5157:
35:
4913:
4447:
5553:
4617:
3433:. The conclusion is that to check nefness of a divisor class, it suffices to compute the intersection number with curves contained in the base locus of the class. So, roughly speaking, the 'smaller' the base locus, the 'more likely' it is that the class is nef.
3081:
2902:
refers to the subvariety of points 'common' to all divisors in the linear system. Geometrically, this corresponds to the common intersection of the varieties. Linear systems may or may not have a base locus – for example, the pencil of affine lines
2839:. The technical demands became quite stringent; later developments clarified a number of issues. The computation of the relevant dimensions — the Riemann–Roch problem as it can be called — can be better phrased in terms of
5187:
Each morphism from an algebraic variety to a projective space determines a base-point-free linear system on the variety; because of this, a base-point-free linear system and a map to a projective space are often used interchangeably.
4194:
4204:
Each linear system on an algebraic variety determines a morphism from the complement of the base locus to a projective space of dimension of the system, as follows. (In a sense, the converse is also true; see the section below)
4784:
4318:
5012:
3899:
2929:
has no common intersection, but given two (nondegenerate) conics in the complex projective plane, they intersect in four points (counting with multiplicity) and thus the pencil they define has these points as base locus.
5377:
737:
1559:
2065:
3750:
1289:
1180:
5439:
2807:. Note a characteristic system need not to be complete; in fact, the question of completeness is something studied extensively by the Italian school without a satisfactory conclusion; nowadays, the
3621:
The notion of the base locus still makes sense for a non-complete linear system as well: the base locus of it is still the intersection of the supports of all the effective divisors in the system.
1706:
3681:
4502:
4309:
2721:
5135:
1003:
2552:
4704:
931:
3579:
1829:
1413:
2989:
235:
140:
4255:
3104:
2674:
2285:
5646:
5221:
4009:
538:
3193:
4742:
2805:
3612:
3517:
2475:
1088:
5434:
5073:
4775:
4038:
3431:
2612:
2504:
1327:
1117:
5245:
4486:
3774:
2823:
is a property of a pencil of cubics, which states that the base locus satisfies an "8 implies 9" property: any cubic containing 8 of the points necessarily contains the 9th.
2583:
2132:
955:
885:
4937:
3131:
2434:
1970:
592:
3948:
3369:
1733:
5044:
1226:
2385:
2349:
2317:
2224:
2108:
283:
827:(this is well defined since two non-zero rational functions have the same divisor if and only if they are non-zero multiples of each other). A complete linear system
4640:
775:
5579:
1921:
1886:
381:
5713:
Hartshorne, R. 'Algebraic
Geometry', proposition II.7.2, page 151, proposition II.7.7, page 157, page 158, exercise IV.1.7, page 298, proposition IV.5.3, page 342
4064:
3464:
3340:
3230:
3165:
2961:
2927:
2250:
1855:
855:
805:
662:
568:
407:
1456:
5405:
5285:
5265:
3537:
3484:
3389:
3310:
3290:
3270:
3250:
2981:
2741:
2192:
2172:
2152:
1990:
1787:
1765:
1630:
1610:
1587:
1433:
1347:
1034:
825:
632:
612:
501:
450:
427:
349:
329:
309:
192:
4072:
1708:. This definition follows from proposition II.7.7 of Hartshorne since every effective divisor in the linear system comes from the zeros of some section of
4908:{\displaystyle \operatorname {Sym} ((V\otimes _{k}{\mathcal {O}}_{X})\otimes _{{\mathcal {O}}_{X}}L^{-1})\to \bigoplus _{n=0}^{\infty }{\mathcal {I}}^{n}}
4442:{\displaystyle \operatorname {Sym} ((V\otimes _{k}{\mathcal {O}}_{X})\otimes _{{\mathcal {O}}_{X}}L^{-1})\to \bigoplus _{n=0}^{\infty }{\mathcal {O}}_{X}}
4949:
3782:
5290:
667:
352:
5788:
5775:. Grundlehren der Mathematischen Wissenschaften. Vol. II, with a contribution by Joseph Daniel Harris. Heidelberg: Springer. p. 3.
1464:
2836:
1995:
5548:{\displaystyle {\mathcal {O}}_{X}(1)={\mathcal {O}}_{X}\otimes _{{\mathcal {O}}_{\mathbb {P} ^{r}}}{\mathcal {O}}_{\mathbb {P} ^{r}}(1)}
3686:
2882:. There was a controversy, one of the final issues in the conflict between 'old' and 'new' points of view in algebraic geometry, over
1231:
1122:
5894:
5876:
5851:
4612:{\displaystyle i:X\hookrightarrow \mathbb {P} (V_{X}^{*}\otimes L)\simeq \mathbb {P} (V_{X}^{*})=\mathbb {P} (V^{*})\times X}
1635:
5144:
is chosen, the above discussion becomes more down-to-earth (and that is the style used in
Hartshorne, Algebraic Geometry).
3641:
5911:
4264:
2682:
2820:
465:
86:
1768:
2808:
5839:
5081:
960:
2509:
5667:
4656:
5657:
1858:
894:
3076:{\displaystyle \operatorname {Bl} (|D|):=\bigcap _{D_{\text{eff}}\in |D|}\operatorname {Supp} D_{\text{eff}}\ }
2844:
453:
3542:
28:
1793:
1352:
2752:
473:
197:
102:
4219:
3089:
2620:
2255:
5686:
5584:
5194:
3953:
506:
3178:
4716:
3199:
2784:
3584:
3489:
2447:
1060:
5410:
5049:
4751:
4014:
3394:
2588:
2480:
1294:
1093:
5226:
4452:
3755:
3391:, and so intersects it properly. Basic facts from intersection theory then tell us that we must have
3168:
2557:
2113:
936:
866:
147:
93:
4918:
3109:
2390:
1926:
573:
3907:
2840:
2832:
1744:
1590:
3345:
887:
is then a projective subspace of a complete linear system, so it corresponds to a vector subspace
3625:
2860:
2852:
1711:
469:
55:
47:
5020:
4040:. Then, the base locus of this system of divisors is the scheme given by the vanishing locus of
1189:
4257:
a finite-dimensional vector subspace. For the sake of clarity, we first consider the case when
2358:
2322:
2290:
2197:
2081:
247:
5890:
5882:
5872:
5847:
5835:
5784:
5768:
5746:
4643:
3615:
2899:
2883:
2867:
2770:
in the family is a linear system formed by the curves in the family that are infinitely near
289:
4625:
742:
81:. It assumed a more general form, through gradual generalisation, so that one could speak of
5860:
5817:
5776:
5764:
5736:
5662:
5558:
4493:
3636:
3106:
denotes the support of a divisor, and the intersection is taken over all effective divisors
1891:
1013:
78:
66:; the dimension of the linear system corresponds to the number of parameters of the family.
63:
43:
5798:
1864:
5868:
5794:
2886:'s characteristic linear system of an algebraic family of curves on an algebraic surface.
2856:
1183:
1037:
503:
is defined as the set of all effective divisors linearly equivalent to some given divisor
457:
357:
74:
4189:{\displaystyle {\text{Bl}}({\mathfrak {X}})={\text{Proj}}\left({\frac {k}{(f,g)}}\right)}
4043:
3439:
3315:
3205:
3140:
2936:
2906:
2229:
1834:
830:
780:
637:
543:
386:
1438:
1228:, it is linearly equivalent to any other divisor defined by the vanishing locus of some
5390:
5270:
5250:
3522:
3469:
3374:
3295:
3275:
3255:
3235:
2966:
2726:
2352:
2177:
2157:
2137:
1975:
1772:
1750:
1743:
One application of linear systems is used in the classification of algebraic curves. A
1615:
1595:
1572:
1418:
1332:
1019:
810:
617:
597:
486:
435:
412:
334:
314:
294:
177:
5156:
5905:
4489:
2778:
5724:
3172:
2848:
461:
97:
34:
5821:
1009:
2762:
The characteristic linear system of a family of curves on an algebraic surface
5780:
5007:{\displaystyle i:{\widetilde {X}}\hookrightarrow \mathbb {P} (V^{*})\times X.}
3894:{\displaystyle {\mathfrak {X}}={\text{Proj}}\left({\frac {k}{(sf+tg)}}\right)}
5750:
1612:
is given by the complete linear system associated with the canonical divisor
5372:{\displaystyle f^{-1}({\mathfrak {d}})=\{f^{-1}(D)|D\in {\mathfrak {d}}\}}
3436:
In the modern formulation of algebraic geometry, a complete linear system
2777:
In modern terms, it is a subsystem of the linear system associated to the
1044:
of two divisors means that the corresponding line bundles are isomorphic.
732:{\displaystyle (\Gamma (X,{\mathcal {L}})\smallsetminus \{0\})/k^{\ast },}
17:
4744:
in the direct sum replaced by an ideal sheaf defining the base locus and
3904:
This has an associated linear system of divisors since each polynomial,
468:). The definition in that case is usually said with greater care (using
5741:
5436:
has a natural linear system determining a map to projective space from
2871:
2319:
from proposition 5.3. Another close set of examples are curves with a
1554:{\displaystyle D=E+\left({\frac {x^{2}+y^{2}+z^{2}+w^{2}}{xy}}\right)}
2060:{\displaystyle \mathbb {P} ^{1}=\mathbb {P} (H^{0}(C,\omega _{C}))}
1569:
One of the important complete linear systems on an algebraic curve
3745:{\displaystyle f,g\in \Gamma (\mathbb {P} ^{n},{\mathcal {O}}(d))}
3202:
of a
Cartier divisor class (i.e. complete linear system). Suppose
33:
4488:
for the trivial vector bundle and passing the surjection to the
1284:{\displaystyle t\in \Gamma (\mathbb {P} ^{3},{\mathcal {O}}(2))}
1175:{\displaystyle s\in \Gamma (\mathbb {P} ^{3},{\mathcal {O}}(2))}
5223:
of algebraic varieties there is a pullback of a linear system
5151:
2870:
to that of linear systems cut out by surfaces in three-space;
5516:
5491:
5472:
5446:
4924:
4894:
4838:
4816:
4723:
4428:
4372:
4350:
4284:
3725:
3590:
3495:
2527:
2453:
1264:
1155:
1066:
912:
688:
579:
162:
A map determined by a linear system is sometimes called the
118:
62:
is an algebraic generalization of the geometric notion of a
5846:. Wiley Classics Library. Wiley Interscience. p. 137.
4713:
is not empty, the above discussion still goes through with
2963:
is a complete linear system of divisors on some variety
2811:
can be used to answer the question of the completeness.
409:
denotes the divisor of zeroes and poles of the function
5168:
5148:
Linear system determined by a map to a projective space
2866:
The
Italian school liked to reduce the geometry on an
1701:{\displaystyle |K|=\mathbb {P} (H^{0}(C,\omega _{C}))}
1008:
Linear systems can also be introduced by means of the
27:"Kodaira map" redirects here. Not to be confused with
5587:
5561:
5442:
5413:
5393:
5293:
5273:
5253:
5229:
5197:
5084:
5052:
5023:
4952:
4921:
4787:
4754:
4719:
4659:
4628:
4505:
4455:
4321:
4267:
4222:
4075:
4046:
4017:
3956:
3910:
3785:
3758:
3689:
3676:{\displaystyle p:{\mathfrak {X}}\to \mathbb {P} ^{1}}
3644:
3587:
3545:
3525:
3492:
3472:
3442:
3397:
3377:
3348:
3318:
3298:
3278:
3258:
3238:
3208:
3181:
3143:
3112:
3092:
2992:
2969:
2939:
2909:
2787:
2729:
2685:
2623:
2591:
2560:
2512:
2483:
2450:
2440:
Linear systems of hypersurfaces in a projective space
2393:
2361:
2325:
2293:
2258:
2232:
2200:
2180:
2160:
2140:
2116:
2084:
1998:
1978:
1929:
1894:
1867:
1837:
1796:
1775:
1753:
1714:
1638:
1618:
1598:
1575:
1467:
1441:
1421:
1355:
1335:
1297:
1234:
1192:
1125:
1096:
1063:
1022:
963:
939:
897:
869:
833:
813:
783:
745:
670:
640:
620:
600:
576:
546:
509:
489:
438:
415:
389:
360:
337:
317:
297:
250:
200:
180:
105:
5725:"Another proof of the existence of special divisors"
4304:{\displaystyle V\otimes _{k}{\mathcal {O}}_{X}\to L}
4261:
is base-point-free; in other words, the natural map
2716:{\displaystyle \mathbb {P} ^{k}\to \mathbb {P} ^{N}}
145:
Linear systems of dimension 1, 2, or 3 are called a
456:, the notion of 'divisor' is inherently ambiguous (
5640:
5573:
5547:
5428:
5399:
5371:
5279:
5259:
5239:
5215:
5129:
5067:
5038:
5006:
4931:
4907:
4769:
4736:
4698:
4634:
4611:
4480:
4441:
4303:
4249:
4188:
4058:
4032:
4003:
3942:
3893:
3768:
3744:
3675:
3606:
3573:
3531:
3511:
3478:
3458:
3425:
3383:
3363:
3334:
3304:
3284:
3264:
3244:
3224:
3198:One application of the notion of base locus is to
3187:
3159:
3125:
3098:
3075:
2975:
2955:
2921:
2799:
2758:Characteristic linear system of a family of curves
2735:
2715:
2668:
2606:
2577:
2546:
2498:
2469:
2428:
2379:
2343:
2311:
2279:
2244:
2218:
2186:
2166:
2146:
2126:
2102:
2059:
1984:
1964:
1915:
1880:
1849:
1823:
1781:
1759:
1727:
1700:
1624:
1604:
1581:
1553:
1450:
1427:
1407:
1341:
1321:
1283:
1220:
1174:
1111:
1082:
1028:
997:
949:
925:
879:
849:
819:
799:
769:
731:
656:
626:
606:
586:
562:
532:
495:
444:
421:
401:
375:
343:
323:
303:
277:
229:
186:
134:
5812:Fulton, William (1998). "§ 4.4. Linear Systems".
2831:In general linear systems became a basic tool of
2654:
2633:
1040:, to be precise) correspond to line bundles, and
2855:generated by codimension-one subvarieties), and
42:algebraicizes the classic geometric notion of a
3581:is the set of common zeroes of all sections of
2878:to try to pull together the methods, involving
4777:of it along the (scheme-theoretic) base locus
3167:(as a set, at least: there may be more subtle
2723:we can construct a linear system of dimension
2287:. In fact, hyperelliptic curves have a unique
957:is its dimension as a projective space. Hence
5130:{\displaystyle f:X-B\to \mathbb {P} (V^{*}).}
4781:. Precisely, as above, there is a surjection
3614:. A simple consequence is that the bundle is
998:{\displaystyle \dim {\mathfrak {d}}=\dim W-1}
634:is a nonsingular projective variety, the set
8:
5366:
5323:
2547:{\displaystyle V=\Gamma ({\mathcal {O}}(d))}
1458:. Then, there is the equivalence of divisors
1329:(Proposition 7.2). For example, the divisor
705:
699:
4699:{\displaystyle f:X\to \mathbb {P} (V^{*}).}
4650:by a projection, there results in the map:
2194:. For example, hyperelliptic curves have a
4646:under a twist by a line bundle. Following
2843:. The effect of working on varieties with
926:{\displaystyle \Gamma (X,{\mathcal {L}}).}
5740:
5632:
5628:
5627:
5614:
5595:
5586:
5560:
5528:
5524:
5523:
5521:
5515:
5514:
5503:
5499:
5498:
5496:
5490:
5489:
5487:
5477:
5471:
5470:
5451:
5445:
5444:
5441:
5420:
5416:
5415:
5412:
5392:
5360:
5359:
5348:
5330:
5311:
5310:
5298:
5292:
5272:
5252:
5231:
5230:
5228:
5196:
5115:
5104:
5103:
5083:
5054:
5053:
5051:
5022:
4986:
4975:
4974:
4960:
4959:
4951:
4923:
4922:
4920:
4899:
4893:
4892:
4885:
4874:
4855:
4843:
4837:
4836:
4834:
4821:
4815:
4814:
4807:
4786:
4756:
4755:
4753:
4728:
4722:
4721:
4718:
4684:
4673:
4672:
4658:
4627:
4594:
4583:
4582:
4570:
4565:
4554:
4553:
4535:
4530:
4519:
4518:
4504:
4460:
4454:
4433:
4427:
4426:
4419:
4408:
4389:
4377:
4371:
4370:
4368:
4355:
4349:
4348:
4341:
4320:
4289:
4283:
4282:
4275:
4266:
4221:
4212:be a line bundle on an algebraic variety
4153:
4134:
4106:
4097:
4085:
4084:
4076:
4074:
4045:
4024:
4020:
4019:
4016:
3995:
3991:
3990:
3977:
3964:
3955:
3931:
3915:
3909:
3852:
3833:
3805:
3796:
3787:
3786:
3784:
3760:
3759:
3757:
3724:
3723:
3714:
3710:
3709:
3688:
3667:
3663:
3662:
3652:
3651:
3643:
3589:
3588:
3586:
3563:
3555:
3544:
3524:
3494:
3493:
3491:
3471:
3451:
3443:
3441:
3406:
3398:
3396:
3376:
3350:
3349:
3347:
3327:
3319:
3317:
3297:
3277:
3257:
3237:
3217:
3209:
3207:
3180:
3152:
3144:
3142:
3117:
3111:
3091:
3064:
3047:
3039:
3030:
3025:
3010:
3002:
2991:
2968:
2948:
2940:
2938:
2908:
2786:
2728:
2707:
2703:
2702:
2692:
2688:
2687:
2684:
2653:
2632:
2630:
2622:
2598:
2594:
2593:
2590:
2562:
2561:
2559:
2526:
2525:
2511:
2490:
2486:
2485:
2482:
2452:
2451:
2449:
2406:
2392:
2371:
2366:
2360:
2335:
2330:
2324:
2303:
2298:
2292:
2271:
2267:
2266:
2257:
2231:
2210:
2205:
2199:
2179:
2159:
2139:
2118:
2117:
2115:
2094:
2089:
2083:
2045:
2026:
2015:
2014:
2005:
2001:
2000:
1997:
1977:
1947:
1934:
1928:
1893:
1872:
1866:
1836:
1815:
1811:
1810:
1795:
1774:
1752:
1719:
1713:
1686:
1667:
1656:
1655:
1647:
1639:
1637:
1617:
1597:
1574:
1530:
1517:
1504:
1491:
1484:
1466:
1440:
1420:
1399:
1386:
1373:
1360:
1354:
1334:
1306:
1296:
1263:
1262:
1253:
1249:
1248:
1233:
1197:
1191:
1154:
1153:
1144:
1140:
1139:
1124:
1103:
1099:
1098:
1095:
1065:
1064:
1062:
1021:
971:
970:
962:
941:
940:
938:
911:
910:
896:
871:
870:
868:
842:
834:
832:
812:
792:
784:
782:
744:
720:
711:
687:
686:
669:
649:
641:
639:
619:
599:
578:
577:
575:
555:
547:
545:
516:
508:
488:
437:
414:
388:
359:
336:
316:
296:
249:
213:
199:
179:
123:
117:
116:
104:
5723:Kleiman, Steven L.; Laksov, Dan (1974).
3618:if and only if the base locus is empty.
3574:{\displaystyle \operatorname {Bl} (|D|)}
2554:, then we can take its projectivization
5679:
1824:{\displaystyle f:C\to \mathbb {P} ^{1}}
1408:{\displaystyle x^{2}+y^{2}+z^{2}+w^{2}}
331:, or in other words a non-zero element
5871:, 1977; corrected 6th printing, 1993.
4642:on the right is the invariance of the
3539:. From this viewpoint, the base locus
3312:is not contained in the base locus of
1415:is linearly equivalent to the divisor
230:{\displaystyle D,E\in {\text{Div}}(X)}
135:{\displaystyle (X,{\mathcal {O}}_{X})}
4250:{\displaystyle V\subset \Gamma (X,L)}
3099:{\displaystyle \operatorname {Supp} }
2880:linear systems with fixed base points
2827:Linear systems in birational geometry
2669:{\displaystyle N={\binom {n+d}{n}}-1}
2280:{\displaystyle C\to \mathbb {P} ^{1}}
1435:associated to the vanishing locus of
1349:associated to the vanishing locus of
7:
5709:
5707:
5705:
5703:
5701:
5699:
5641:{\displaystyle \in \mathbb {P} ^{r}}
5216:{\displaystyle f:Y\hookrightarrow X}
4315:= the base field). Or equivalently,
4004:{\displaystyle \in \mathbb {P} ^{1}}
3371:in the class which does not contain
2898:of a linear system of divisors on a
2837:Italian school of algebraic geometry
807:to the set of non-zero multiples of
533:{\displaystyle D\in {\text{Div}}(X)}
5361:
5312:
5232:
4200:A map determined by a linear system
4086:
3788:
3761:
3653:
3466:of (Cartier) divisors on a variety
3188:{\displaystyle \operatorname {Bl} }
2847:is to show up a difference between
2119:
1016:language. In those terms, divisors
972:
942:
933:The dimension of the linear system
872:
69:These arose first in the form of a
5887:Positivity in Algebraic Geometry I
4886:
4737:{\displaystyle {\mathcal {O}}_{X}}
4420:
4229:
3702:
3133:in the linear system. This is the
2800:{\displaystyle C\hookrightarrow Y}
2637:
2519:
1857:all curves are hyperelliptic: the
1241:
1132:
898:
674:
25:
3607:{\displaystyle {\mathcal {O}}(D)}
3512:{\displaystyle {\mathcal {O}}(D)}
3342:, then there exists some divisor
2470:{\displaystyle {\mathcal {O}}(d)}
1083:{\displaystyle {\mathcal {O}}(2)}
857:is therefore a projective space.
594:be the line bundle associated to
5844:Principles of Algebraic Geometry
5429:{\displaystyle \mathbb {P} ^{r}}
5155:
5068:{\displaystyle {\widetilde {X}}}
4770:{\displaystyle {\widetilde {X}}}
4033:{\displaystyle \mathbb {P} ^{n}}
3426:{\displaystyle |D|\cdot C\geq 0}
2607:{\displaystyle \mathbb {P} ^{N}}
2499:{\displaystyle \mathbb {P} ^{n}}
1322:{\displaystyle \left(t/s\right)}
1112:{\displaystyle \mathbb {P} ^{3}}
5240:{\displaystyle {\mathfrak {d}}}
4481:{\displaystyle V_{X}=V\times X}
3769:{\displaystyle {\mathfrak {X}}}
2578:{\displaystyle \mathbb {P} (V)}
2127:{\displaystyle {\mathfrak {d}}}
950:{\displaystyle {\mathfrak {d}}}
880:{\displaystyle {\mathfrak {d}}}
5620:
5588:
5542:
5536:
5463:
5457:
5349:
5345:
5339:
5317:
5307:
5207:
5121:
5108:
5100:
4992:
4979:
4971:
4932:{\displaystyle {\mathcal {I}}}
4867:
4864:
4827:
4797:
4794:
4690:
4677:
4669:
4600:
4587:
4576:
4558:
4547:
4523:
4515:
4449:is surjective. Hence, writing
4401:
4398:
4361:
4331:
4328:
4295:
4244:
4232:
4176:
4164:
4159:
4127:
4124:
4112:
4091:
4081:
3983:
3957:
3881:
3863:
3858:
3826:
3823:
3811:
3739:
3736:
3730:
3705:
3683:given by two generic sections
3658:
3601:
3595:
3568:
3564:
3556:
3552:
3506:
3500:
3452:
3444:
3407:
3399:
3355:
3328:
3320:
3218:
3210:
3171:considerations as to what the
3153:
3145:
3126:{\displaystyle D_{\text{eff}}}
3048:
3040:
3015:
3011:
3003:
2999:
2949:
2941:
2791:
2698:
2572:
2566:
2541:
2538:
2532:
2522:
2464:
2458:
2429:{\displaystyle d\geq (1/2)g+1}
2414:
2400:
2262:
2054:
2051:
2032:
2019:
1965:{\displaystyle h^{0}(K_{C})=2}
1953:
1940:
1806:
1695:
1692:
1673:
1660:
1648:
1640:
1278:
1275:
1269:
1244:
1215:
1209:
1169:
1166:
1160:
1135:
1077:
1071:
917:
901:
843:
835:
793:
785:
764:
758:
708:
693:
677:
671:
650:
642:
587:{\displaystyle {\mathcal {L}}}
556:
548:
527:
521:
396:
390:
370:
364:
269:
263:
224:
218:
129:
106:
1:
3943:{\displaystyle s_{0}f+t_{0}g}
3232:is such a class on a variety
2933:More precisely, suppose that
2506:. If we take global sections
1186:. For the associated divisor
664:is in natural bijection with
5773:Geometry of algebraic curves
5383:O(1) on a projective variety
5075:, there results in the map:
3364:{\displaystyle {\tilde {D}}}
2983:. Consider the intersection
1291:using the rational function
466:divisor (algebraic geometry)
5822:10.1007/978-1-4612-1700-8_5
5581:to its corresponding point
3486:is viewed as a line bundle
2355:. In fact, any curve has a
1728:{\displaystyle \omega _{C}}
739:by associating the element
5928:
5039:{\displaystyle X-B\simeq }
3623:
2874:wrote his celebrated book
2750:
2679:Then, using any embedding
1972:, hence there is a degree
1221:{\displaystyle D_{s}=Z(s)}
26:
5889:, Springer-Verlag, 2004.
5781:10.1007/978-1-4757-5323-3
5668:bundle of principal parts
5140:Finally, when a basis of
2444:Consider the line bundle
2380:{\displaystyle g_{1}^{d}}
2344:{\displaystyle g_{1}^{3}}
2312:{\displaystyle g_{2}^{1}}
2219:{\displaystyle g_{2}^{1}}
2103:{\displaystyle g_{d}^{r}}
1861:then gives the degree of
1057:Consider the line bundle
278:{\displaystyle E=D+(f)\ }
60:linear system of divisors
40:linear system of divisors
4943:and that gives rise to
4748:replaced by the blow-up
3272:an irreducible curve on
2859:coming from sections of
2821:Cayley–Bacharach theorem
2585:. This is isomorphic to
2226:which is induced by the
1565:Linear systems on curves
474:holomorphic line bundles
174:Given a general variety
5687:Grothendieck, Alexandre
5191:For a closed immersion
4709:When the base locus of
4635:{\displaystyle \simeq }
2753:Linear system of conics
2747:Linear system of conics
770:{\displaystyle E=D+(f)}
31:from cohomology theory.
5767:; Cornalba, Maurizio;
5642:
5575:
5574:{\displaystyle x\in X}
5549:
5430:
5401:
5373:
5281:
5261:
5241:
5217:
5131:
5069:
5040:
5008:
4939:is the ideal sheaf of
4933:
4909:
4890:
4771:
4738:
4700:
4636:
4613:
4482:
4443:
4424:
4305:
4251:
4197:
4190:
4060:
4034:
4005:
3944:
3902:
3895:
3770:
3746:
3677:
3608:
3575:
3533:
3513:
3480:
3460:
3427:
3385:
3365:
3336:
3306:
3286:
3266:
3246:
3226:
3189:
3161:
3127:
3100:
3077:
2977:
2957:
2923:
2809:Kodaira–Spencer theory
2801:
2737:
2717:
2670:
2608:
2579:
2548:
2500:
2471:
2430:
2381:
2345:
2313:
2281:
2246:
2220:
2188:
2168:
2148:
2128:
2104:
2061:
1986:
1966:
1917:
1916:{\displaystyle 2g-2=2}
1882:
1851:
1825:
1783:
1761:
1729:
1702:
1626:
1606:
1583:
1562:
1555:
1452:
1429:
1409:
1343:
1323:
1285:
1222:
1176:
1113:
1084:
1030:
999:
951:
927:
881:
851:
821:
801:
771:
733:
658:
628:
608:
588:
564:
534:
497:
481:complete linear system
446:
423:
403:
377:
345:
325:
305:
279:
231:
188:
136:
51:
5643:
5576:
5555:. This sends a point
5550:
5431:
5402:
5387:A projective variety
5374:
5282:
5262:
5242:
5218:
5132:
5070:
5041:
5009:
4934:
4910:
4870:
4772:
4739:
4701:
4637:
4614:
4483:
4444:
4404:
4311:is surjective (here,
4306:
4252:
4191:
4068:
4061:
4035:
4006:
3945:
3896:
3778:
3771:
3747:
3678:
3609:
3576:
3534:
3514:
3481:
3461:
3428:
3386:
3366:
3337:
3307:
3287:
3267:
3247:
3227:
3190:
3162:
3128:
3101:
3078:
2978:
2958:
2924:
2802:
2738:
2718:
2671:
2609:
2580:
2549:
2501:
2472:
2431:
2382:
2346:
2314:
2282:
2247:
2221:
2189:
2169:
2149:
2129:
2105:
2062:
1987:
1967:
1918:
1883:
1881:{\displaystyle K_{C}}
1852:
1826:
1784:
1762:
1730:
1703:
1627:
1607:
1584:
1556:
1460:
1453:
1430:
1410:
1344:
1324:
1286:
1223:
1177:
1114:
1085:
1031:
1000:
952:
928:
882:
852:
822:
802:
772:
734:
659:
629:
609:
589:
565:
535:
498:
447:
424:
404:
378:
346:
326:
306:
280:
232:
189:
137:
37:
5912:Geometry of divisors
5658:Brill–Noether theory
5585:
5559:
5440:
5411:
5391:
5291:
5271:
5251:
5227:
5195:
5082:
5050:
5021:
4950:
4919:
4785:
4752:
4717:
4657:
4626:
4503:
4453:
4319:
4265:
4220:
4073:
4044:
4015:
3954:
3908:
3783:
3756:
3687:
3642:
3585:
3543:
3523:
3490:
3470:
3440:
3395:
3375:
3346:
3316:
3296:
3276:
3256:
3236:
3206:
3179:
3141:
3110:
3090:
2990:
2967:
2937:
2907:
2835:as practised by the
2785:
2727:
2683:
2621:
2589:
2558:
2510:
2481:
2448:
2391:
2359:
2323:
2291:
2256:
2230:
2198:
2178:
2158:
2138:
2114:
2082:
1996:
1976:
1927:
1892:
1865:
1859:Riemann–Roch theorem
1835:
1794:
1773:
1751:
1739:Hyperelliptic curves
1712:
1636:
1616:
1596:
1573:
1465:
1439:
1419:
1353:
1333:
1295:
1232:
1190:
1123:
1094:
1061:
1020:
961:
937:
895:
867:
831:
811:
781:
743:
668:
638:
618:
598:
574:
544:
507:
487:
436:
413:
387:
376:{\displaystyle k(X)}
358:
335:
315:
295:
248:
198:
178:
103:
5814:Intersection Theory
5689:; Dieudonné, Jean.
4575:
4540:
4059:{\displaystyle f,g}
3776:given by the scheme
3459:{\displaystyle |D|}
3335:{\displaystyle |D|}
3225:{\displaystyle |D|}
3160:{\displaystyle |D|}
2956:{\displaystyle |D|}
2922:{\displaystyle x=a}
2841:homological algebra
2833:birational geometry
2376:
2340:
2308:
2245:{\displaystyle 2:1}
2215:
2154:which is of degree
2110:is a linear system
2099:
1850:{\displaystyle g=2}
1745:hyperelliptic curve
850:{\displaystyle |D|}
800:{\displaystyle |D|}
657:{\displaystyle |D|}
614:. In the case that
563:{\displaystyle |D|}
402:{\displaystyle (f)}
239:linearly equivalent
29:Kodaira–Spencer map
5865:Algebraic Geometry
5769:Griffiths, Phillip
5742:10.1007/BF02392112
5638:
5571:
5545:
5426:
5397:
5369:
5277:
5257:
5237:
5213:
5167:. You can help by
5127:
5065:
5046:an open subset of
5036:
5004:
4929:
4905:
4767:
4734:
4696:
4632:
4609:
4561:
4526:
4478:
4439:
4301:
4247:
4186:
4056:
4030:
4001:
3940:
3891:
3766:
3742:
3673:
3626:Theorem of Bertini
3616:globally generated
3604:
3571:
3529:
3509:
3476:
3456:
3423:
3381:
3361:
3332:
3302:
3282:
3262:
3242:
3222:
3185:
3157:
3123:
3096:
3073:
3053:
2973:
2953:
2919:
2876:Algebraic Surfaces
2861:invertible sheaves
2853:free abelian group
2797:
2733:
2713:
2666:
2604:
2575:
2544:
2496:
2467:
2426:
2377:
2362:
2341:
2326:
2309:
2294:
2277:
2242:
2216:
2201:
2184:
2164:
2144:
2124:
2100:
2085:
2057:
1982:
1962:
1913:
1878:
1847:
1821:
1779:
1757:
1725:
1698:
1622:
1602:
1579:
1551:
1451:{\displaystyle xy}
1448:
1425:
1405:
1339:
1319:
1281:
1218:
1172:
1109:
1080:
1053:Linear equivalence
1042:linear equivalence
1026:
995:
947:
923:
877:
847:
817:
797:
767:
729:
654:
624:
604:
584:
560:
530:
493:
470:invertible sheaves
442:
419:
399:
373:
341:
321:
301:
288:for some non-zero
275:
227:
184:
132:
83:linear equivalence
56:algebraic geometry
52:
48:Apollonian circles
5790:978-1-4419-2825-2
5765:Arbarello, Enrico
5400:{\displaystyle X}
5280:{\displaystyle Y}
5260:{\displaystyle X}
5185:
5184:
5062:
4968:
4764:
4644:projective bundle
4180:
4100:
4079:
3885:
3799:
3532:{\displaystyle X}
3479:{\displaystyle X}
3384:{\displaystyle C}
3358:
3305:{\displaystyle C}
3285:{\displaystyle X}
3265:{\displaystyle C}
3245:{\displaystyle X}
3120:
3072:
3067:
3033:
3021:
2976:{\displaystyle X}
2868:algebraic surface
2736:{\displaystyle k}
2652:
2351:which are called
2187:{\displaystyle r}
2167:{\displaystyle d}
2147:{\displaystyle C}
1985:{\displaystyle 2}
1782:{\displaystyle 2}
1760:{\displaystyle C}
1625:{\displaystyle K}
1605:{\displaystyle g}
1582:{\displaystyle C}
1545:
1428:{\displaystyle E}
1342:{\displaystyle D}
1029:{\displaystyle D}
820:{\displaystyle f}
627:{\displaystyle X}
607:{\displaystyle D}
519:
496:{\displaystyle X}
445:{\displaystyle X}
422:{\displaystyle f}
344:{\displaystyle f}
324:{\displaystyle X}
304:{\displaystyle f}
290:rational function
274:
216:
187:{\displaystyle X}
16:(Redirected from
5919:
5857:
5826:
5825:
5809:
5803:
5802:
5761:
5755:
5754:
5744:
5729:Acta Mathematica
5720:
5714:
5711:
5694:
5684:
5663:Lefschetz pencil
5647:
5645:
5644:
5639:
5637:
5636:
5631:
5619:
5618:
5600:
5599:
5580:
5578:
5577:
5572:
5554:
5552:
5551:
5546:
5535:
5534:
5533:
5532:
5527:
5520:
5519:
5512:
5511:
5510:
5509:
5508:
5507:
5502:
5495:
5494:
5482:
5481:
5476:
5475:
5456:
5455:
5450:
5449:
5435:
5433:
5432:
5427:
5425:
5424:
5419:
5406:
5404:
5403:
5398:
5378:
5376:
5375:
5370:
5365:
5364:
5352:
5338:
5337:
5316:
5315:
5306:
5305:
5286:
5284:
5283:
5278:
5266:
5264:
5263:
5258:
5246:
5244:
5243:
5238:
5236:
5235:
5222:
5220:
5219:
5214:
5180:
5177:
5159:
5152:
5136:
5134:
5133:
5128:
5120:
5119:
5107:
5074:
5072:
5071:
5066:
5064:
5063:
5055:
5045:
5043:
5042:
5037:
5013:
5011:
5010:
5005:
4991:
4990:
4978:
4970:
4969:
4961:
4938:
4936:
4935:
4930:
4928:
4927:
4914:
4912:
4911:
4906:
4904:
4903:
4898:
4897:
4889:
4884:
4863:
4862:
4850:
4849:
4848:
4847:
4842:
4841:
4826:
4825:
4820:
4819:
4812:
4811:
4776:
4774:
4773:
4768:
4766:
4765:
4757:
4743:
4741:
4740:
4735:
4733:
4732:
4727:
4726:
4705:
4703:
4702:
4697:
4689:
4688:
4676:
4641:
4639:
4638:
4633:
4618:
4616:
4615:
4610:
4599:
4598:
4586:
4574:
4569:
4557:
4539:
4534:
4522:
4494:closed immersion
4487:
4485:
4484:
4479:
4465:
4464:
4448:
4446:
4445:
4440:
4438:
4437:
4432:
4431:
4423:
4418:
4397:
4396:
4384:
4383:
4382:
4381:
4376:
4375:
4360:
4359:
4354:
4353:
4346:
4345:
4310:
4308:
4307:
4302:
4294:
4293:
4288:
4287:
4280:
4279:
4256:
4254:
4253:
4248:
4195:
4193:
4192:
4187:
4185:
4181:
4179:
4162:
4158:
4157:
4139:
4138:
4107:
4101:
4098:
4090:
4089:
4080:
4077:
4065:
4063:
4062:
4057:
4039:
4037:
4036:
4031:
4029:
4028:
4023:
4011:is a divisor in
4010:
4008:
4007:
4002:
4000:
3999:
3994:
3982:
3981:
3969:
3968:
3949:
3947:
3946:
3941:
3936:
3935:
3920:
3919:
3900:
3898:
3897:
3892:
3890:
3886:
3884:
3861:
3857:
3856:
3838:
3837:
3806:
3800:
3797:
3792:
3791:
3775:
3773:
3772:
3767:
3765:
3764:
3751:
3749:
3748:
3743:
3729:
3728:
3719:
3718:
3713:
3682:
3680:
3679:
3674:
3672:
3671:
3666:
3657:
3656:
3637:Lefschetz pencil
3613:
3611:
3610:
3605:
3594:
3593:
3580:
3578:
3577:
3572:
3567:
3559:
3538:
3536:
3535:
3530:
3518:
3516:
3515:
3510:
3499:
3498:
3485:
3483:
3482:
3477:
3465:
3463:
3462:
3457:
3455:
3447:
3432:
3430:
3429:
3424:
3410:
3402:
3390:
3388:
3387:
3382:
3370:
3368:
3367:
3362:
3360:
3359:
3351:
3341:
3339:
3338:
3333:
3331:
3323:
3311:
3309:
3308:
3303:
3291:
3289:
3288:
3283:
3271:
3269:
3268:
3263:
3251:
3249:
3248:
3243:
3231:
3229:
3228:
3223:
3221:
3213:
3194:
3192:
3191:
3186:
3169:scheme-theoretic
3166:
3164:
3163:
3158:
3156:
3148:
3132:
3130:
3129:
3124:
3122:
3121:
3118:
3105:
3103:
3102:
3097:
3082:
3080:
3079:
3074:
3070:
3069:
3068:
3065:
3052:
3051:
3043:
3035:
3034:
3031:
3014:
3006:
2982:
2980:
2979:
2974:
2962:
2960:
2959:
2954:
2952:
2944:
2928:
2926:
2925:
2920:
2857:Cartier divisors
2806:
2804:
2803:
2798:
2742:
2740:
2739:
2734:
2722:
2720:
2719:
2714:
2712:
2711:
2706:
2697:
2696:
2691:
2675:
2673:
2672:
2667:
2659:
2658:
2657:
2648:
2636:
2613:
2611:
2610:
2605:
2603:
2602:
2597:
2584:
2582:
2581:
2576:
2565:
2553:
2551:
2550:
2545:
2531:
2530:
2505:
2503:
2502:
2497:
2495:
2494:
2489:
2476:
2474:
2473:
2468:
2457:
2456:
2435:
2433:
2432:
2427:
2410:
2386:
2384:
2383:
2378:
2375:
2370:
2350:
2348:
2347:
2342:
2339:
2334:
2318:
2316:
2315:
2310:
2307:
2302:
2286:
2284:
2283:
2278:
2276:
2275:
2270:
2251:
2249:
2248:
2243:
2225:
2223:
2222:
2217:
2214:
2209:
2193:
2191:
2190:
2185:
2173:
2171:
2170:
2165:
2153:
2151:
2150:
2145:
2133:
2131:
2130:
2125:
2123:
2122:
2109:
2107:
2106:
2101:
2098:
2093:
2066:
2064:
2063:
2058:
2050:
2049:
2031:
2030:
2018:
2010:
2009:
2004:
1991:
1989:
1988:
1983:
1971:
1969:
1968:
1963:
1952:
1951:
1939:
1938:
1922:
1920:
1919:
1914:
1887:
1885:
1884:
1879:
1877:
1876:
1856:
1854:
1853:
1848:
1830:
1828:
1827:
1822:
1820:
1819:
1814:
1788:
1786:
1785:
1780:
1766:
1764:
1763:
1758:
1734:
1732:
1731:
1726:
1724:
1723:
1707:
1705:
1704:
1699:
1691:
1690:
1672:
1671:
1659:
1651:
1643:
1631:
1629:
1628:
1623:
1611:
1609:
1608:
1603:
1588:
1586:
1585:
1580:
1560:
1558:
1557:
1552:
1550:
1546:
1544:
1536:
1535:
1534:
1522:
1521:
1509:
1508:
1496:
1495:
1485:
1457:
1455:
1454:
1449:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1404:
1403:
1391:
1390:
1378:
1377:
1365:
1364:
1348:
1346:
1345:
1340:
1328:
1326:
1325:
1320:
1318:
1314:
1310:
1290:
1288:
1287:
1282:
1268:
1267:
1258:
1257:
1252:
1227:
1225:
1224:
1219:
1202:
1201:
1184:quadric surfaces
1181:
1179:
1178:
1173:
1159:
1158:
1149:
1148:
1143:
1118:
1116:
1115:
1110:
1108:
1107:
1102:
1089:
1087:
1086:
1081:
1070:
1069:
1038:Cartier divisors
1035:
1033:
1032:
1027:
1014:invertible sheaf
1004:
1002:
1001:
996:
976:
975:
956:
954:
953:
948:
946:
945:
932:
930:
929:
924:
916:
915:
886:
884:
883:
878:
876:
875:
856:
854:
853:
848:
846:
838:
826:
824:
823:
818:
806:
804:
803:
798:
796:
788:
776:
774:
773:
768:
738:
736:
735:
730:
725:
724:
715:
692:
691:
663:
661:
660:
655:
653:
645:
633:
631:
630:
625:
613:
611:
610:
605:
593:
591:
590:
585:
583:
582:
569:
567:
566:
561:
559:
551:
540:. It is denoted
539:
537:
536:
531:
520:
517:
502:
500:
499:
494:
458:Cartier divisors
451:
449:
448:
443:
428:
426:
425:
420:
408:
406:
405:
400:
382:
380:
379:
374:
350:
348:
347:
342:
330:
328:
327:
322:
310:
308:
307:
302:
284:
282:
281:
276:
272:
236:
234:
233:
228:
217:
214:
193:
191:
190:
185:
159:, respectively.
141:
139:
138:
133:
128:
127:
122:
121:
79:projective plane
75:algebraic curves
64:family of curves
44:family of curves
21:
5927:
5926:
5922:
5921:
5920:
5918:
5917:
5916:
5902:
5901:
5900:
5869:Springer-Verlag
5854:
5834:
5830:
5829:
5811:
5810:
5806:
5791:
5763:
5762:
5758:
5722:
5721:
5717:
5712:
5697:
5685:
5681:
5676:
5654:
5626:
5610:
5591:
5583:
5582:
5557:
5556:
5522:
5513:
5497:
5488:
5483:
5469:
5443:
5438:
5437:
5414:
5409:
5408:
5389:
5388:
5385:
5326:
5294:
5289:
5288:
5269:
5268:
5249:
5248:
5225:
5224:
5193:
5192:
5181:
5175:
5172:
5165:needs expansion
5150:
5111:
5080:
5079:
5048:
5047:
5019:
5018:
4982:
4948:
4947:
4917:
4916:
4891:
4851:
4835:
4830:
4813:
4803:
4783:
4782:
4750:
4749:
4720:
4715:
4714:
4680:
4655:
4654:
4624:
4623:
4590:
4501:
4500:
4456:
4451:
4450:
4425:
4385:
4369:
4364:
4347:
4337:
4317:
4316:
4281:
4271:
4263:
4262:
4218:
4217:
4202:
4163:
4149:
4130:
4108:
4102:
4071:
4070:
4042:
4041:
4018:
4013:
4012:
3989:
3973:
3960:
3952:
3951:
3927:
3911:
3906:
3905:
3862:
3848:
3829:
3807:
3801:
3781:
3780:
3754:
3753:
3708:
3685:
3684:
3661:
3640:
3639:
3633:
3628:
3583:
3582:
3541:
3540:
3521:
3520:
3488:
3487:
3468:
3467:
3438:
3437:
3393:
3392:
3373:
3372:
3344:
3343:
3314:
3313:
3294:
3293:
3274:
3273:
3254:
3253:
3234:
3233:
3204:
3203:
3177:
3176:
3173:structure sheaf
3139:
3138:
3113:
3108:
3107:
3088:
3087:
3060:
3026:
2988:
2987:
2965:
2964:
2935:
2934:
2905:
2904:
2892:
2845:singular points
2829:
2817:
2783:
2782:
2760:
2755:
2749:
2725:
2724:
2701:
2686:
2681:
2680:
2638:
2631:
2619:
2618:
2592:
2587:
2586:
2556:
2555:
2508:
2507:
2484:
2479:
2478:
2446:
2445:
2442:
2389:
2388:
2357:
2356:
2353:trigonal curves
2321:
2320:
2289:
2288:
2265:
2254:
2253:
2228:
2227:
2196:
2195:
2176:
2175:
2156:
2155:
2136:
2135:
2112:
2111:
2080:
2079:
2076:
2074:
2041:
2022:
1999:
1994:
1993:
1974:
1973:
1943:
1930:
1925:
1924:
1890:
1889:
1868:
1863:
1862:
1833:
1832:
1831:. For the case
1809:
1792:
1791:
1771:
1770:
1749:
1748:
1741:
1715:
1710:
1709:
1682:
1663:
1634:
1633:
1614:
1613:
1594:
1593:
1571:
1570:
1567:
1537:
1526:
1513:
1500:
1487:
1486:
1480:
1463:
1462:
1437:
1436:
1417:
1416:
1395:
1382:
1369:
1356:
1351:
1350:
1331:
1330:
1302:
1298:
1293:
1292:
1247:
1230:
1229:
1193:
1188:
1187:
1138:
1121:
1120:
1119:whose sections
1097:
1092:
1091:
1059:
1058:
1055:
1050:
1018:
1017:
959:
958:
935:
934:
893:
892:
865:
864:
829:
828:
809:
808:
779:
778:
741:
740:
716:
666:
665:
636:
635:
616:
615:
596:
595:
572:
571:
542:
541:
505:
504:
485:
484:
454:singular points
434:
433:
411:
410:
385:
384:
356:
355:
333:
332:
313:
312:
293:
292:
246:
245:
196:
195:
194:, two divisors
176:
175:
172:
115:
101:
100:
32:
23:
22:
15:
12:
11:
5:
5925:
5923:
5915:
5914:
5904:
5903:
5899:
5898:
5883:Lazarsfeld, R.
5880:
5861:Hartshorne, R.
5858:
5852:
5831:
5828:
5827:
5804:
5789:
5756:
5715:
5695:
5678:
5677:
5675:
5672:
5671:
5670:
5665:
5660:
5653:
5650:
5635:
5630:
5625:
5622:
5617:
5613:
5609:
5606:
5603:
5598:
5594:
5590:
5570:
5567:
5564:
5544:
5541:
5538:
5531:
5526:
5518:
5506:
5501:
5493:
5486:
5480:
5474:
5468:
5465:
5462:
5459:
5454:
5448:
5423:
5418:
5396:
5384:
5381:
5368:
5363:
5358:
5355:
5351:
5347:
5344:
5341:
5336:
5333:
5329:
5325:
5322:
5319:
5314:
5309:
5304:
5301:
5297:
5276:
5256:
5234:
5212:
5209:
5206:
5203:
5200:
5183:
5182:
5162:
5160:
5149:
5146:
5138:
5137:
5126:
5123:
5118:
5114:
5110:
5106:
5102:
5099:
5096:
5093:
5090:
5087:
5061:
5058:
5035:
5032:
5029:
5026:
5015:
5014:
5003:
5000:
4997:
4994:
4989:
4985:
4981:
4977:
4973:
4967:
4964:
4958:
4955:
4926:
4902:
4896:
4888:
4883:
4880:
4877:
4873:
4869:
4866:
4861:
4858:
4854:
4846:
4840:
4833:
4829:
4824:
4818:
4810:
4806:
4802:
4799:
4796:
4793:
4790:
4763:
4760:
4731:
4725:
4707:
4706:
4695:
4692:
4687:
4683:
4679:
4675:
4671:
4668:
4665:
4662:
4631:
4620:
4619:
4608:
4605:
4602:
4597:
4593:
4589:
4585:
4581:
4578:
4573:
4568:
4564:
4560:
4556:
4552:
4549:
4546:
4543:
4538:
4533:
4529:
4525:
4521:
4517:
4514:
4511:
4508:
4477:
4474:
4471:
4468:
4463:
4459:
4436:
4430:
4422:
4417:
4414:
4411:
4407:
4403:
4400:
4395:
4392:
4388:
4380:
4374:
4367:
4363:
4358:
4352:
4344:
4340:
4336:
4333:
4330:
4327:
4324:
4300:
4297:
4292:
4286:
4278:
4274:
4270:
4246:
4243:
4240:
4237:
4234:
4231:
4228:
4225:
4201:
4198:
4184:
4178:
4175:
4172:
4169:
4166:
4161:
4156:
4152:
4148:
4145:
4142:
4137:
4133:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4105:
4096:
4093:
4088:
4083:
4055:
4052:
4049:
4027:
4022:
3998:
3993:
3988:
3985:
3980:
3976:
3972:
3967:
3963:
3959:
3939:
3934:
3930:
3926:
3923:
3918:
3914:
3889:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3860:
3855:
3851:
3847:
3844:
3841:
3836:
3832:
3828:
3825:
3822:
3819:
3816:
3813:
3810:
3804:
3795:
3790:
3763:
3741:
3738:
3735:
3732:
3727:
3722:
3717:
3712:
3707:
3704:
3701:
3698:
3695:
3692:
3670:
3665:
3660:
3655:
3650:
3647:
3632:
3629:
3603:
3600:
3597:
3592:
3570:
3566:
3562:
3558:
3554:
3551:
3548:
3528:
3508:
3505:
3502:
3497:
3475:
3454:
3450:
3446:
3422:
3419:
3416:
3413:
3409:
3405:
3401:
3380:
3357:
3354:
3330:
3326:
3322:
3301:
3281:
3261:
3241:
3220:
3216:
3212:
3184:
3155:
3151:
3147:
3116:
3095:
3084:
3083:
3063:
3059:
3056:
3050:
3046:
3042:
3038:
3029:
3024:
3020:
3017:
3013:
3009:
3005:
3001:
2998:
2995:
2972:
2951:
2947:
2943:
2918:
2915:
2912:
2891:
2888:
2884:Henri Poincaré
2828:
2825:
2816:
2815:Other examples
2813:
2796:
2793:
2790:
2759:
2756:
2751:Main article:
2748:
2745:
2732:
2710:
2705:
2700:
2695:
2690:
2677:
2676:
2665:
2662:
2656:
2651:
2647:
2644:
2641:
2635:
2629:
2626:
2601:
2596:
2574:
2571:
2568:
2564:
2543:
2540:
2537:
2534:
2529:
2524:
2521:
2518:
2515:
2493:
2488:
2466:
2463:
2460:
2455:
2441:
2438:
2425:
2422:
2419:
2416:
2413:
2409:
2405:
2402:
2399:
2396:
2374:
2369:
2365:
2338:
2333:
2329:
2306:
2301:
2297:
2274:
2269:
2264:
2261:
2241:
2238:
2235:
2213:
2208:
2204:
2183:
2174:and dimension
2163:
2143:
2121:
2097:
2092:
2088:
2075:
2072:
2069:
2056:
2053:
2048:
2044:
2040:
2037:
2034:
2029:
2025:
2021:
2017:
2013:
2008:
2003:
1981:
1961:
1958:
1955:
1950:
1946:
1942:
1937:
1933:
1912:
1909:
1906:
1903:
1900:
1897:
1875:
1871:
1846:
1843:
1840:
1818:
1813:
1808:
1805:
1802:
1799:
1778:
1756:
1740:
1737:
1722:
1718:
1697:
1694:
1689:
1685:
1681:
1678:
1675:
1670:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1621:
1601:
1578:
1566:
1563:
1549:
1543:
1540:
1533:
1529:
1525:
1520:
1516:
1512:
1507:
1503:
1499:
1494:
1490:
1483:
1479:
1476:
1473:
1470:
1447:
1444:
1424:
1402:
1398:
1394:
1389:
1385:
1381:
1376:
1372:
1368:
1363:
1359:
1338:
1317:
1313:
1309:
1305:
1301:
1280:
1277:
1274:
1271:
1266:
1261:
1256:
1251:
1246:
1243:
1240:
1237:
1217:
1214:
1211:
1208:
1205:
1200:
1196:
1171:
1168:
1165:
1162:
1157:
1152:
1147:
1142:
1137:
1134:
1131:
1128:
1106:
1101:
1079:
1076:
1073:
1068:
1054:
1051:
1049:
1046:
1025:
994:
991:
988:
985:
982:
979:
974:
969:
966:
944:
922:
919:
914:
909:
906:
903:
900:
874:
845:
841:
837:
816:
795:
791:
787:
766:
763:
760:
757:
754:
751:
748:
728:
723:
719:
714:
710:
707:
704:
701:
698:
695:
690:
685:
682:
679:
676:
673:
652:
648:
644:
623:
603:
581:
558:
554:
550:
529:
526:
523:
515:
512:
492:
476:); see below.
441:
418:
398:
395:
392:
372:
369:
366:
363:
353:function field
340:
320:
300:
286:
285:
271:
268:
265:
262:
259:
256:
253:
226:
223:
220:
212:
209:
206:
203:
183:
171:
168:
131:
126:
120:
114:
111:
108:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5924:
5913:
5910:
5909:
5907:
5896:
5895:3-540-22533-1
5892:
5888:
5884:
5881:
5878:
5877:0-387-90244-9
5874:
5870:
5866:
5862:
5859:
5855:
5853:0-471-05059-8
5849:
5845:
5841:
5837:
5833:
5832:
5823:
5819:
5815:
5808:
5805:
5800:
5796:
5792:
5786:
5782:
5778:
5774:
5770:
5766:
5760:
5757:
5752:
5748:
5743:
5738:
5734:
5730:
5726:
5719:
5716:
5710:
5708:
5706:
5704:
5702:
5700:
5696:
5692:
5688:
5683:
5680:
5673:
5669:
5666:
5664:
5661:
5659:
5656:
5655:
5651:
5649:
5633:
5623:
5615:
5611:
5607:
5604:
5601:
5596:
5592:
5568:
5565:
5562:
5539:
5529:
5504:
5484:
5478:
5466:
5460:
5452:
5421:
5394:
5382:
5380:
5356:
5353:
5342:
5334:
5331:
5327:
5320:
5302:
5299:
5295:
5287:, defined as
5274:
5254:
5210:
5204:
5201:
5198:
5189:
5179:
5170:
5166:
5163:This section
5161:
5158:
5154:
5153:
5147:
5145:
5143:
5124:
5116:
5112:
5097:
5094:
5091:
5088:
5085:
5078:
5077:
5076:
5059:
5056:
5033:
5030:
5027:
5024:
5001:
4998:
4995:
4987:
4983:
4965:
4962:
4956:
4953:
4946:
4945:
4944:
4942:
4900:
4881:
4878:
4875:
4871:
4859:
4856:
4852:
4844:
4831:
4822:
4808:
4804:
4800:
4791:
4788:
4780:
4761:
4758:
4747:
4729:
4712:
4693:
4685:
4681:
4666:
4663:
4660:
4653:
4652:
4651:
4649:
4645:
4629:
4606:
4603:
4595:
4591:
4579:
4571:
4566:
4562:
4550:
4544:
4541:
4536:
4531:
4527:
4512:
4509:
4506:
4499:
4498:
4497:
4495:
4492:, there is a
4491:
4490:relative Proj
4475:
4472:
4469:
4466:
4461:
4457:
4434:
4415:
4412:
4409:
4405:
4393:
4390:
4386:
4378:
4365:
4356:
4342:
4338:
4334:
4325:
4322:
4314:
4298:
4290:
4276:
4272:
4268:
4260:
4241:
4238:
4235:
4226:
4223:
4215:
4211:
4206:
4199:
4196:
4182:
4173:
4170:
4167:
4154:
4150:
4146:
4143:
4140:
4135:
4131:
4121:
4118:
4115:
4109:
4103:
4094:
4067:
4053:
4050:
4047:
4025:
3996:
3986:
3978:
3974:
3970:
3965:
3961:
3937:
3932:
3928:
3924:
3921:
3916:
3912:
3901:
3887:
3878:
3875:
3872:
3869:
3866:
3853:
3849:
3845:
3842:
3839:
3834:
3830:
3820:
3817:
3814:
3808:
3802:
3793:
3777:
3733:
3720:
3715:
3699:
3696:
3693:
3690:
3668:
3648:
3645:
3638:
3635:Consider the
3630:
3627:
3622:
3619:
3617:
3598:
3560:
3549:
3546:
3526:
3503:
3473:
3448:
3434:
3420:
3417:
3414:
3411:
3403:
3378:
3352:
3324:
3299:
3279:
3259:
3239:
3214:
3201:
3196:
3182:
3174:
3170:
3149:
3136:
3114:
3093:
3061:
3057:
3054:
3044:
3036:
3027:
3022:
3018:
3007:
2996:
2993:
2986:
2985:
2984:
2970:
2945:
2931:
2916:
2913:
2910:
2901:
2897:
2889:
2887:
2885:
2881:
2877:
2873:
2869:
2864:
2862:
2858:
2854:
2850:
2849:Weil divisors
2846:
2842:
2838:
2834:
2826:
2824:
2822:
2814:
2812:
2810:
2794:
2788:
2780:
2779:normal bundle
2775:
2773:
2769:
2765:
2757:
2754:
2746:
2744:
2730:
2708:
2693:
2663:
2660:
2649:
2645:
2642:
2639:
2627:
2624:
2617:
2616:
2615:
2599:
2569:
2535:
2516:
2513:
2491:
2461:
2439:
2437:
2423:
2420:
2417:
2411:
2407:
2403:
2397:
2394:
2372:
2367:
2363:
2354:
2336:
2331:
2327:
2304:
2299:
2295:
2272:
2259:
2239:
2236:
2233:
2211:
2206:
2202:
2181:
2161:
2141:
2095:
2090:
2086:
2070:
2068:
2046:
2042:
2038:
2035:
2027:
2023:
2011:
2006:
1979:
1959:
1956:
1948:
1944:
1935:
1931:
1910:
1907:
1904:
1901:
1898:
1895:
1873:
1869:
1860:
1844:
1841:
1838:
1816:
1803:
1800:
1797:
1790:
1776:
1754:
1746:
1738:
1736:
1720:
1716:
1687:
1683:
1679:
1676:
1668:
1664:
1652:
1644:
1619:
1599:
1592:
1576:
1564:
1561:
1547:
1541:
1538:
1531:
1527:
1523:
1518:
1514:
1510:
1505:
1501:
1497:
1492:
1488:
1481:
1477:
1474:
1471:
1468:
1459:
1445:
1442:
1422:
1400:
1396:
1392:
1387:
1383:
1379:
1374:
1370:
1366:
1361:
1357:
1336:
1315:
1311:
1307:
1303:
1299:
1272:
1259:
1254:
1238:
1235:
1212:
1206:
1203:
1198:
1194:
1185:
1163:
1150:
1145:
1129:
1126:
1104:
1074:
1052:
1047:
1045:
1043:
1039:
1023:
1015:
1011:
1006:
992:
989:
986:
983:
980:
977:
967:
964:
920:
907:
904:
890:
863:
862:linear system
858:
839:
814:
789:
761:
755:
752:
749:
746:
726:
721:
717:
712:
702:
696:
683:
680:
646:
621:
601:
552:
524:
513:
510:
490:
482:
477:
475:
471:
467:
463:
462:Weil divisors
459:
455:
439:
432:Note that if
430:
416:
393:
367:
361:
354:
338:
318:
298:
291:
266:
260:
257:
254:
251:
244:
243:
242:
240:
221:
210:
207:
204:
201:
181:
169:
167:
165:
160:
158:
154:
150:
149:
143:
124:
112:
109:
99:
95:
92:on a general
91:
88:
84:
80:
76:
72:
71:linear system
67:
65:
61:
57:
49:
45:
41:
36:
30:
19:
5886:
5864:
5843:
5836:P. Griffiths
5816:. Springer.
5813:
5807:
5772:
5759:
5732:
5728:
5718:
5690:
5682:
5407:embedded in
5386:
5379:(page 158).
5190:
5186:
5173:
5169:adding to it
5164:
5141:
5139:
5016:
4940:
4778:
4745:
4710:
4708:
4647:
4621:
4312:
4258:
4213:
4209:
4207:
4203:
4069:
3950:for a fixed
3903:
3779:
3634:
3620:
3435:
3197:
3195:should be).
3134:
3085:
2932:
2895:
2893:
2879:
2875:
2865:
2830:
2818:
2776:
2771:
2767:
2766:for a curve
2763:
2761:
2678:
2443:
2077:
1742:
1568:
1461:
1056:
1041:
1007:
888:
861:
859:
480:
478:
431:
287:
238:
173:
163:
161:
156:
152:
146:
144:
98:ringed space
89:
82:
70:
68:
59:
53:
46:, as in the
39:
5735:: 163–176.
5176:August 2019
2134:on a curve
1747:is a curve
1010:line bundle
170:Definitions
164:Kodaira map
5674:References
3624:See also:
3135:base locus
2896:base locus
2890:Base locus
1632:, denoted
96:or even a
18:Base locus
5840:J. Harris
5751:0001-5962
5624:∈
5605:⋯
5566:∈
5485:⊗
5357:∈
5332:−
5300:−
5208:↪
5117:∗
5101:→
5095:−
5060:~
5034:≃
5028:−
4996:×
4988:∗
4972:↪
4966:~
4887:∞
4872:⨁
4868:→
4857:−
4832:⊗
4805:⊗
4792:
4762:~
4686:∗
4670:→
4630:≃
4604:×
4596:∗
4572:∗
4551:≃
4542:⊗
4537:∗
4516:↪
4473:×
4421:∞
4406:⨁
4402:→
4391:−
4366:⊗
4339:⊗
4326:
4296:→
4273:⊗
4230:Γ
4227:⊂
4144:…
3987:∈
3843:…
3703:Γ
3700:∈
3659:→
3550:
3418:≥
3412:⋅
3356:~
3058:
3037:∈
3023:⋂
2997:
2792:↪
2699:→
2661:−
2520:Γ
2398:≥
2263:→
2043:ω
1902:−
1807:→
1717:ω
1684:ω
1242:Γ
1239:∈
1133:Γ
1130:∈
990:−
984:
968:
899:Γ
722:∗
697:∖
675:Γ
514:∈
211:∈
5906:Category
5842:(1994).
5771:(2011).
5652:See also
2851:(in the
1789:morphism
1048:Examples
87:divisors
5799:2807457
5693:, 21.3.
3631:Example
3200:nefness
2900:variety
2872:Zariski
1992:map to
1767:with a
1182:define
383:. Here
351:of the
155:, or a
77:in the
5893:
5875:
5850:
5797:
5787:
5749:
5691:EGA IV
5017:Since
4915:where
4622:where
3252:, and
3086:where
3071:
2614:where
1769:degre
570:. Let
464:: see
273:
148:pencil
94:scheme
3752:, so
3292:. If
2477:over
2252:-map
1591:genus
5891:ISBN
5873:ISBN
5848:ISBN
5785:ISBN
5747:ISSN
4216:and
4208:Let
4099:Proj
4066:, so
3798:Proj
3094:Supp
3055:Supp
2894:The
2819:The
2387:for
1923:and
452:has
237:are
151:, a
58:, a
5818:doi
5777:doi
5737:doi
5733:132
5267:to
5247:on
5171:.
4789:Sym
4323:Sym
3519:on
3175:of
3137:of
3119:eff
3066:eff
3032:eff
2781:to
1888:is
1589:of
1090:on
1012:or
981:dim
965:dim
891:of
777:of
518:Div
483:on
472:or
311:on
241:if
215:Div
157:web
153:net
85:of
73:of
54:In
5908::
5885:,
5867:,
5863:,
5838:;
5795:MR
5793:.
5783:.
5745:.
5731:.
5727:.
5698:^
5648:.
4496::
4078:Bl
3547:Bl
3183:Bl
3019::=
2994:Bl
2863:.
2774:.
2743:.
2436:.
2078:A
2067:.
1735:.
1005:.
860:A
479:A
460:,
429:.
166:.
142:.
38:A
5897:.
5879:.
5856:.
5824:.
5820::
5801:.
5779::
5753:.
5739::
5634:r
5629:P
5621:]
5616:r
5612:x
5608::
5602::
5597:0
5593:x
5589:[
5569:X
5563:x
5543:)
5540:1
5537:(
5530:r
5525:P
5517:O
5505:r
5500:P
5492:O
5479:X
5473:O
5467:=
5464:)
5461:1
5458:(
5453:X
5447:O
5422:r
5417:P
5395:X
5367:}
5362:d
5354:D
5350:|
5346:)
5343:D
5340:(
5335:1
5328:f
5324:{
5321:=
5318:)
5313:d
5308:(
5303:1
5296:f
5275:Y
5255:X
5233:d
5211:X
5205:Y
5202::
5199:f
5178:)
5174:(
5142:V
5125:.
5122:)
5113:V
5109:(
5105:P
5098:B
5092:X
5089::
5086:f
5057:X
5031:B
5025:X
5002:.
4999:X
4993:)
4984:V
4980:(
4976:P
4963:X
4957::
4954:i
4941:B
4925:I
4901:n
4895:I
4882:0
4879:=
4876:n
4865:)
4860:1
4853:L
4845:X
4839:O
4828:)
4823:X
4817:O
4809:k
4801:V
4798:(
4795:(
4779:B
4759:X
4746:X
4730:X
4724:O
4711:V
4694:.
4691:)
4682:V
4678:(
4674:P
4667:X
4664::
4661:f
4648:i
4607:X
4601:)
4592:V
4588:(
4584:P
4580:=
4577:)
4567:X
4563:V
4559:(
4555:P
4548:)
4545:L
4532:X
4528:V
4524:(
4520:P
4513:X
4510::
4507:i
4476:X
4470:V
4467:=
4462:X
4458:V
4435:X
4429:O
4416:0
4413:=
4410:n
4399:)
4394:1
4387:L
4379:X
4373:O
4362:)
4357:X
4351:O
4343:k
4335:V
4332:(
4329:(
4313:k
4299:L
4291:X
4285:O
4277:k
4269:V
4259:V
4245:)
4242:L
4239:,
4236:X
4233:(
4224:V
4214:X
4210:L
4183:)
4177:)
4174:g
4171:,
4168:f
4165:(
4160:]
4155:n
4151:x
4147:,
4141:,
4136:0
4132:x
4128:[
4125:]
4122:t
4119:,
4116:s
4113:[
4110:k
4104:(
4095:=
4092:)
4087:X
4082:(
4054:g
4051:,
4048:f
4026:n
4021:P
3997:1
3992:P
3984:]
3979:0
3975:t
3971::
3966:0
3962:s
3958:[
3938:g
3933:0
3929:t
3925:+
3922:f
3917:0
3913:s
3888:)
3882:)
3879:g
3876:t
3873:+
3870:f
3867:s
3864:(
3859:]
3854:n
3850:x
3846:,
3840:,
3835:0
3831:x
3827:[
3824:]
3821:t
3818:,
3815:s
3812:[
3809:k
3803:(
3794:=
3789:X
3762:X
3740:)
3737:)
3734:d
3731:(
3726:O
3721:,
3716:n
3711:P
3706:(
3697:g
3694:,
3691:f
3669:1
3664:P
3654:X
3649::
3646:p
3602:)
3599:D
3596:(
3591:O
3569:)
3565:|
3561:D
3557:|
3553:(
3527:X
3507:)
3504:D
3501:(
3496:O
3474:X
3453:|
3449:D
3445:|
3421:0
3415:C
3408:|
3404:D
3400:|
3379:C
3353:D
3329:|
3325:D
3321:|
3300:C
3280:X
3260:C
3240:X
3219:|
3215:D
3211:|
3154:|
3150:D
3146:|
3115:D
3062:D
3049:|
3045:D
3041:|
3028:D
3016:)
3012:|
3008:D
3004:|
3000:(
2971:X
2950:|
2946:D
2942:|
2917:a
2914:=
2911:x
2795:Y
2789:C
2772:C
2768:C
2764:Y
2731:k
2709:N
2704:P
2694:k
2689:P
2664:1
2655:)
2650:n
2646:d
2643:+
2640:n
2634:(
2628:=
2625:N
2600:N
2595:P
2573:)
2570:V
2567:(
2563:P
2542:)
2539:)
2536:d
2533:(
2528:O
2523:(
2517:=
2514:V
2492:n
2487:P
2465:)
2462:d
2459:(
2454:O
2424:1
2421:+
2418:g
2415:)
2412:2
2408:/
2404:1
2401:(
2395:d
2373:d
2368:1
2364:g
2337:3
2332:1
2328:g
2305:1
2300:2
2296:g
2273:1
2268:P
2260:C
2240:1
2237::
2234:2
2212:1
2207:2
2203:g
2182:r
2162:d
2142:C
2120:d
2096:r
2091:d
2087:g
2073:d
2071:g
2055:)
2052:)
2047:C
2039:,
2036:C
2033:(
2028:0
2024:H
2020:(
2016:P
2012:=
2007:1
2002:P
1980:2
1960:2
1957:=
1954:)
1949:C
1945:K
1941:(
1936:0
1932:h
1911:2
1908:=
1905:2
1899:g
1896:2
1874:C
1870:K
1845:2
1842:=
1839:g
1817:1
1812:P
1804:C
1801::
1798:f
1777:2
1755:C
1721:C
1696:)
1693:)
1688:C
1680:,
1677:C
1674:(
1669:0
1665:H
1661:(
1657:P
1653:=
1649:|
1645:K
1641:|
1620:K
1600:g
1577:C
1548:)
1542:y
1539:x
1532:2
1528:w
1524:+
1519:2
1515:z
1511:+
1506:2
1502:y
1498:+
1493:2
1489:x
1482:(
1478:+
1475:E
1472:=
1469:D
1446:y
1443:x
1423:E
1401:2
1397:w
1393:+
1388:2
1384:z
1380:+
1375:2
1371:y
1367:+
1362:2
1358:x
1337:D
1316:)
1312:s
1308:/
1304:t
1300:(
1279:)
1276:)
1273:2
1270:(
1265:O
1260:,
1255:3
1250:P
1245:(
1236:t
1216:)
1213:s
1210:(
1207:Z
1204:=
1199:s
1195:D
1170:)
1167:)
1164:2
1161:(
1156:O
1151:,
1146:3
1141:P
1136:(
1127:s
1105:3
1100:P
1078:)
1075:2
1072:(
1067:O
1036:(
1024:D
993:1
987:W
978:=
973:d
943:d
921:.
918:)
913:L
908:,
905:X
902:(
889:W
873:d
844:|
840:D
836:|
815:f
794:|
790:D
786:|
765:)
762:f
759:(
756:+
753:D
750:=
747:E
727:,
718:k
713:/
709:)
706:}
703:0
700:{
694:)
689:L
684:,
681:X
678:(
672:(
651:|
647:D
643:|
622:X
602:D
580:L
557:|
553:D
549:|
528:)
525:X
522:(
511:D
491:X
440:X
417:f
397:)
394:f
391:(
371:)
368:X
365:(
362:k
339:f
319:X
299:f
270:)
267:f
264:(
261:+
258:D
255:=
252:E
225:)
222:X
219:(
208:E
205:,
202:D
182:X
130:)
125:X
119:O
113:,
110:X
107:(
90:D
50:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
