: 4456: 4430: 4426: 4417: 4413: 4405: 4402: 4401: 4381: 4376: 4370: 4369: 4363: 4357: 4352: 4346: 4345: 4342: 4339: 4338: 4331:Cartier divisor 4304: 4299: 4293: 4292: 4286: 4280: 4275: 4269: 4268: 4258: 4253: 4247: 4246: 4236: 4231: 4225: 4224: 4215: 4212: 4211: 4188: 4182: 4181: 4180: 4178: 4175: 4174: 4163: 4070: 4065: 4064: 4063: 4059: 4057: 4054: 4053: 4030: 4026: 4011: 4007: 3984: 3981: 3980: 3975: 3959: 3952: 3931: 3924: 3919: 3918: 3917: 3911: 3908: 3907: 3878: 3874: 3867: 3863: 3859: 3857: 3840: 3836: 3829: 3825: 3821: 3819: 3811: 3808: 3807: 3802: 3794: 3787: 3773: 3761: 3755: 3704: 3699: 3689: 3685: 3683: 3680: 3679: 3652: 3648: 3639: 3638: 3636: 3633: 3632: 3622: 3526: 3522: 3520: 3517: 3516: 3478: 3241: 3232: 3228: 3222: 3219: 3218: 3194: 3185: 3181: 3175: 3172: 3171: 3116:abelian variety 3106:-points on the 2968:, generated by 2910: 2871: 2865: 2864: 2863: 2855: 2852: 2851: 2830: 2824: 2823: 2822: 2820: 2817: 2816: 2790: 2789: 2787: 2784: 2783: 2757: 2756: 2754: 2751: 2750: 2670: 2630: 2604: 2598: 2597: 2596: 2594: 2591: 2590: 2564: 2563: 2561: 2558: 2557: 2554:reflexive sheaf 2527: 2526: 2524: 2521: 2520: 2492:vanishes along 2467: 2466: 2464: 2461: 2460: 2457: 2417: 2411: 2410: 2409: 2395: 2392: 2391: 2359: 2353: 2352: 2351: 2342: 2336: 2335: 2334: 2326: 2323: 2322: 2276: 2270: 2269: 2268: 2253: 2249: 2247: 2244: 2243: 2213: 2207: 2206: 2205: 2196: 2190: 2189: 2188: 2167: 2161: 2160: 2159: 2151: 2148: 2147: 2112: 2111: 2109: 2106: 2105: 2060: 2059: 2057: 2054: 2053: 2028: 2022: 2021: 2020: 2018: 2015: 2014: 1997: 1991: 1990: 1989: 1987: 1984: 1983: 1957: 1956: 1954: 1951: 1950: 1927: 1924: 1923: 1892: 1889: 1888: 1868: 1867: 1852: 1851: 1845: 1839: 1838: 1837: 1819: 1818: 1811: 1810: 1808: 1805: 1804: 1778: 1754: 1750: 1726: 1722: 1720: 1717: 1716: 1675: 1669: 1668: 1667: 1665: 1662: 1661: 1629: 1597: 1543: 1537: 1536: 1535: 1521: 1518: 1517: 1484: 1478: 1477: 1476: 1474: 1471: 1470: 1452: 1397: 1394: 1393: 1378: 1336: 1332: 1326: 1308: 1305: 1304: 1282: 1276: 1249: 1243: 1242: 1241: 1239: 1236: 1235: 1212: 1206: 1198: 1188: 1182: 1152: 1146: 1145: 1144: 1142: 1139: 1138: 1120: 1105: 1081: 1071: 1061: 1055: 1027: 1015: 1009: 1008: 1007: 1005: 1002: 1001: 989: 983: 948: 942: 941: 940: 932: 929: 928: 925:Krull dimension 901: 895: 894: 893: 891: 888: 887: 852: 846: 845: 844: 836: 833: 832: 821: 808: 798: 782: 756: 752: 741: 738: 737: 707: 703: 692: 689: 688: 662: 658: 652: 646: 643: 642: 577: 535: 515: 308: 304: 292: 274: 271: 270: 245: 169:Riemann surface 165: 157:algebraic cycle 85:equations when 28: 23: 22: 15: 12: 11: 5: 9262: 9252: 9251: 9237: 9236: 9224: 9223:External links 9221: 9220: 9219: 9206: 9182: 9169: 9141: 9100: 9087: 9058: 9045: 9003: 8961: 8957:978-1107602724 8956: 8943: 8930: 8904: 8901: 8899: 8898: 8889: 8876: 8867: 8855: 8846: 8825:For a variety 8818: 8805: 8798: 8772: 8756: 8747: 8738: 8729: 8720: 8711: 8695: 8683: 8674: 8662: 8646: 8630: 8614: 8604: 8602: 8599: 8521: 8518: 8506: 8503: 8500: 8497: 8494: 8491: 8486: 8464: 8461: 8458: 8453: 8427: 8424: 8421: 8410: 8409: 8398: 8393: 8389: 8385: 8380: 8376: 8372: 8369: 8366: 8363: 8360: 8357: 8335: 8334: 8321: 8317: 8311: 8307: 8301: 8297: 8293: 8290: 8216: 8213: 8194: 8173: 8144: 8085:-vector space 8069:For a divisor 8060:ample divisors 8011: 8008: 8005: 8000: 7910: 7907: 7904: 7899: 7850: 7847: 7844: 7839: 7819: 7806: 7795: 7792: 7780: 7779: 7764: 7760: 7756: 7753: 7750: 7745: 7740: 7737: 7734: 7731: 7727: 7723: 7721: 7718: 7716: 7713: 7710: 7707: 7704: 7701: 7700: 7697: 7694: 7691: 7688: 7687: 7684: 7680: 7676: 7673: 7670: 7665: 7661: 7657: 7655: 7652: 7650: 7647: 7644: 7641: 7638: 7635: 7634: 7612: 7611: 7600: 7597: 7593: 7589: 7586: 7583: 7578: 7574: 7570: 7567: 7564: 7561: 7558: 7555: 7525: 7524: 7513: 7510: 7506: 7502: 7499: 7496: 7491: 7486: 7483: 7480: 7477: 7473: 7469: 7466: 7463: 7460: 7457: 7454: 7408: 7315: 7314: 7303: 7300: 7297: 7294: 7291: 7288: 7285: 7282: 7279: 7276: 7273: 7270: 7267: 7262: 7258: 7238: 7235: 7173: 7170: 7167: 7164: 7159: 7155: 7151: 7148: 7143: 7139: 7135: 7130: 7127: 7123: 7119: 7116: 7096: 7093: 7088: 7084: 7080: 7075: 7071: 7067: 7064: 7042: 7036: 7030: 7025: 7019: 7011: 7008: 7004: 6983: 6978: 6974: 6917: 6913: 6910: 6907: 6904: 6886: 6864: 6860: 6857: 6854: 6851: 6802: 6799: 6782: 6779: 6776: 6773: 6770: 6767: 6764: 6761: 6750: 6749: 6737: 6734: 6731: 6728: 6725: 6722: 6719: 6716: 6713: 6708: 6702: 6696: 6693: 6690: 6685: 6681: 6637: 6634: 6621: 6617: 6614: 6608: 6604: 6600: 6576: 6573: 6570: 6566: 6563: 6468: 6434: 6431: 6418: 6415: 6412: 6407: 6358: 6351: 6337: 6309: 6306: 6301: 6297: 6293: 6288: 6284: 6280: 6277: 6255: 6250: 6246: 6241: 6235: 6231: 6202: 6199: 6196: 6191: 6176: 6154: 6151: 6148: 6143: 6112: 6109: 6104: 6100: 6096: 6091: 6087: 6083: 6080: 6067: 6053: 6050: 6047: 6042: 6027: 6013: 6010: 6007: 6002: 5980: 5975: 5969: 5963: 5958: 5954: 5950: 5947: 5944: 5941: 5934: 5930: 5923: 5917: 5912: 5908: 5904: 5901: 5898: 5895: 5875: 5868: 5864: 5858: 5853: 5850: 5847: 5842: 5837: 5830: 5826: 5819: 5803: 5789: 5786: 5783: 5778: 5763: 5749: 5746: 5743: 5738: 5723: 5709: 5706: 5703: 5698: 5676: 5673: 5670: 5665: 5650:is said to be 5643: 5640: 5619: 5616: 5613: 5608: 5604: 5579: 5576: 5573: 5568: 5546: 5543: 5540: 5535: 5506: 5505: 5494: 5491: 5488: 5485: 5482: 5477: 5471: 5465: 5462: 5459: 5456: 5451: 5445: 5439: 5434: 5428: 5422: 5419: 5360: 5357: 5352: 5347: 5341: 5334: 5328: 5323: 5317: 5311: 5308: 5305: 5300: 5296: 5292: 5289: 5284: 5279: 5273: 5267: 5264: 5261: 5256: 5252: 5236: 5235: 5224: 5221: 5218: 5215: 5212: 5209: 5206: 5203: 5198: 5193: 5187: 5181: 5178: 5175: 5170: 5166: 5162: 5159: 5154: 5149: 5143: 5136: 5130: 5125: 5119: 5113: 5110: 5107: 5102: 5098: 5094: 5091: 5086: 5081: 5075: 5069: 5066: 5063: 5058: 5054: 5015: 5012: 5009: 5004: 4978: 4975: 4972: 4967: 4963: 4959: 4954: 4950: 4946: 4943: 4923: 4918: 4912: 4889: 4886: 4883: 4878: 4873: 4867: 4861: 4858: 4838: 4835: 4832: 4827: 4821: 4766: 4761: 4755: 4730: 4724: 4695: 4690: 4685: 4679: 4654: 4650: 4646: 4641: 4637: 4614: 4610: 4606: 4601: 4597: 4576: 4571: 4567: 4544: 4539: 4533: 4508: 4504: 4500: 4497: 4477: 4472: 4468: 4464: 4444: 4441: 4438: 4433: 4429: 4425: 4420: 4416: 4412: 4409: 4389: 4384: 4379: 4373: 4366: 4360: 4355: 4349: 4327: 4326: 4315: 4312: 4307: 4302: 4296: 4289: 4283: 4278: 4272: 4266: 4261: 4256: 4250: 4244: 4239: 4234: 4228: 4222: 4219: 4196: 4191: 4185: 4162: 4159: 4155:Euler sequence 4143: 4142: 4131: 4128: 4125: 4122: 4119: 4116: 4113: 4110: 4107: 4104: 4101: 4098: 4095: 4092: 4089: 4086: 4083: 4080: 4073: 4068: 4062: 4047: 4046: 4033: 4029: 4025: 4022: 4019: 4014: 4010: 4006: 4003: 4000: 3997: 3994: 3991: 3988: 3973: 3957: 3950: 3934: 3927: 3922: 3916: 3900: 3899: 3888: 3881: 3877: 3870: 3866: 3862: 3856: 3853: 3850: 3843: 3839: 3832: 3828: 3824: 3818: 3815: 3800: 3792: 3785: 3771: 3759: 3753: 3712: 3707: 3702: 3698: 3692: 3688: 3660: 3655: 3651: 3647: 3642: 3618: 3597: 3596: 3585: 3582: 3579: 3576: 3573: 3570: 3567: 3564: 3561: 3558: 3555: 3552: 3549: 3546: 3543: 3540: 3537: 3534: 3529: 3525: 3477: 3474: 3473: 3472: 3408: 3407: 3304: 3303: 3267: 3266: 3259:Picard variety 3244: 3239: 3235: 3231: 3227: 3202: 3197: 3192: 3188: 3184: 3180: 3132: 3131: 3130:abelian group. 3124:elliptic curve 3114:, which is an 3100:rational point 3071: 3070: 2978: 2977: 2909: 2906: 2885: 2882: 2879: 2874: 2868: 2862: 2859: 2833: 2827: 2804: 2801: 2798: 2793: 2771: 2768: 2765: 2760: 2665: 2629: 2626: 2612: 2607: 2601: 2578: 2575: 2572: 2567: 2552:is a rank one 2541: 2538: 2535: 2530: 2481: 2478: 2475: 2470: 2455: 2437: 2434: 2431: 2428: 2425: 2420: 2414: 2408: 2405: 2402: 2399: 2388: 2387: 2376: 2373: 2370: 2367: 2362: 2356: 2350: 2345: 2339: 2333: 2330: 2296: 2293: 2290: 2287: 2284: 2279: 2273: 2267: 2264: 2261: 2256: 2252: 2236: 2235: 2224: 2221: 2216: 2210: 2204: 2199: 2193: 2187: 2184: 2181: 2178: 2175: 2170: 2164: 2158: 2155: 2132: 2129: 2126: 2123: 2120: 2115: 2074: 2071: 2068: 2063: 2044:-module, then 2031: 2025: 2000: 1994: 1971: 1968: 1965: 1960: 1934: 1931: 1911: 1908: 1905: 1902: 1899: 1896: 1885: 1884: 1871: 1866: 1863: 1860: 1857: 1854: 1853: 1848: 1842: 1836: 1833: 1830: 1827: 1822: 1817: 1816: 1814: 1776: 1771: 1770: 1757: 1753: 1749: 1746: 1743: 1740: 1737: 1734: 1729: 1725: 1689: 1686: 1683: 1678: 1672: 1654: 1653: 1642: 1639: 1636: 1631: on 1628: 1625: 1622: 1619: 1616: 1613: 1610: 1607: 1604: 1599: or 1596: 1593: 1590: 1587: 1584: 1581: 1578: 1575: 1572: 1569: 1566: 1563: 1560: 1557: 1554: 1551: 1546: 1540: 1534: 1531: 1528: 1525: 1498: 1495: 1492: 1487: 1481: 1468:coherent sheaf 1449: 1448: 1437: 1434: 1431: 1428: 1425: 1422: 1419: 1416: 1413: 1410: 1407: 1404: 1401: 1371: 1370: 1359: 1356: 1353: 1350: 1347: 1344: 1339: 1335: 1329: 1325: 1321: 1318: 1315: 1312: 1296:associated to 1278: 1258: 1255: 1252: 1246: 1208: 1194: 1184: 1166: 1161: 1158: 1155: 1149: 1077: 1067: 1057: 1043: 1040: 1037: 1034: 1030: 1024: 1021: 1018: 1012: 985: 957: 954: 951: 945: 939: 936: 910: 907: 904: 898: 860: 855: 849: 843: 840: 817:is effective. 770: 767: 764: 759: 755: 751: 748: 745: 721: 718: 715: 710: 706: 702: 699: 696: 685: 684: 673: 670: 665: 661: 655: 651: 576: 573: 566:Riemann sphere 556:with positive 531: 511: 416:associated to 343: 342: 331: 328: 325: 322: 319: 316: 311: 307: 301: 298: 295: 291: 287: 284: 281: 278: 266:is defined as 254:has a pole at 241: 164: 161: 119:regular scheme 95:smooth variety 48:Pierre Cartier 26: 9: 6: 4: 3: 2: 9261: 9250: 9247: 9246: 9244: 9234: 9233: 9227: 9226: 9217: 9213: 9209: 9207:3-540-22533-1 9203: 9199: 9195: 9191: 9187: 9183: 9180: 9176: 9172: 9166: 9162: 9158: 9154: 9150: 9146: 9145:Kollár, János 9142: 9139: 9135: 9131: 9127: 9122: 9117: 9113: 9109: 9105: 9101: 9098: 9094: 9090: 9088:0-387-90244-9 9084: 9080: 9076: 9072: 9068: 9064: 9059: 9056: 9052: 9048: 9042: 9038: 9034: 9029: 9024: 9020: 9016: 9012: 9008: 9004: 9000: 8996: 8992: 8988: 8984: 8980: 8979: 8974: 8970: 8966: 8962: 8959: 8953: 8949: 8944: 8941: 8937: 8933: 8931:0-534-03723-2 8927: 8923: 8918: 8917: 8911: 8907: 8906: 8893: 8886: 8880: 8871: 8864: 8859: 8852: 8844: 8840: 8836: 8832: 8828: 8822: 8815: 8809: 8801: 8795: 8791: 8787: 8783: 8776: 8768: 8767: 8760: 8751: 8742: 8733: 8724: 8715: 8707: 8706: 8699: 8690: 8688: 8678: 8669: 8667: 8658: 8657: 8650: 8642: 8641: 8634: 8626: 8625: 8618: 8609: 8605: 8598: 8596: 8592: 8588: 8584: 8580: 8576: 8572: 8570: 8566: 8562: 8558: 8554: 8550: 8546: 8542: 8538: 8535: 8534:ample divisor 8531: 8527: 8517: 8504: 8495: 8459: 8441: 8422: 8396: 8391: 8387: 8378: 8374: 8367: 8364: 8358: 8348: 8347: 8346: 8344: 8340: 8319: 8315: 8309: 8305: 8299: 8295: 8291: 8288: 8281: 8280: 8279: 8276: 8274: 8270: 8266: 8262: 8258: 8254: 8250: 8246: 8242: 8238: 8234: 8230: 8226: 8222: 8212: 8210: 8206: 8202: 8197: 8193: 8189: 8185: 8181: 8176: 8172: 8168: 8164: 8160: 8156: 8152: 8147: 8143: 8138: 8136: 8132: 8128: 8124: 8120: 8116: 8112: 8108: 8104: 8100: 8096: 8092: 8088: 8084: 8080: 8077:over a field 8076: 8072: 8067: 8065: 8061: 8057: 8053: 8049: 8045: 8041: 8037: 8033: 8029: 8025: 8006: 7988: 7984: 7980: 7976: 7973:over a field 7972: 7968: 7962: 7960: 7956: 7952: 7949:, called the 7948: 7944: 7940: 7936: 7932: 7928: 7924: 7905: 7887: 7884:over a field 7883: 7879: 7874: 7872: 7868: 7864: 7845: 7827: 7822: 7818: 7814: 7809: 7805: 7801: 7791: 7789: 7785: 7754: 7751: 7743: 7738: 7735: 7732: 7729: 7725: 7711: 7705: 7702: 7674: 7671: 7663: 7659: 7645: 7639: 7636: 7625: 7624: 7623: 7621: 7617: 7598: 7587: 7584: 7576: 7572: 7562: 7556: 7553: 7546: 7545: 7544: 7542: 7538: 7534: 7530: 7511: 7500: 7497: 7489: 7484: 7481: 7478: 7475: 7471: 7461: 7455: 7452: 7445: 7444: 7443: 7441: 7437: 7433: 7429: 7426:of dimension 7425: 7420: 7418: 7414: 7407: 7403: 7399: 7395: 7391: 7387: 7383: 7379: 7375: 7371: 7367: 7363: 7359: 7355: 7351: 7347: 7343: 7339: 7334: 7332: 7328: 7324: 7320: 7301: 7295: 7289: 7286: 7277: 7271: 7268: 7265: 7260: 7256: 7248: 7247: 7246: 7244: 7234: 7232: 7228: 7222: 7218: 7212: 7207: 7205: 7201: 7197: 7192: 7188: 7165: 7162: 7157: 7153: 7149: 7141: 7137: 7128: 7125: 7121: 7086: 7082: 7078: 7073: 7069: 7040: 7023: 7009: 7006: 7002: 6981: 6976: 6972: 6964: 6960: 6955: 6953: 6949: 6943: 6939: 6933: 6908: 6902: 6891: 6884: 6880: 6855: 6849: 6838: 6834: 6829: 6827: 6823: 6819: 6814: 6810: 6801:Functoriality 6798: 6794: 6777: 6774: 6771: 6765: 6762: 6759: 6735: 6732: 6723: 6720: 6717: 6714: 6706: 6694: 6691: 6683: 6679: 6671: 6670: 6669: 6651: 6641: 6633: 6619: 6615: 6612: 6606: 6602: 6598: 6590: 6574: 6571: 6564: 6561: 6552: 6548: 6544: 6540: 6536: 6532: 6531: 6525: 6521: 6514: 6512: 6508: 6502: 6498: 6494: 6490: 6484: 6480: 6475: 6471: 6464: 6460: 6456: 6452: 6448: 6444: 6439: 6430: 6413: 6395: 6391: 6387: 6383: 6379: 6375: 6370: 6368: 6363: 6361: 6354: 6335: 6327: 6323: 6299: 6295: 6291: 6286: 6282: 6266: 6253: 6248: 6244: 6239: 6233: 6229: 6220: 6216: 6197: 6179: 6172: 6168: 6149: 6131: 6127: 6102: 6098: 6094: 6089: 6085: 6070: 6048: 6030: 6008: 5973: 5961: 5956: 5952: 5942: 5932: 5928: 5915: 5910: 5906: 5896: 5893: 5886:The image of 5873: 5866: 5862: 5848: 5828: 5824: 5806: 5784: 5766: 5744: 5726: 5704: 5671: 5653: 5649: 5639: 5637: 5633: 5614: 5606: 5602: 5593: 5574: 5541: 5523: 5519: 5515: 5511: 5492: 5483: 5475: 5457: 5449: 5432: 5417: 5410: 5409: 5408: 5406: 5401: 5399: 5394: 5390: 5386: 5382: 5378: 5374: 5358: 5350: 5345: 5332: 5326: 5321: 5309: 5306: 5298: 5294: 5282: 5277: 5265: 5262: 5254: 5250: 5241: 5222: 5216: 5210: 5207: 5204: 5196: 5191: 5179: 5176: 5168: 5164: 5152: 5147: 5134: 5128: 5123: 5111: 5108: 5100: 5096: 5084: 5079: 5067: 5064: 5056: 5052: 5044: 5043: 5042: 5040: 5035: 5033: 5029: 5010: 4992: 4976: 4965: 4961: 4957: 4952: 4948: 4921: 4916: 4884: 4876: 4871: 4859: 4856: 4836: 4833: 4830: 4825: 4808: 4804: 4800: 4796: 4792: 4788: 4785:if, for each 4784: 4780: 4764: 4759: 4728: 4711: 4706: 4693: 4688: 4683: 4652: 4648: 4644: 4639: 4635: 4612: 4608: 4604: 4599: 4595: 4574: 4569: 4565: 4542: 4537: 4506: 4502: 4498: 4495: 4470: 4466: 4442: 4431: 4427: 4423: 4418: 4414: 4387: 4382: 4377: 4364: 4358: 4353: 4336: 4332: 4313: 4305: 4300: 4287: 4281: 4276: 4259: 4254: 4237: 4232: 4217: 4210: 4209: 4208: 4194: 4189: 4172: 4168: 4158: 4156: 4152: 4148: 4126: 4117: 4114: 4111: 4105: 4102: 4093: 4087: 4084: 4078: 4071: 4060: 4052: 4051: 4050: 4031: 4027: 4023: 4020: 4017: 4012: 4008: 4004: 4001: 3995: 3989: 3986: 3979: 3978: 3977: 3972: 3968: 3964: 3960: 3953: 3932: 3925: 3905: 3886: 3879: 3875: 3868: 3864: 3860: 3854: 3851: 3848: 3841: 3837: 3830: 3826: 3822: 3816: 3813: 3806: 3805: 3804: 3799: 3795: 3788: 3781: 3777: 3770: 3766: 3762: 3752: 3748: 3744: 3740: 3736: 3732: 3730: 3726: 3710: 3705: 3700: 3690: 3686: 3678: 3674: 3653: 3649: 3630: 3626: 3621: 3617: 3614: 3610: 3606: 3602: 3580: 3574: 3571: 3568: 3562: 3556: 3553: 3544: 3538: 3535: 3532: 3527: 3523: 3515: 3514: 3513: 3511: 3507: 3503: 3499: 3495: 3491: 3487: 3486:perfect field 3483: 3470: 3466: 3462: 3458: 3454: 3450: 3446: 3442: 3438: 3434: 3430: 3426: 3422: 3418: 3414: 3410: 3409: 3405: 3401: 3397: 3393: 3389: 3385: 3381: 3377: 3373: 3369: 3365: 3362: 3358: 3354: 3350: 3346: 3342: 3338: 3334: 3330: 3326: 3319: 3315: 3310: 3306: 3305: 3301: 3297: 3293: 3289: 3285: 3281: 3277: 3273: 3269: 3268: 3264: 3260: 3242: 3237: 3233: 3229: 3225: 3216: 3200: 3195: 3190: 3186: 3182: 3178: 3170: 3166: 3162: 3158: 3154: 3150: 3146: 3142: 3139:over a field 3138: 3134: 3133: 3129: 3125: 3121: 3117: 3113: 3109: 3105: 3101: 3097: 3093: 3089: 3085: 3081: 3078:over a field 3077: 3073: 3072: 3068: 3064: 3060: 3056: 3052: 3048: 3044: 3040: 3036: 3032: 3028: 3024: 3020: 3016: 3012: 3008: 3004: 3000: 2996: 2992: 2988: 2984: 2980: 2979: 2975: 2971: 2967: 2963: 2959: 2955: 2951: 2947: 2943: 2939: 2936: 2932: 2928: 2924: 2920: 2916: 2912: 2911: 2905: 2903: 2899: 2880: 2872: 2857: 2849: 2831: 2799: 2766: 2748: 2744: 2740: 2735: 2733: 2729: 2725: 2721: 2717: 2713: 2709: 2705: 2701: 2697: 2693: 2689: 2685: 2680: 2678: 2674: 2668: 2663: 2660:; namely, Cl( 2659: 2655: 2652:of dimension 2651: 2647: 2643: 2639: 2635: 2625: 2610: 2605: 2573: 2555: 2536: 2518: 2514: 2509: 2507: 2503: 2499: 2495: 2476: 2458: 2451: 2435: 2426: 2418: 2406: 2403: 2374: 2368: 2360: 2343: 2328: 2321: 2320: 2319: 2316: 2314: 2310: 2288: 2285: 2277: 2265: 2262: 2254: 2250: 2241: 2222: 2214: 2197: 2179: 2176: 2168: 2153: 2146: 2145: 2144: 2130: 2124: 2121: 2103: 2099: 2096:(for example 2095: 2091: 2086: 2069: 2051: 2047: 2029: 1998: 1966: 1948: 1932: 1929: 1906: 1903: 1897: 1894: 1864: 1861: 1855: 1846: 1828: 1812: 1803: 1802: 1801: 1799: 1795: 1791: 1787: 1783: 1779: 1755: 1751: 1747: 1744: 1738: 1732: 1727: 1723: 1715: 1714: 1713: 1711: 1708:intersecting 1707: 1703: 1684: 1676: 1659: 1640: 1634: 1626: 1623: 1620: 1617: 1611: 1605: 1602: 1594: 1591: 1588: 1585: 1579: 1573: 1570: 1567: 1561: 1552: 1544: 1532: 1529: 1516: 1515: 1514: 1512: 1493: 1485: 1469: 1466:determines a 1465: 1461: 1456: 1451:Consequently 1435: 1432: 1429: 1426: 1423: 1420: 1417: 1414: 1411: 1408: 1405: 1402: 1399: 1392: 1391: 1390: 1388: 1382: 1376: 1357: 1354: 1348: 1342: 1337: 1333: 1327: 1323: 1319: 1316: 1313: 1310: 1303: 1302: 1301: 1299: 1295: 1291: 1287: 1281: 1274: 1256: 1253: 1250: 1233: 1229: 1224: 1220: 1216: 1211: 1202: 1197: 1192: 1187: 1180: 1164: 1159: 1156: 1153: 1136: 1132: 1127: 1123: 1116: 1112: 1108: 1103: 1099: 1095: 1091: 1085: 1080: 1075: 1070: 1065: 1060: 1041: 1035: 1028: 1022: 1019: 1016: 999: 993: 988: 981: 977: 973: 955: 952: 949: 937: 934: 926: 908: 905: 902: 885: 881: 876: 874: 858: 853: 841: 838: 830: 825: 818: 815: 811: 805: 801: 796: 792: 786: 765: 762: 757: 753: 749: 746: 735: 716: 713: 708: 704: 700: 697: 671: 668: 663: 659: 653: 649: 641: 640: 639: 637: 633: 629: 625: 621: 617: 613: 609: 605: 601: 597: 593: 592:prime divisor 589: 586: 582: 575:Weil divisors 572: 570: 567: 563: 559: 555: 554:Kähler metric 551: 547: 543: 539: 534: 530: 526: 522: 519: 514: 510: 506: 502: 498: 494: 490: 485: 483: 479: 475: 471: 467: 463: 459: 455: 451: 447: 443: 439: 435: 431: 427: 423: 419: 415: 411: 407: 403: 399: 395: 391: 387: 383: 379: 374: 370: 368: 364: 360: 356: 352: 348: 329: 326: 320: 314: 309: 305: 299: 296: 293: 289: 285: 279: 269: 268: 267: 265: 261: 257: 253: 249: 244: 239: 235: 231: 227: 223: 220: 215: 213: 209: 205: 201: 198:of points of 197: 193: 188: 186: 182: 178: 174: 170: 160: 158: 153: 151: 147: 143: 139: 135: 131: 126: 124: 120: 116: 112: 107: 103: 101: 96: 92: 88: 84: 80: 76: 72: 67: 65: 61: 57: 56:David Mumford 53: 49: 45: 41: 37: 33: 19: 18:Divisor class 9231: 9189: 9148: 9121:math/0504020 9107: 9066: 9028:math/0511279 9014: 9011:Laszlo, Yves 8982: 8976: 8947: 8915: 8892: 8887:, Chapter 1) 8879: 8870: 8858: 8850: 8842: 8838: 8830: 8826: 8821: 8808: 8781: 8775: 8765: 8759: 8750: 8741: 8732: 8723: 8714: 8704: 8698: 8677: 8655: 8649: 8639: 8633: 8623: 8617: 8608: 8594: 8590: 8586: 8578: 8575:Grothendieck 8573: 8568: 8564: 8560: 8555:is a smooth 8552: 8548: 8544: 8540: 8536: 8529: 8523: 8442:. The sheaf 8439: 8411: 8342: 8338: 8336: 8277: 8272: 8271:-divisor is 8268: 8264: 8260: 8256: 8252: 8248: 8244: 8240: 8239:-divisor is 8236: 8232: 8228: 8224: 8220: 8218: 8208: 8204: 8200: 8195: 8191: 8187: 8183: 8179: 8174: 8170: 8166: 8162: 8154: 8145: 8141: 8139: 8134: 8130: 8126: 8122: 8118: 8106: 8098: 8094: 8090: 8086: 8082: 8078: 8074: 8070: 8068: 8064:nef divisors 8055: 8051: 8047: 8043: 8039: 8031: 8027: 8026:. Moreover, 8023: 7982: 7978: 7974: 7970: 7966: 7963: 7954: 7950: 7946: 7942: 7938: 7934: 7930: 7926: 7922: 7885: 7877: 7875: 7870: 7866: 7862: 7825: 7820: 7816: 7812: 7807: 7803: 7799: 7797: 7787: 7786:smooth over 7783: 7781: 7619: 7613: 7536: 7532: 7528: 7526: 7431: 7427: 7423: 7421: 7416: 7412: 7405: 7401: 7397: 7393: 7389: 7385: 7381: 7377: 7373: 7369: 7365: 7361: 7357: 7353: 7349: 7345: 7341: 7337: 7335: 7330: 7326: 7322: 7316: 7242: 7240: 7226: 7220: 7216: 7210: 7208: 7203: 7199: 7195: 7190: 7186: 6962: 6958: 6956: 6951: 6947: 6941: 6937: 6931: 6889: 6882: 6878: 6836: 6832: 6830: 6825: 6821: 6817: 6812: 6808: 6804: 6796: 6751: 6643: 6639: 6588: 6550: 6546: 6542: 6538: 6534: 6528: 6523: 6519: 6515: 6510: 6506: 6500: 6496: 6492: 6488: 6482: 6478: 6473: 6466: 6465:, the stalk 6462: 6458: 6454: 6450: 6446: 6442: 6440: 6436: 6393: 6389: 6381: 6377: 6373: 6371: 6366: 6364: 6356: 6349: 6325: 6321: 6267: 6218: 6214: 6174: 6170: 6166: 6129: 6125: 6065: 6025: 5801: 5767:} such that 5761: 5721: 5651: 5647: 5645: 5635: 5631: 5591: 5521: 5517: 5513: 5509: 5507: 5404: 5402: 5397: 5392: 5389:Picard group 5384: 5380: 5376: 5372: 5239: 5237: 5036: 5031: 5027: 4990: 4809:is equal to 4806: 4802: 4798: 4794: 4790: 4786: 4782: 4778: 4709: 4707: 4334: 4330: 4328: 4170: 4166: 4164: 4150: 4146: 4144: 4048: 3970: 3966: 3962: 3955: 3948: 3903: 3901: 3797: 3790: 3783: 3779: 3775: 3768: 3764: 3757: 3750: 3746: 3742: 3738: 3734: 3733: 3728: 3724: 3672: 3628: 3624: 3619: 3615: 3608: 3604: 3600: 3598: 3509: 3505: 3501: 3497: 3493: 3481: 3479: 3468: 3464: 3460: 3456: 3452: 3448: 3444: 3440: 3436: 3432: 3428: 3424: 3420: 3416: 3412: 3403: 3399: 3395: 3391: 3387: 3383: 3379: 3375: 3371: 3367: 3363: 3360: 3356: 3352: 3348: 3344: 3340: 3336: 3332: 3324: 3317: 3313: 3295: 3287: 3283: 3280:number field 3271: 3262: 3214: 3169:group scheme 3164: 3152: 3148: 3144: 3140: 3136: 3119: 3111: 3103: 3095: 3091: 3087: 3083: 3079: 3075: 3066: 3062: 3058: 3054: 3050: 3046: 3038: 3034: 3030: 3026: 3022: 3014: 3010: 3006: 3002: 2998: 2994: 2990: 2986: 2982: 2973: 2969: 2965: 2961: 2957: 2953: 2949: 2945: 2941: 2937: 2930: 2926: 2922: 2918: 2914: 2901: 2897: 2847: 2746: 2742: 2738: 2736: 2727: 2723: 2719: 2715: 2711: 2707: 2703: 2699: 2695: 2691: 2687: 2683: 2681: 2676: 2672: 2666: 2661: 2653: 2649: 2645: 2641: 2637: 2633: 2631: 2556:, and since 2516: 2512: 2511:Assume that 2510: 2501: 2497: 2493: 2453: 2449: 2389: 2317: 2312: 2308: 2237: 2104:is equal to 2101: 2097: 2093: 2089: 2087: 2049: 2045: 1946: 1886: 1797: 1793: 1789: 1785: 1781: 1774: 1772: 1709: 1705: 1701: 1657: 1655: 1510: 1463: 1459: 1457: 1450: 1386: 1380: 1374: 1372: 1297: 1293: 1289: 1285: 1279: 1227: 1222: 1218: 1214: 1209: 1200: 1195: 1190: 1185: 1178: 1134: 1130: 1125: 1121: 1114: 1110: 1106: 1101: 1093: 1089: 1083: 1078: 1073: 1068: 1063: 1058: 991: 986: 979: 975: 971: 883: 879: 877: 872: 828: 823: 819: 813: 809: 803: 799: 794: 790: 784: 733: 686: 635: 631: 623: 620:Weil divisor 619: 615: 607: 599: 595: 591: 580: 578: 568: 561: 549: 545: 541: 537: 536:has degree 2 532: 528: 524: 520: 512: 508: 504: 500: 486: 481: 477: 473: 469: 465: 461: 453: 449: 445: 441: 437: 433: 429: 425: 421: 417: 413: 409: 405: 401: 397: 393: 389: 386:vector space 381: 377: 375: 371: 366: 362: 358: 354: 350: 346: 344: 263: 259: 255: 251: 247: 242: 237: 233: 229: 225: 221: 216: 211: 207: 199: 194:is a finite 191: 189: 184: 180: 166: 154: 127: 108: 104: 100:line bundles 86: 82: 78: 68: 35: 29: 8950:, C. U.P., 8835:cap product 8178:) (meaning 8030:comes with 7319:Chern class 6948:pushforward 6946:called the 5594:is smooth, 4743:-module of 4145:where = , 3774:≠ 0}. Then 3431:defined by 3347:defined by 3128:uncountable 3045:curve over 612:codimension 232:at a point 40:codimension 8903:References 8343:round-down 8275:-Cartier. 8215:Q-divisors 8159:birational 8056:positivity 8036:base locus 6486:such that 6376:is called 6031:. Because 5520:. Because 4783:invertible 4149:= 0, ..., 3965:= 1, ..., 3143:such that 3043:projective 3009:, and the 2658:Chow group 982:, written 628:formal sum 412:)) or the 115:cohomology 52:André Weil 8985:: 5–361. 8499:⌋ 8493:⌊ 8426:⌋ 8420:⌊ 8384:⌋ 8371:⌊ 8368:∑ 8362:⌋ 8356:⌊ 8296:∑ 8253:Q-Cartier 8247:-divisor 8241:effective 8157:is a key 7800:effective 7736:− 7720:⟶ 7706:⁡ 7696:↓ 7690:↓ 7654:⟶ 7640:⁡ 7569:→ 7557:⁡ 7482:− 7468:→ 7456:⁡ 7436:cycle map 7333:regular. 7290:⁡ 7284:→ 7272:⁡ 7166:φ 7163:∘ 7126:− 7122:φ 7029:→ 7007:− 7003:φ 6977:∗ 6973:φ 6916:¯ 6903:φ 6863:¯ 6850:φ 6807:φ : 6763:∈ 6733:≠ 6721:− 6603:× 6569:→ 6518:φ : 6378:factorial 5946:Γ 5900:Γ 5897:∈ 5836:→ 5490:→ 5464:→ 5438:→ 5421:→ 5351:× 5327:× 5291:→ 5283:× 5240:principal 5211:⁡ 5197:× 5161:→ 5153:× 5129:× 5093:→ 5085:× 4877:× 4860:∈ 4831:⋅ 4712:is a sub- 4689:× 4645:∩ 4543:× 4383:× 4359:× 4311:→ 4306:× 4282:× 4265:→ 4260:× 4243:→ 4238:× 4221:→ 4106:− 4094:ω 4088:⁡ 4024:− 4021:⋯ 4018:− 4005:− 3996:ω 3990:⁡ 3915:Ω 3855:∧ 3852:⋯ 3849:∧ 3814:ω 3697:Ω 3691:∗ 3575:⁡ 3557:⁡ 3551:→ 3539:⁡ 3528:∗ 3449:Q-Cartier 3370:, namely 2861:↦ 2398:Γ 2349:→ 2332:→ 2286:− 2220:→ 2203:→ 2186:→ 2177:− 2157:→ 2122:− 1898:⁡ 1859:↦ 1835:→ 1748:− 1745:≥ 1733:⁡ 1624:≥ 1606:⁡ 1571:∈ 1524:Γ 1430:⁡ 1418:⁡ 1403:⁡ 1343:⁡ 1324:∑ 1314:⁡ 1096:) is the 996:, is the 938:∈ 842:⁡ 795:effective 763:≠ 714:≠ 650:∑ 558:curvature 396:, called 353:. Since ( 315:⁡ 297:∈ 290:∑ 9243:Category 9188:(2004), 9147:(2013), 9065:(1977), 8971:(1967). 8912:(1985), 8865:, § 1.4. 8547:) → Pic( 8113:and the 7987:pullback 7415:) in Cl( 7202:, then φ 6963:pullback 6940:) → Div( 6660:and let 6652:and let 6616:′ 6565:′ 6505:, where 5041:groups: 2908:Examples 1213: : 1129:, where 927:one. If 604:integral 585:integral 130:Dedekind 111:homology 60:integers 36:divisors 9216:2095471 9179:3057950 9138:2223410 9126:Bibcode 9097:0463157 9055:2171939 9033:Bibcode 9013:(ed.), 8999:0238860 8940:0780183 8833:act by 8199:))) as 7865:; then 6839:, then 6668:. Then 6495:= Spec 6481:= Spec 5652:Cartier 5403:Assume 3756:, ..., 3735:Example 3675:is the 3447:is not 3329:quadric 3327:be the 3094:with a 2722:) → Cl( 2508:below. 2500:. When 1193:) − ord 1137:are in 1076:) + ord 1066:) = ord 204:integer 177:compact 9214: 9204: 9177: 9167: 9136: 9095: 9085: 9053: 9043: 8997: 8954: 8938: 8928: 8796: 8412:where 8081:, the 7985:, the 7344:, let 6950:. (If 4849:where 4455:where 3961:= 0}, 3803:. Let 3763:. Let 3737:: Let 3723:where 3492:locus 3490:smooth 3488:. The 3159:, the 3147:has a 3126:is an 3031:degree 3019:degree 3011:degree 2675:) of ( 2013:as an 1887:since 1773:where 1292:, the 1232:normal 998:length 978:along 871:where 602:is an 583:be an 552:has a 516:. The 493:1-form 208:degree 9116:arXiv 9023:arXiv 8853:) ∩ . 8601:Notes 8581:is a 8337:is a 8263:. If 8042:with 7880:be a 7388:) + ( 7384:) = ( 7364:) on 7198:into 6537:over 6165:with 3278:of a 3041:is a 3037:. If 3025:over 2940:over 2929:over 2710:. If 2690:. If 1788:. If 1700:over 1385:. If 1271:is a 1226:. If 1088:. If 822:Spec 626:is a 614:1 in 518:genus 497:field 361:) + ( 357:) = ( 240:, ord 202:with 134:Weber 9202:ISBN 9165:ISBN 9083:ISBN 9041:ISBN 8952:ISBN 8926:ISBN 8794:ISBN 8524:The 8278:If 8219:Let 8062:and 7876:Let 7782:For 7219:= φ( 6936:Div( 6805:Let 6644:Let 6533:for 6441:Let 6384:are 6213:and 5512:and 4587:and 4165:Let 3480:Let 3411:Let 3323:Let 3274:the 3270:For 3213:For 3086:) ≅ 3065:) → 2981:Let 2913:Let 2782:and 2682:Let 2632:The 2238:The 1458:Let 1221:) → 1133:and 923:has 839:Spec 783:Div( 618:. A 590:. A 579:Let 452:for 132:and 62:and 50:and 9194:doi 9157:doi 9075:doi 8987:doi 8845:↦ c 8786:doi 8539:in 8255:if 8251:is 8153:of 8022:on 7981:on 7953:of 7637:Pic 7554:Pic 7442:: 7400:of 7269:Pic 7213:is 6957:If 6831:If 6824:to 6591:to 6545:on 6499:/ ( 6461:in 6449:on 6365:If 6336:div 6024:on 5634:in 5208:Pic 5034:). 5026:or 4805:to 4797:of 4789:in 4781:is 4627:on 4557:on 4333:on 4157:.) 4085:div 3987:div 3954:= { 3767:= { 3671:on 3623:of 3572:Pic 3496:of 3451:on 3427:in 3378:on 3364:can 3343:in 3294:of 3261:of 3226:Pic 3179:Pic 3110:of 3053:on 3021:of 3013:of 3005:of 2993:in 2636:Cl( 2088:If 1895:div 1784:in 1724:ord 1603:div 1509:on 1453:div 1427:div 1415:div 1400:div 1334:ord 1311:div 1288:on 1277:ord 1230:is 1207:ord 1183:ord 1100:on 1056:ord 1000:of 984:ord 974:of 878:If 793:is 634:of 622:on 610:of 598:on 594:or 523:of 503:of 306:ord 236:in 224:on 155:An 54:by 30:In 9245:: 9212:MR 9210:, 9200:, 9175:MR 9173:, 9163:, 9155:, 9151:, 9134:MR 9132:, 9124:, 9093:MR 9091:, 9081:, 9073:, 9051:MR 9049:, 9039:, 9031:, 9021:, 8995:MR 8993:. 8983:32 8981:. 8975:. 8967:; 8936:MR 8934:, 8792:. 8686:^ 8665:^ 8257:mD 8192:mK 8186:, 8171:mK 8169:, 8125:, 8093:, 8066:. 8050:→ 7961:. 7937:, 7873:. 7744:BM 7703:Cl 7490:BM 7453:Cl 7382:fs 7287:Cl 7233:. 7189:= 6811:→ 6522:→ 6491:∩ 6362:. 5638:. 5493:0. 4329:A 4314:0. 3789:= 3741:= 3731:. 3603:− 3554:Cl 3536:Cl 3508:→ 3504:: 3435:= 3421:zw 3419:= 3417:xy 3351:= 3335:= 3333:xy 3316:= 3314:xy 2904:. 2745:, 2726:− 2698:− 2669:−1 2315:. 2223:0. 1712:, 1124:/ 1109:∈ 1064:fg 882:⊂ 814:D′ 812:− 804:D′ 802:≥ 638:, 571:. 569:CP 507:, 468:, 446:mD 440:, 404:, 369:. 355:fg 286::= 187:. 167:A 102:. 66:. 34:, 9196:: 9159:: 9128:: 9118:: 9077:: 9035:: 9025:: 9001:. 8989:: 8883:( 8851:L 8849:( 8847:1 8843:L 8839:X 8831:X 8827:X 8812:( 8802:. 8788:: 8770:. 8709:. 8660:. 8644:. 8628:. 8595:R 8591:R 8587:R 8579:R 8569:O 8565:Z 8561:Y 8553:Y 8549:Y 8545:X 8541:X 8537:Y 8530:X 8505:. 8502:) 8496:D 8490:( 8485:O 8463:) 8460:D 8457:( 8452:O 8440:a 8423:a 8397:, 8392:j 8388:Z 8379:j 8375:a 8365:= 8359:D 8339:Q 8320:j 8316:Z 8310:j 8306:a 8300:j 8292:= 8289:D 8273:Q 8269:Q 8265:X 8261:m 8249:D 8245:Q 8237:Q 8233:R 8229:X 8225:Q 8221:X 8209:n 8205:n 8201:m 8196:X 8190:( 8188:O 8184:X 8182:( 8180:H 8175:X 8167:X 8165:( 8163:H 8155:X 8146:X 8142:K 8135:X 8131:D 8129:( 8127:O 8123:X 8121:( 8119:H 8107:X 8099:D 8097:( 8095:O 8091:X 8089:( 8087:H 8083:k 8079:k 8075:X 8071:D 8052:P 8048:X 8044:n 8040:L 8032:n 8028:L 8024:P 8010:) 8007:1 8004:( 7999:O 7983:X 7979:L 7975:k 7971:P 7967:X 7955:D 7947:D 7943:D 7941:( 7939:O 7935:X 7933:( 7931:H 7927:k 7923:k 7909:) 7906:D 7903:( 7898:O 7886:k 7878:X 7871:s 7867:D 7863:s 7849:) 7846:D 7843:( 7838:O 7826:D 7821:i 7817:f 7813:X 7808:i 7804:f 7788:C 7784:X 7763:) 7759:Z 7755:, 7752:X 7749:( 7739:2 7733:n 7730:2 7726:H 7715:) 7712:X 7709:( 7683:) 7679:Z 7675:, 7672:X 7669:( 7664:2 7660:H 7649:) 7646:X 7643:( 7620:X 7599:. 7596:) 7592:Z 7588:, 7585:X 7582:( 7577:2 7573:H 7566:) 7563:X 7560:( 7537:X 7533:C 7531:( 7529:X 7512:. 7509:) 7505:Z 7501:, 7498:X 7495:( 7485:2 7479:n 7476:2 7472:H 7465:) 7462:X 7459:( 7432:C 7428:n 7424:X 7417:X 7413:L 7411:( 7409:1 7406:c 7402:L 7398:s 7394:f 7390:s 7386:f 7378:s 7374:s 7370:L 7366:X 7362:s 7358:L 7354:L 7350:L 7346:s 7342:X 7338:L 7331:X 7327:X 7323:X 7302:, 7299:) 7296:X 7293:( 7281:) 7278:X 7275:( 7266:: 7261:1 7257:c 7243:X 7227:Z 7223:) 7221:Z 7217:Z 7215:φ 7211:Z 7204:Z 7200:Y 7196:Z 7191:Z 7187:X 7172:} 7169:) 7158:i 7154:f 7150:, 7147:) 7142:i 7138:U 7134:( 7129:1 7118:( 7115:{ 7095:} 7092:) 7087:i 7083:f 7079:, 7074:i 7070:U 7066:( 7063:{ 7041:X 7035:M 7024:Y 7018:M 7010:1 6982:Z 6959:Z 6952:X 6944:) 6942:Y 6938:X 6932:X 6912:) 6909:Z 6906:( 6890:Z 6887:* 6883:Z 6879:Y 6859:) 6856:Z 6853:( 6837:X 6833:Z 6826:Y 6822:X 6818:D 6813:Y 6809:X 6793:. 6781:) 6778:D 6775:, 6772:X 6769:( 6766:N 6760:m 6748:. 6736:0 6730:) 6727:) 6724:H 6718:D 6715:m 6712:( 6707:X 6701:O 6695:, 6692:X 6689:( 6684:0 6680:H 6666:X 6662:H 6658:X 6654:D 6646:X 6620:, 6613:S 6607:S 6599:X 6589:D 6575:, 6572:S 6562:S 6551:S 6547:X 6543:D 6539:S 6535:X 6524:S 6520:X 6511:A 6507:f 6503:) 6501:f 6497:A 6493:D 6489:U 6483:A 6479:U 6474:x 6469:x 6467:I 6463:X 6459:x 6455:I 6451:X 6443:X 6417:) 6414:D 6411:( 6406:O 6394:D 6390:X 6382:X 6374:X 6367:X 6359:i 6357:U 6352:i 6350:f 6326:X 6322:X 6308:} 6305:) 6300:i 6296:f 6292:, 6287:i 6283:U 6279:( 6276:{ 6254:. 6249:j 6245:f 6240:/ 6234:i 6230:f 6219:D 6217:( 6215:L 6201:) 6198:D 6195:( 6190:O 6177:i 6175:U 6171:D 6169:( 6167:L 6153:) 6150:D 6147:( 6142:O 6130:D 6128:( 6126:L 6111:} 6108:) 6103:i 6099:f 6095:, 6090:i 6086:U 6082:( 6079:{ 6068:i 6066:f 6052:) 6049:D 6046:( 6041:O 6028:i 6026:U 6012:) 6009:D 6006:( 6001:O 5979:) 5974:X 5968:O 5962:, 5957:i 5953:U 5949:( 5943:= 5940:) 5933:i 5929:U 5922:O 5916:, 5911:i 5907:U 5903:( 5894:1 5874:. 5867:i 5863:U 5857:| 5852:) 5849:D 5846:( 5841:O 5829:i 5825:U 5818:O 5804:i 5802:U 5788:) 5785:D 5782:( 5777:O 5764:i 5762:U 5748:) 5745:D 5742:( 5737:O 5724:X 5722:M 5708:) 5705:D 5702:( 5697:O 5675:) 5672:D 5669:( 5664:O 5648:D 5636:X 5632:D 5618:) 5615:D 5612:( 5607:D 5603:O 5592:D 5578:) 5575:D 5572:( 5567:O 5545:) 5542:D 5539:( 5534:O 5522:D 5518:D 5514:D 5510:X 5487:) 5484:D 5481:( 5476:D 5470:O 5461:) 5458:D 5455:( 5450:X 5444:O 5433:X 5427:O 5418:0 5405:D 5398:C 5393:X 5385:X 5381:L 5373:X 5359:, 5356:) 5346:X 5340:O 5333:/ 5322:X 5316:M 5310:, 5307:X 5304:( 5299:0 5295:H 5288:) 5278:X 5272:M 5266:, 5263:X 5260:( 5255:0 5251:H 5223:. 5220:) 5217:X 5214:( 5205:= 5202:) 5192:X 5186:O 5180:, 5177:X 5174:( 5169:1 5165:H 5158:) 5148:X 5142:O 5135:/ 5124:X 5118:M 5112:, 5109:X 5106:( 5101:0 5097:H 5090:) 5080:X 5074:M 5068:, 5065:X 5062:( 5057:0 5053:H 5032:D 5030:( 5028:L 5014:) 5011:D 5008:( 5003:O 4991:D 4977:, 4974:} 4971:) 4966:i 4962:f 4958:, 4953:i 4949:U 4945:( 4942:{ 4922:. 4917:X 4911:M 4888:) 4885:U 4882:( 4872:X 4866:M 4857:f 4837:, 4834:f 4826:U 4820:O 4807:U 4803:J 4799:x 4795:U 4791:X 4787:x 4779:J 4765:. 4760:X 4754:M 4729:X 4723:O 4694:. 4684:X 4678:O 4653:j 4649:U 4640:i 4636:U 4613:j 4609:f 4605:= 4600:i 4596:f 4575:, 4570:i 4566:U 4538:X 4532:M 4507:i 4503:f 4499:, 4496:X 4476:} 4471:i 4467:U 4463:{ 4443:, 4440:} 4437:) 4432:i 4428:f 4424:, 4419:i 4415:U 4411:( 4408:{ 4388:. 4378:X 4372:O 4365:/ 4354:X 4348:M 4335:X 4301:X 4295:O 4288:/ 4277:X 4271:M 4255:X 4249:M 4233:X 4227:O 4218:0 4195:. 4190:X 4184:M 4171:X 4167:X 4151:n 4147:i 4130:] 4127:H 4124:[ 4121:) 4118:1 4115:+ 4112:n 4109:( 4103:= 4100:] 4097:) 4091:( 4082:[ 4079:= 4072:n 4067:P 4061:K 4032:n 4028:Z 4013:0 4009:Z 4002:= 3999:) 3993:( 3974:0 3971:Z 3967:n 3963:i 3958:i 3956:x 3951:i 3949:Z 3933:n 3926:n 3921:P 3904:U 3887:. 3880:n 3876:y 3869:n 3865:y 3861:d 3842:1 3838:y 3831:1 3827:y 3823:d 3817:= 3801:0 3798:x 3796:/ 3793:i 3791:x 3786:i 3784:y 3780:n 3776:U 3772:0 3769:x 3765:U 3760:n 3758:x 3754:0 3751:x 3747:n 3743:P 3739:X 3729:X 3725:n 3711:, 3706:n 3701:U 3687:j 3673:X 3659:) 3654:X 3650:K 3646:( 3641:O 3629:U 3625:X 3620:X 3616:K 3609:X 3605:U 3601:X 3584:) 3581:U 3578:( 3569:= 3566:) 3563:U 3560:( 3548:) 3545:X 3542:( 3533:: 3524:j 3510:X 3506:U 3502:j 3498:X 3494:U 3482:X 3471:. 3469:D 3465:Z 3461:X 3457:D 3453:X 3445:D 3441:X 3437:z 3433:x 3429:X 3425:D 3413:X 3406:. 3404:D 3400:Z 3396:X 3392:X 3388:D 3384:D 3380:X 3376:x 3372:x 3368:X 3361:D 3357:X 3353:z 3349:x 3345:X 3341:D 3337:z 3325:X 3320:. 3318:z 3302:. 3296:R 3288:R 3284:R 3272:R 3265:. 3263:X 3243:0 3238:k 3234:/ 3230:X 3215:k 3201:. 3196:0 3191:k 3187:/ 3183:X 3165:k 3153:X 3149:k 3145:X 3141:k 3137:X 3120:X 3112:X 3104:k 3098:- 3096:k 3092:X 3088:Z 3084:P 3080:k 3076:P 3069:. 3067:Z 3063:X 3059:X 3055:X 3051:f 3047:k 3039:X 3035:X 3027:k 3023:E 3015:p 3007:k 3003:E 2999:E 2995:X 2991:p 2987:k 2983:X 2974:P 2970:H 2966:Z 2962:P 2958:H 2954:P 2950:A 2946:H 2942:k 2938:P 2931:k 2927:A 2923:k 2919:n 2915:k 2902:X 2898:X 2884:) 2881:D 2878:( 2873:X 2867:O 2858:D 2848:X 2832:X 2826:O 2803:) 2800:E 2797:( 2792:O 2770:) 2767:D 2764:( 2759:O 2747:E 2743:D 2739:X 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Knowledge

Divisor (algebraic geometry)

Index