5146:
24:
4902:
4436:
5542:
4606:
3422:. 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.
3070:
2891:
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
2828:. 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
5176:
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.
4183:
4193:
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)
4773:
4307:
5001:
3888:
2918:
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.
5366:
726:
1548:
2054:
3739:
1278:
1169:
5428:
2796:. 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
3610:
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.
1695:
3670:
4491:
4298:
2710:
5124:
992:
2541:
4693:
920:
3568:
1818:
1402:
2978:
224:
129:
4244:
3093:
2663:
2274:
5635:
5210:
3998:
527:
3182:
4731:
2794:
3601:
3506:
2464:
1077:
5423:
5062:
4764:
4027:
3420:
2601:
2493:
1316:
1106:
5234:
4475:
3763:
2812:
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.
2572:
2121:
944:
874:
4926:
3120:
2423:
1959:
581:
3937:
3358:
1722:
5033:
1215:
2374:
2338:
2306:
2213:
2097:
272:
816:(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
4629:
764:
5568:
1910:
1875:
370:
5702:
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
4053:
3453:
3329:
3219:
3154:
2950:
2916:
2239:
1844:
844:
794:
651:
557:
396:
1445:
5394:
5274:
5254:
3526:
3473:
3378:
3299:
3279:
3259:
3239:
2970:
2730:
2181:
2161:
2141:
1979:
1776:
1754:
1619:
1599:
1576:
1422:
1336:
1023:
814:
621:
601:
490:
439:
416:
338:
318:
298:
181:
4061:
1697:. 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
4897:{\displaystyle \operatorname {Sym} ((V\otimes _{k}{\mathcal {O}}_{X})\otimes _{{\mathcal {O}}_{X}}L^{-1})\to \bigoplus _{n=0}^{\infty }{\mathcal {I}}^{n}}
4431:{\displaystyle \operatorname {Sym} ((V\otimes _{k}{\mathcal {O}}_{X})\otimes _{{\mathcal {O}}_{X}}L^{-1})\to \bigoplus _{n=0}^{\infty }{\mathcal {O}}_{X}}
4938:
3771:
5279:
656:
341:
5777:
5764:. Grundlehren der Mathematischen Wissenschaften. Vol. II, with a contribution by Joseph Daniel Harris. Heidelberg: Springer. p. 3.
1453:
2825:
1984:
5537:{\displaystyle {\mathcal {O}}_{X}(1)={\mathcal {O}}_{X}\otimes _{{\mathcal {O}}_{\mathbb {P} ^{r}}}{\mathcal {O}}_{\mathbb {P} ^{r}}(1)}
3675:
2871:. There was a controversy, one of the final issues in the conflict between 'old' and 'new' points of view in algebraic geometry, over
1220:
1111:
5883:
5865:
5840:
4601:{\displaystyle i:X\hookrightarrow \mathbb {P} (V_{X}^{*}\otimes L)\simeq \mathbb {P} (V_{X}^{*})=\mathbb {P} (V^{*})\times X}
1624:
5133:
is chosen, the above discussion becomes more down-to-earth (and that is the style used in
Hartshorne, Algebraic Geometry).
3630:
5900:
4253:
2671:
2809:
454:
75:
1757:
2797:
5828:
5070:
949:
2498:
5656:
4645:
5646:
1847:
883:
3065:{\displaystyle \operatorname {Bl} (|D|):=\bigcap _{D_{\text{eff}}\in |D|}\operatorname {Supp} D_{\text{eff}}\ }
2833:
442:
3531:
17:
1782:
1341:
2741:
462:
186:
91:
4208:
3078:
2609:
2244:
5675:
5573:
5183:
3942:
495:
3167:
4705:
3188:
2773:
3573:
3478:
2436:
1049:
5399:
5038:
4740:
4003:
3383:
2577:
2469:
1283:
1082:
5215:
4441:
3744:
3380:, and so intersects it properly. Basic facts from intersection theory then tell us that we must have
3157:
2546:
2102:
925:
855:
136:
82:
4907:
3098:
2379:
1915:
562:
3896:
2829:
2821:
1733:
1579:
3334:
876:
is then a projective subspace of a complete linear system, so it corresponds to a vector subspace
3614:
2849:
2841:
1700:
458:
44:
36:
5009:
4029:. Then, the base locus of this system of divisors is the scheme given by the vanishing locus of
1178:
4246:
a finite-dimensional vector subspace. For the sake of clarity, we first consider the case when
2347:
2311:
2279:
2186:
2070:
236:
5879:
5871:
5861:
5836:
5824:
5773:
5757:
5735:
4632:
3604:
2888:
2872:
2856:
2759:
in the family is a linear system formed by the curves in the family that are infinitely near
278:
4614:
731:
70:. It assumed a more general form, through gradual generalisation, so that one could speak of
5849:
5806:
5765:
5753:
5725:
5651:
5547:
4482:
3625:
3095:
denotes the support of a divisor, and the intersection is taken over all effective divisors
1880:
1002:
67:
55:; the dimension of the linear system corresponds to the number of parameters of the family.
52:
32:
5787:
1853:
5857:
5783:
2875:'s characteristic linear system of an algebraic family of curves on an algebraic surface.
2845:
1172:
1026:
492:
is defined as the set of all effective divisors linearly equivalent to some given divisor
446:
346:
63:
4178:{\displaystyle {\text{Bl}}({\mathfrak {X}})={\text{Proj}}\left({\frac {k}{(f,g)}}\right)}
4032:
3428:
3304:
3194:
3129:
2925:
2895:
2218:
1823:
819:
769:
626:
532:
375:
1427:
1217:, it is linearly equivalent to any other divisor defined by the vanishing locus of some
5379:
5259:
5239:
3511:
3458:
3363:
3284:
3264:
3244:
3224:
2955:
2715:
2341:
2166:
2146:
2126:
1964:
1761:
1739:
1732:
One application of linear systems is used in the classification of algebraic curves. A
1604:
1584:
1561:
1407:
1321:
1008:
799:
606:
586:
475:
424:
401:
323:
303:
283:
166:
5145:
5894:
4478:
2767:
5713:
3161:
2837:
450:
86:
23:
5810:
998:
2751:
The characteristic linear system of a family of curves on an algebraic surface
5769:
4996:{\displaystyle i:{\widetilde {X}}\hookrightarrow \mathbb {P} (V^{*})\times X.}
3883:{\displaystyle {\mathfrak {X}}={\text{Proj}}\left({\frac {k}{(sf+tg)}}\right)}
5739:
1601:
is given by the complete linear system associated with the canonical divisor
5361:{\displaystyle f^{-1}({\mathfrak {d}})=\{f^{-1}(D)|D\in {\mathfrak {d}}\}}
3425:
In the modern formulation of algebraic geometry, a complete linear system
2766:
In modern terms, it is a subsystem of the linear system associated to the
1033:
of two divisors means that the corresponding line bundles are isomorphic.
721:{\displaystyle (\Gamma (X,{\mathcal {L}})\smallsetminus \{0\})/k^{\ast },}
4733:
in the direct sum replaced by an ideal sheaf defining the base locus and
3893:
This has an associated linear system of divisors since each polynomial,
457:). The definition in that case is usually said with greater care (using
5730:
5425:
has a natural linear system determining a map to projective space from
2860:
2308:
from proposition 5.3. Another close set of examples are curves with a
1543:{\displaystyle D=E+\left({\frac {x^{2}+y^{2}+z^{2}+w^{2}}{xy}}\right)}
2049:{\displaystyle \mathbb {P} ^{1}=\mathbb {P} (H^{0}(C,\omega _{C}))}
1558:
One of the important complete linear systems on an algebraic curve
3734:{\displaystyle f,g\in \Gamma (\mathbb {P} ^{n},{\mathcal {O}}(d))}
3191:
of a
Cartier divisor class (i.e. complete linear system). Suppose
22:
4477:
for the trivial vector bundle and passing the surjection to the
1273:{\displaystyle t\in \Gamma (\mathbb {P} ^{3},{\mathcal {O}}(2))}
1164:{\displaystyle s\in \Gamma (\mathbb {P} ^{3},{\mathcal {O}}(2))}
5212:
of algebraic varieties there is a pullback of a linear system
5140:
2859:
to that of linear systems cut out by surfaces in three-space;
5505:
5480:
5461:
5435:
4913:
4883:
4827:
4805:
4712:
4417:
4361:
4339:
4273:
3714:
3579:
3484:
2516:
2442:
1253:
1144:
1055:
901:
677:
568:
151:
A map determined by a linear system is sometimes called the
107:
51:
is an algebraic generalization of the geometric notion of a
5835:. Wiley Classics Library. Wiley Interscience. p. 137.
4702:
is not empty, the above discussion still goes through with
2952:
is a complete linear system of divisors on some variety
2800:
can be used to answer the question of the completeness.
398:
denotes the divisor of zeroes and poles of the function
5157:
5137:
Linear system determined by a map to a projective space
2855:
The
Italian school liked to reduce the geometry on an
1690:{\displaystyle |K|=\mathbb {P} (H^{0}(C,\omega _{C}))}
997:
Linear systems can also be introduced by means of the
16:"Kodaira map" redirects here. Not to be confused with
5576:
5550:
5431:
5402:
5382:
5282:
5262:
5242:
5218:
5186:
5073:
5041:
5012:
4941:
4910:
4776:
4743:
4708:
4648:
4617:
4494:
4444:
4310:
4256:
4211:
4064:
4035:
4006:
3945:
3899:
3774:
3747:
3678:
3665:{\displaystyle p:{\mathfrak {X}}\to \mathbb {P} ^{1}}
3633:
3576:
3534:
3514:
3481:
3461:
3431:
3386:
3366:
3337:
3307:
3287:
3267:
3247:
3227:
3197:
3170:
3132:
3101:
3081:
2981:
2958:
2928:
2898:
2776:
2718:
2674:
2612:
2580:
2549:
2501:
2472:
2439:
2429:
Linear systems of hypersurfaces in a projective space
2382:
2350:
2314:
2282:
2247:
2221:
2189:
2169:
2149:
2129:
2105:
2073:
1987:
1967:
1918:
1883:
1856:
1826:
1785:
1764:
1742:
1703:
1627:
1607:
1587:
1564:
1456:
1430:
1410:
1344:
1324:
1286:
1223:
1181:
1114:
1085:
1052:
1011:
952:
928:
886:
858:
822:
802:
772:
734:
659:
629:
609:
589:
565:
535:
498:
478:
427:
404:
378:
349:
326:
306:
286:
239:
189:
169:
94:
5714:"Another proof of the existence of special divisors"
4293:{\displaystyle V\otimes _{k}{\mathcal {O}}_{X}\to L}
4250:
is base-point-free; in other words, the natural map
2705:{\displaystyle \mathbb {P} ^{k}\to \mathbb {P} ^{N}}
134:
Linear systems of dimension 1, 2, or 3 are called a
445:, the notion of 'divisor' is inherently ambiguous (
5629:
5562:
5536:
5417:
5388:
5360:
5268:
5248:
5228:
5204:
5118:
5056:
5027:
4995:
4920:
4896:
4758:
4725:
4687:
4623:
4600:
4469:
4430:
4292:
4238:
4177:
4047:
4021:
3992:
3931:
3882:
3757:
3733:
3664:
3595:
3562:
3520:
3500:
3467:
3447:
3414:
3372:
3352:
3323:
3293:
3273:
3253:
3233:
3213:
3187:One application of the notion of base locus is to
3176:
3148:
3114:
3087:
3064:
2964:
2944:
2910:
2788:
2747:Characteristic linear system of a family of curves
2724:
2704:
2657:
2595:
2566:
2535:
2487:
2458:
2417:
2368:
2332:
2300:
2268:
2233:
2207:
2175:
2155:
2135:
2115:
2091:
2048:
1973:
1953:
1904:
1869:
1838:
1812:
1770:
1748:
1716:
1689:
1613:
1593:
1570:
1542:
1439:
1416:
1396:
1330:
1310:
1272:
1209:
1163:
1100:
1071:
1017:
986:
938:
914:
868:
838:
808:
788:
758:
720:
645:
615:
595:
575:
551:
521:
484:
433:
410:
390:
364:
332:
312:
292:
266:
218:
175:
123:
5801:Fulton, William (1998). "§ 4.4. Linear Systems".
2820:In general linear systems became a basic tool of
2643:
2622:
1029:, to be precise) correspond to line bundles, and
2844:generated by codimension-one subvarieties), and
31:algebraicizes the classic geometric notion of a
3570:is the set of common zeroes of all sections of
2867:to try to pull together the methods, involving
4766:of it along the (scheme-theoretic) base locus
3156:(as a set, at least: there may be more subtle
2712:we can construct a linear system of dimension
2276:. In fact, hyperelliptic curves have a unique
946:is its dimension as a projective space. Hence
5119:{\displaystyle f:X-B\to \mathbb {P} (V^{*}).}
4770:. Precisely, as above, there is a surjection
3603:. A simple consequence is that the bundle is
987:{\displaystyle \dim {\mathfrak {d}}=\dim W-1}
623:is a nonsingular projective variety, the set
8:
5355:
5312:
2536:{\displaystyle V=\Gamma ({\mathcal {O}}(d))}
1447:. Then, there is the equivalence of divisors
1318:(Proposition 7.2). For example, the divisor
694:
688:
4688:{\displaystyle f:X\to \mathbb {P} (V^{*}).}
4639:by a projection, there results in the map:
2183:. For example, hyperelliptic curves have a
4635:under a twist by a line bundle. Following
2832:. The effect of working on varieties with
915:{\displaystyle \Gamma (X,{\mathcal {L}}).}
5729:
5621:
5617:
5616:
5603:
5584:
5575:
5549:
5517:
5513:
5512:
5510:
5504:
5503:
5492:
5488:
5487:
5485:
5479:
5478:
5476:
5466:
5460:
5459:
5440:
5434:
5433:
5430:
5409:
5405:
5404:
5401:
5381:
5349:
5348:
5337:
5319:
5300:
5299:
5287:
5281:
5261:
5241:
5220:
5219:
5217:
5185:
5104:
5093:
5092:
5072:
5043:
5042:
5040:
5011:
4975:
4964:
4963:
4949:
4948:
4940:
4912:
4911:
4909:
4888:
4882:
4881:
4874:
4863:
4844:
4832:
4826:
4825:
4823:
4810:
4804:
4803:
4796:
4775:
4745:
4744:
4742:
4717:
4711:
4710:
4707:
4673:
4662:
4661:
4647:
4616:
4583:
4572:
4571:
4559:
4554:
4543:
4542:
4524:
4519:
4508:
4507:
4493:
4449:
4443:
4422:
4416:
4415:
4408:
4397:
4378:
4366:
4360:
4359:
4357:
4344:
4338:
4337:
4330:
4309:
4278:
4272:
4271:
4264:
4255:
4210:
4201:be a line bundle on an algebraic variety
4142:
4123:
4095:
4086:
4074:
4073:
4065:
4063:
4034:
4013:
4009:
4008:
4005:
3984:
3980:
3979:
3966:
3953:
3944:
3920:
3904:
3898:
3841:
3822:
3794:
3785:
3776:
3775:
3773:
3749:
3748:
3746:
3713:
3712:
3703:
3699:
3698:
3677:
3656:
3652:
3651:
3641:
3640:
3632:
3578:
3577:
3575:
3552:
3544:
3533:
3513:
3483:
3482:
3480:
3460:
3440:
3432:
3430:
3395:
3387:
3385:
3365:
3339:
3338:
3336:
3316:
3308:
3306:
3286:
3266:
3246:
3226:
3206:
3198:
3196:
3169:
3141:
3133:
3131:
3106:
3100:
3080:
3053:
3036:
3028:
3019:
3014:
2999:
2991:
2980:
2957:
2937:
2929:
2927:
2897:
2775:
2717:
2696:
2692:
2691:
2681:
2677:
2676:
2673:
2642:
2621:
2619:
2611:
2587:
2583:
2582:
2579:
2551:
2550:
2548:
2515:
2514:
2500:
2479:
2475:
2474:
2471:
2441:
2440:
2438:
2395:
2381:
2360:
2355:
2349:
2324:
2319:
2313:
2292:
2287:
2281:
2260:
2256:
2255:
2246:
2220:
2199:
2194:
2188:
2168:
2148:
2128:
2107:
2106:
2104:
2083:
2078:
2072:
2034:
2015:
2004:
2003:
1994:
1990:
1989:
1986:
1966:
1936:
1923:
1917:
1882:
1861:
1855:
1825:
1804:
1800:
1799:
1784:
1763:
1741:
1708:
1702:
1675:
1656:
1645:
1644:
1636:
1628:
1626:
1606:
1586:
1563:
1519:
1506:
1493:
1480:
1473:
1455:
1429:
1409:
1388:
1375:
1362:
1349:
1343:
1323:
1295:
1285:
1252:
1251:
1242:
1238:
1237:
1222:
1186:
1180:
1143:
1142:
1133:
1129:
1128:
1113:
1092:
1088:
1087:
1084:
1054:
1053:
1051:
1010:
960:
959:
951:
930:
929:
927:
900:
899:
885:
860:
859:
857:
831:
823:
821:
801:
781:
773:
771:
733:
709:
700:
676:
675:
658:
638:
630:
628:
608:
588:
567:
566:
564:
544:
536:
534:
505:
497:
477:
426:
403:
377:
348:
325:
305:
285:
238:
202:
188:
168:
112:
106:
105:
93:
5712:Kleiman, Steven L.; Laksov, Dan (1974).
3607:if and only if the base locus is empty.
3563:{\displaystyle \operatorname {Bl} (|D|)}
2543:, then we can take its projectivization
5668:
1813:{\displaystyle f:C\to \mathbb {P} ^{1}}
1397:{\displaystyle x^{2}+y^{2}+z^{2}+w^{2}}
320:, or in other words a non-zero element
5860:, 1977; corrected 6th printing, 1993.
4631:on the right is the invariance of the
3528:. From this viewpoint, the base locus
3301:is not contained in the base locus of
1404:is linearly equivalent to the divisor
219:{\displaystyle D,E\in {\text{Div}}(X)}
124:{\displaystyle (X,{\mathcal {O}}_{X})}
4239:{\displaystyle V\subset \Gamma (X,L)}
3088:{\displaystyle \operatorname {Supp} }
2869:linear systems with fixed base points
2816:Linear systems in birational geometry
2658:{\displaystyle N={\binom {n+d}{n}}-1}
2269:{\displaystyle C\to \mathbb {P} ^{1}}
1424:associated to the vanishing locus of
1338:associated to the vanishing locus of
7:
5698:
5696:
5694:
5692:
5690:
5688:
5630:{\displaystyle \in \mathbb {P} ^{r}}
5205:{\displaystyle f:Y\hookrightarrow X}
4304:= the base field). Or equivalently,
3993:{\displaystyle \in \mathbb {P} ^{1}}
3360:in the class which does not contain
2887:of a linear system of divisors on a
2826:Italian school of algebraic geometry
796:to the set of non-zero multiples of
522:{\displaystyle D\in {\text{Div}}(X)}
5350:
5301:
5221:
4189:A map determined by a linear system
4075:
3777:
3750:
3642:
3455:of (Cartier) divisors on a variety
3177:{\displaystyle \operatorname {Bl} }
2836:is to show up a difference between
2108:
1005:language. In those terms, divisors
961:
931:
922:The dimension of the linear system
861:
58:These arose first in the form of a
5876:Positivity in Algebraic Geometry I
4875:
4726:{\displaystyle {\mathcal {O}}_{X}}
4409:
4218:
3691:
3122:in the linear system. This is the
2789:{\displaystyle C\hookrightarrow Y}
2626:
2508:
1846:all curves are hyperelliptic: the
1230:
1121:
887:
663:
14:
3596:{\displaystyle {\mathcal {O}}(D)}
3501:{\displaystyle {\mathcal {O}}(D)}
3331:, then there exists some divisor
2459:{\displaystyle {\mathcal {O}}(d)}
1072:{\displaystyle {\mathcal {O}}(2)}
846:is therefore a projective space.
583:be the line bundle associated to
5833:Principles of Algebraic Geometry
5418:{\displaystyle \mathbb {P} ^{r}}
5144:
5057:{\displaystyle {\widetilde {X}}}
4759:{\displaystyle {\widetilde {X}}}
4022:{\displaystyle \mathbb {P} ^{n}}
3415:{\displaystyle |D|\cdot C\geq 0}
2596:{\displaystyle \mathbb {P} ^{N}}
2488:{\displaystyle \mathbb {P} ^{n}}
1311:{\displaystyle \left(t/s\right)}
1101:{\displaystyle \mathbb {P} ^{3}}
5229:{\displaystyle {\mathfrak {d}}}
4470:{\displaystyle V_{X}=V\times X}
3758:{\displaystyle {\mathfrak {X}}}
2567:{\displaystyle \mathbb {P} (V)}
2116:{\displaystyle {\mathfrak {d}}}
939:{\displaystyle {\mathfrak {d}}}
869:{\displaystyle {\mathfrak {d}}}
5609:
5577:
5531:
5525:
5452:
5446:
5338:
5334:
5328:
5306:
5296:
5196:
5110:
5097:
5089:
4981:
4968:
4960:
4921:{\displaystyle {\mathcal {I}}}
4856:
4853:
4816:
4786:
4783:
4679:
4666:
4658:
4589:
4576:
4565:
4547:
4536:
4512:
4504:
4438:is surjective. Hence, writing
4390:
4387:
4350:
4320:
4317:
4284:
4233:
4221:
4165:
4153:
4148:
4116:
4113:
4101:
4080:
4070:
3972:
3946:
3870:
3852:
3847:
3815:
3812:
3800:
3728:
3725:
3719:
3694:
3672:given by two generic sections
3647:
3590:
3584:
3557:
3553:
3545:
3541:
3495:
3489:
3441:
3433:
3396:
3388:
3344:
3317:
3309:
3207:
3199:
3160:considerations as to what the
3142:
3134:
3115:{\displaystyle D_{\text{eff}}}
3037:
3029:
3004:
3000:
2992:
2988:
2938:
2930:
2780:
2687:
2561:
2555:
2530:
2527:
2521:
2511:
2453:
2447:
2418:{\displaystyle d\geq (1/2)g+1}
2403:
2389:
2251:
2043:
2040:
2021:
2008:
1954:{\displaystyle h^{0}(K_{C})=2}
1942:
1929:
1795:
1684:
1681:
1662:
1649:
1637:
1629:
1267:
1264:
1258:
1233:
1204:
1198:
1158:
1155:
1149:
1124:
1066:
1060:
906:
890:
832:
824:
782:
774:
753:
747:
697:
682:
666:
660:
639:
631:
576:{\displaystyle {\mathcal {L}}}
545:
537:
516:
510:
385:
379:
359:
353:
258:
252:
213:
207:
118:
95:
1:
3932:{\displaystyle s_{0}f+t_{0}g}
3221:is such a class on a variety
2922:More precisely, suppose that
2495:. If we take global sections
1175:. For the associated divisor
653:is in natural bijection with
5762:Geometry of algebraic curves
5372:O(1) on a projective variety
5064:, there results in the map:
3353:{\displaystyle {\tilde {D}}}
2972:. Consider the intersection
1280:using the rational function
455:divisor (algebraic geometry)
5811:10.1007/978-1-4612-1700-8_5
5570:to its corresponding point
3475:is viewed as a line bundle
2344:. In fact, any curve has a
1717:{\displaystyle \omega _{C}}
728:by associating the element
5917:
5028:{\displaystyle X-B\simeq }
3612:
2863:wrote his celebrated book
2739:
2668:Then, using any embedding
1961:, hence there is a degree
1210:{\displaystyle D_{s}=Z(s)}
15:
5878:, Springer-Verlag, 2004.
5770:10.1007/978-1-4757-5323-3
5657:bundle of principal parts
5129:Finally, when a basis of
2433:Consider the line bundle
2369:{\displaystyle g_{1}^{d}}
2333:{\displaystyle g_{1}^{3}}
2301:{\displaystyle g_{2}^{1}}
2208:{\displaystyle g_{2}^{1}}
2092:{\displaystyle g_{d}^{r}}
1850:then gives the degree of
1046:Consider the line bundle
267:{\displaystyle E=D+(f)\ }
49:linear system of divisors
29:linear system of divisors
4932:and that gives rise to
4737:replaced by the blow-up
3261:an irreducible curve on
2848:coming from sections of
2810:Cayley–Bacharach theorem
2574:. This is isomorphic to
2215:which is induced by the
1554:Linear systems on curves
463:holomorphic line bundles
163:Given a general variety
5676:Grothendieck, Alexandre
5180:For a closed immersion
4698:When the base locus of
4624:{\displaystyle \simeq }
2742:Linear system of conics
2736:Linear system of conics
759:{\displaystyle E=D+(f)}
20:from cohomology theory.
5756:; Cornalba, Maurizio;
5631:
5564:
5563:{\displaystyle x\in X}
5538:
5419:
5390:
5362:
5270:
5250:
5230:
5206:
5120:
5058:
5029:
4997:
4928:is the ideal sheaf of
4922:
4898:
4879:
4760:
4727:
4689:
4625:
4602:
4471:
4432:
4413:
4294:
4240:
4186:
4179:
4049:
4023:
3994:
3933:
3891:
3884:
3759:
3735:
3666:
3597:
3564:
3522:
3502:
3469:
3449:
3416:
3374:
3354:
3325:
3295:
3275:
3255:
3235:
3215:
3178:
3150:
3116:
3089:
3066:
2966:
2946:
2912:
2798:Kodaira–Spencer theory
2790:
2726:
2706:
2659:
2597:
2568:
2537:
2489:
2460:
2419:
2370:
2334:
2302:
2270:
2235:
2209:
2177:
2157:
2137:
2117:
2093:
2050:
1975:
1955:
1906:
1905:{\displaystyle 2g-2=2}
1871:
1840:
1814:
1772:
1750:
1718:
1691:
1615:
1595:
1572:
1551:
1544:
1441:
1418:
1398:
1332:
1312:
1274:
1211:
1165:
1102:
1073:
1019:
988:
940:
916:
870:
840:
810:
790:
760:
722:
647:
617:
597:
577:
553:
523:
486:
470:complete linear system
435:
412:
392:
366:
334:
314:
294:
268:
220:
177:
125:
40:
5632:
5565:
5544:. This sends a point
5539:
5420:
5391:
5376:A projective variety
5363:
5271:
5251:
5231:
5207:
5121:
5059:
5030:
4998:
4923:
4899:
4859:
4761:
4728:
4690:
4626:
4603:
4472:
4433:
4393:
4300:is surjective (here,
4295:
4241:
4180:
4057:
4050:
4024:
3995:
3934:
3885:
3767:
3760:
3736:
3667:
3598:
3565:
3523:
3503:
3470:
3450:
3417:
3375:
3355:
3326:
3296:
3276:
3256:
3236:
3216:
3179:
3151:
3117:
3090:
3067:
2967:
2947:
2913:
2791:
2727:
2707:
2660:
2598:
2569:
2538:
2490:
2461:
2420:
2371:
2335:
2303:
2271:
2236:
2210:
2178:
2158:
2138:
2118:
2094:
2051:
1976:
1956:
1907:
1872:
1870:{\displaystyle K_{C}}
1841:
1815:
1773:
1751:
1719:
1692:
1616:
1596:
1573:
1545:
1449:
1442:
1419:
1399:
1333:
1313:
1275:
1212:
1166:
1103:
1074:
1020:
989:
941:
917:
871:
841:
811:
791:
761:
723:
648:
618:
598:
578:
554:
524:
487:
436:
413:
393:
367:
335:
315:
295:
269:
221:
178:
126:
26:
5901:Geometry of divisors
5647:Brill–Noether theory
5574:
5548:
5429:
5400:
5380:
5280:
5260:
5240:
5216:
5184:
5071:
5039:
5010:
4939:
4908:
4774:
4741:
4706:
4646:
4615:
4492:
4442:
4308:
4254:
4209:
4062:
4033:
4004:
3943:
3897:
3772:
3745:
3676:
3631:
3574:
3532:
3512:
3479:
3459:
3429:
3384:
3364:
3335:
3305:
3285:
3265:
3245:
3225:
3195:
3168:
3130:
3099:
3079:
2979:
2956:
2926:
2896:
2824:as practised by the
2774:
2716:
2672:
2610:
2578:
2547:
2499:
2470:
2437:
2380:
2348:
2312:
2280:
2245:
2219:
2187:
2167:
2147:
2127:
2103:
2071:
1985:
1965:
1916:
1881:
1854:
1848:Riemann–Roch theorem
1824:
1783:
1762:
1740:
1728:Hyperelliptic curves
1701:
1625:
1605:
1585:
1562:
1454:
1428:
1408:
1342:
1322:
1284:
1221:
1179:
1112:
1083:
1050:
1009:
950:
926:
884:
856:
820:
800:
770:
732:
657:
627:
607:
587:
563:
533:
496:
476:
425:
402:
376:
365:{\displaystyle k(X)}
347:
324:
304:
284:
237:
187:
167:
92:
5803:Intersection Theory
5678:; Dieudonné, Jean.
4564:
4529:
4048:{\displaystyle f,g}
3765:given by the scheme
3448:{\displaystyle |D|}
3324:{\displaystyle |D|}
3214:{\displaystyle |D|}
3149:{\displaystyle |D|}
2945:{\displaystyle |D|}
2911:{\displaystyle x=a}
2830:homological algebra
2822:birational geometry
2365:
2329:
2297:
2234:{\displaystyle 2:1}
2204:
2143:which is of degree
2099:is a linear system
2088:
1839:{\displaystyle g=2}
1734:hyperelliptic curve
839:{\displaystyle |D|}
789:{\displaystyle |D|}
646:{\displaystyle |D|}
603:. In the case that
552:{\displaystyle |D|}
391:{\displaystyle (f)}
228:linearly equivalent
18:Kodaira–Spencer map
5854:Algebraic Geometry
5758:Griffiths, Phillip
5731:10.1007/BF02392112
5627:
5560:
5534:
5415:
5386:
5358:
5266:
5246:
5226:
5202:
5156:. You can help by
5116:
5054:
5035:an open subset of
5025:
4993:
4918:
4894:
4756:
4723:
4685:
4621:
4598:
4550:
4515:
4467:
4428:
4290:
4236:
4175:
4045:
4019:
3990:
3929:
3880:
3755:
3731:
3662:
3615:Theorem of Bertini
3605:globally generated
3593:
3560:
3518:
3498:
3465:
3445:
3412:
3370:
3350:
3321:
3291:
3271:
3251:
3231:
3211:
3174:
3146:
3112:
3085:
3062:
3042:
2962:
2942:
2908:
2865:Algebraic Surfaces
2850:invertible sheaves
2842:free abelian group
2786:
2722:
2702:
2655:
2593:
2564:
2533:
2485:
2456:
2415:
2366:
2351:
2330:
2315:
2298:
2283:
2266:
2231:
2205:
2190:
2173:
2153:
2133:
2113:
2089:
2074:
2046:
1971:
1951:
1902:
1867:
1836:
1810:
1768:
1746:
1714:
1687:
1611:
1591:
1568:
1540:
1440:{\displaystyle xy}
1437:
1414:
1394:
1328:
1308:
1270:
1207:
1161:
1098:
1069:
1042:Linear equivalence
1031:linear equivalence
1015:
984:
936:
912:
866:
836:
806:
786:
756:
718:
643:
613:
593:
573:
549:
519:
482:
459:invertible sheaves
431:
408:
388:
362:
330:
310:
290:
277:for some non-zero
264:
216:
173:
121:
72:linear equivalence
45:algebraic geometry
41:
37:Apollonian circles
5779:978-1-4419-2825-2
5754:Arbarello, Enrico
5389:{\displaystyle X}
5269:{\displaystyle Y}
5249:{\displaystyle X}
5174:
5173:
5051:
4957:
4753:
4633:projective bundle
4169:
4089:
4068:
3874:
3788:
3521:{\displaystyle X}
3468:{\displaystyle X}
3373:{\displaystyle C}
3347:
3294:{\displaystyle C}
3274:{\displaystyle X}
3254:{\displaystyle C}
3234:{\displaystyle X}
3109:
3061:
3056:
3022:
3010:
2965:{\displaystyle X}
2857:algebraic surface
2725:{\displaystyle k}
2641:
2340:which are called
2176:{\displaystyle r}
2156:{\displaystyle d}
2136:{\displaystyle C}
1974:{\displaystyle 2}
1771:{\displaystyle 2}
1749:{\displaystyle C}
1614:{\displaystyle K}
1594:{\displaystyle g}
1571:{\displaystyle C}
1534:
1417:{\displaystyle E}
1331:{\displaystyle D}
1018:{\displaystyle D}
809:{\displaystyle f}
616:{\displaystyle X}
596:{\displaystyle D}
508:
485:{\displaystyle X}
434:{\displaystyle X}
411:{\displaystyle f}
333:{\displaystyle f}
313:{\displaystyle X}
293:{\displaystyle f}
279:rational function
263:
205:
176:{\displaystyle X}
5908:
5846:
5815:
5814:
5798:
5792:
5791:
5750:
5744:
5743:
5733:
5718:Acta Mathematica
5709:
5703:
5700:
5683:
5673:
5652:Lefschetz pencil
5636:
5634:
5633:
5628:
5626:
5625:
5620:
5608:
5607:
5589:
5588:
5569:
5567:
5566:
5561:
5543:
5541:
5540:
5535:
5524:
5523:
5522:
5521:
5516:
5509:
5508:
5501:
5500:
5499:
5498:
5497:
5496:
5491:
5484:
5483:
5471:
5470:
5465:
5464:
5445:
5444:
5439:
5438:
5424:
5422:
5421:
5416:
5414:
5413:
5408:
5395:
5393:
5392:
5387:
5367:
5365:
5364:
5359:
5354:
5353:
5341:
5327:
5326:
5305:
5304:
5295:
5294:
5275:
5273:
5272:
5267:
5255:
5253:
5252:
5247:
5235:
5233:
5232:
5227:
5225:
5224:
5211:
5209:
5208:
5203:
5169:
5166:
5148:
5141:
5125:
5123:
5122:
5117:
5109:
5108:
5096:
5063:
5061:
5060:
5055:
5053:
5052:
5044:
5034:
5032:
5031:
5026:
5002:
5000:
4999:
4994:
4980:
4979:
4967:
4959:
4958:
4950:
4927:
4925:
4924:
4919:
4917:
4916:
4903:
4901:
4900:
4895:
4893:
4892:
4887:
4886:
4878:
4873:
4852:
4851:
4839:
4838:
4837:
4836:
4831:
4830:
4815:
4814:
4809:
4808:
4801:
4800:
4765:
4763:
4762:
4757:
4755:
4754:
4746:
4732:
4730:
4729:
4724:
4722:
4721:
4716:
4715:
4694:
4692:
4691:
4686:
4678:
4677:
4665:
4630:
4628:
4627:
4622:
4607:
4605:
4604:
4599:
4588:
4587:
4575:
4563:
4558:
4546:
4528:
4523:
4511:
4483:closed immersion
4476:
4474:
4473:
4468:
4454:
4453:
4437:
4435:
4434:
4429:
4427:
4426:
4421:
4420:
4412:
4407:
4386:
4385:
4373:
4372:
4371:
4370:
4365:
4364:
4349:
4348:
4343:
4342:
4335:
4334:
4299:
4297:
4296:
4291:
4283:
4282:
4277:
4276:
4269:
4268:
4245:
4243:
4242:
4237:
4184:
4182:
4181:
4176:
4174:
4170:
4168:
4151:
4147:
4146:
4128:
4127:
4096:
4090:
4087:
4079:
4078:
4069:
4066:
4054:
4052:
4051:
4046:
4028:
4026:
4025:
4020:
4018:
4017:
4012:
4000:is a divisor in
3999:
3997:
3996:
3991:
3989:
3988:
3983:
3971:
3970:
3958:
3957:
3938:
3936:
3935:
3930:
3925:
3924:
3909:
3908:
3889:
3887:
3886:
3881:
3879:
3875:
3873:
3850:
3846:
3845:
3827:
3826:
3795:
3789:
3786:
3781:
3780:
3764:
3762:
3761:
3756:
3754:
3753:
3740:
3738:
3737:
3732:
3718:
3717:
3708:
3707:
3702:
3671:
3669:
3668:
3663:
3661:
3660:
3655:
3646:
3645:
3626:Lefschetz pencil
3602:
3600:
3599:
3594:
3583:
3582:
3569:
3567:
3566:
3561:
3556:
3548:
3527:
3525:
3524:
3519:
3507:
3505:
3504:
3499:
3488:
3487:
3474:
3472:
3471:
3466:
3454:
3452:
3451:
3446:
3444:
3436:
3421:
3419:
3418:
3413:
3399:
3391:
3379:
3377:
3376:
3371:
3359:
3357:
3356:
3351:
3349:
3348:
3340:
3330:
3328:
3327:
3322:
3320:
3312:
3300:
3298:
3297:
3292:
3280:
3278:
3277:
3272:
3260:
3258:
3257:
3252:
3240:
3238:
3237:
3232:
3220:
3218:
3217:
3212:
3210:
3202:
3183:
3181:
3180:
3175:
3158:scheme-theoretic
3155:
3153:
3152:
3147:
3145:
3137:
3121:
3119:
3118:
3113:
3111:
3110:
3107:
3094:
3092:
3091:
3086:
3071:
3069:
3068:
3063:
3059:
3058:
3057:
3054:
3041:
3040:
3032:
3024:
3023:
3020:
3003:
2995:
2971:
2969:
2968:
2963:
2951:
2949:
2948:
2943:
2941:
2933:
2917:
2915:
2914:
2909:
2846:Cartier divisors
2795:
2793:
2792:
2787:
2731:
2729:
2728:
2723:
2711:
2709:
2708:
2703:
2701:
2700:
2695:
2686:
2685:
2680:
2664:
2662:
2661:
2656:
2648:
2647:
2646:
2637:
2625:
2602:
2600:
2599:
2594:
2592:
2591:
2586:
2573:
2571:
2570:
2565:
2554:
2542:
2540:
2539:
2534:
2520:
2519:
2494:
2492:
2491:
2486:
2484:
2483:
2478:
2465:
2463:
2462:
2457:
2446:
2445:
2424:
2422:
2421:
2416:
2399:
2375:
2373:
2372:
2367:
2364:
2359:
2339:
2337:
2336:
2331:
2328:
2323:
2307:
2305:
2304:
2299:
2296:
2291:
2275:
2273:
2272:
2267:
2265:
2264:
2259:
2240:
2238:
2237:
2232:
2214:
2212:
2211:
2206:
2203:
2198:
2182:
2180:
2179:
2174:
2162:
2160:
2159:
2154:
2142:
2140:
2139:
2134:
2122:
2120:
2119:
2114:
2112:
2111:
2098:
2096:
2095:
2090:
2087:
2082:
2055:
2053:
2052:
2047:
2039:
2038:
2020:
2019:
2007:
1999:
1998:
1993:
1980:
1978:
1977:
1972:
1960:
1958:
1957:
1952:
1941:
1940:
1928:
1927:
1911:
1909:
1908:
1903:
1876:
1874:
1873:
1868:
1866:
1865:
1845:
1843:
1842:
1837:
1819:
1817:
1816:
1811:
1809:
1808:
1803:
1777:
1775:
1774:
1769:
1755:
1753:
1752:
1747:
1723:
1721:
1720:
1715:
1713:
1712:
1696:
1694:
1693:
1688:
1680:
1679:
1661:
1660:
1648:
1640:
1632:
1620:
1618:
1617:
1612:
1600:
1598:
1597:
1592:
1577:
1575:
1574:
1569:
1549:
1547:
1546:
1541:
1539:
1535:
1533:
1525:
1524:
1523:
1511:
1510:
1498:
1497:
1485:
1484:
1474:
1446:
1444:
1443:
1438:
1423:
1421:
1420:
1415:
1403:
1401:
1400:
1395:
1393:
1392:
1380:
1379:
1367:
1366:
1354:
1353:
1337:
1335:
1334:
1329:
1317:
1315:
1314:
1309:
1307:
1303:
1299:
1279:
1277:
1276:
1271:
1257:
1256:
1247:
1246:
1241:
1216:
1214:
1213:
1208:
1191:
1190:
1173:quadric surfaces
1170:
1168:
1167:
1162:
1148:
1147:
1138:
1137:
1132:
1107:
1105:
1104:
1099:
1097:
1096:
1091:
1078:
1076:
1075:
1070:
1059:
1058:
1027:Cartier divisors
1024:
1022:
1021:
1016:
1003:invertible sheaf
993:
991:
990:
985:
965:
964:
945:
943:
942:
937:
935:
934:
921:
919:
918:
913:
905:
904:
875:
873:
872:
867:
865:
864:
845:
843:
842:
837:
835:
827:
815:
813:
812:
807:
795:
793:
792:
787:
785:
777:
765:
763:
762:
757:
727:
725:
724:
719:
714:
713:
704:
681:
680:
652:
650:
649:
644:
642:
634:
622:
620:
619:
614:
602:
600:
599:
594:
582:
580:
579:
574:
572:
571:
558:
556:
555:
550:
548:
540:
529:. It is denoted
528:
526:
525:
520:
509:
506:
491:
489:
488:
483:
447:Cartier divisors
440:
438:
437:
432:
417:
415:
414:
409:
397:
395:
394:
389:
371:
369:
368:
363:
339:
337:
336:
331:
319:
317:
316:
311:
299:
297:
296:
291:
273:
271:
270:
265:
261:
225:
223:
222:
217:
206:
203:
182:
180:
179:
174:
148:, respectively.
130:
128:
127:
122:
117:
116:
111:
110:
68:projective plane
64:algebraic curves
53:family of curves
33:family of curves
5916:
5915:
5911:
5910:
5909:
5907:
5906:
5905:
5891:
5890:
5889:
5858:Springer-Verlag
5843:
5823:
5819:
5818:
5800:
5799:
5795:
5780:
5752:
5751:
5747:
5711:
5710:
5706:
5701:
5686:
5674:
5670:
5665:
5643:
5615:
5599:
5580:
5572:
5571:
5546:
5545:
5511:
5502:
5486:
5477:
5472:
5458:
5432:
5427:
5426:
5403:
5398:
5397:
5378:
5377:
5374:
5315:
5283:
5278:
5277:
5258:
5257:
5238:
5237:
5214:
5213:
5182:
5181:
5170:
5164:
5161:
5154:needs expansion
5139:
5100:
5069:
5068:
5037:
5036:
5008:
5007:
4971:
4937:
4936:
4906:
4905:
4880:
4840:
4824:
4819:
4802:
4792:
4772:
4771:
4739:
4738:
4709:
4704:
4703:
4669:
4644:
4643:
4613:
4612:
4579:
4490:
4489:
4445:
4440:
4439:
4414:
4374:
4358:
4353:
4336:
4326:
4306:
4305:
4270:
4260:
4252:
4251:
4207:
4206:
4191:
4152:
4138:
4119:
4097:
4091:
4060:
4059:
4031:
4030:
4007:
4002:
4001:
3978:
3962:
3949:
3941:
3940:
3916:
3900:
3895:
3894:
3851:
3837:
3818:
3796:
3790:
3770:
3769:
3743:
3742:
3697:
3674:
3673:
3650:
3629:
3628:
3622:
3617:
3572:
3571:
3530:
3529:
3510:
3509:
3477:
3476:
3457:
3456:
3427:
3426:
3382:
3381:
3362:
3361:
3333:
3332:
3303:
3302:
3283:
3282:
3263:
3262:
3243:
3242:
3223:
3222:
3193:
3192:
3166:
3165:
3162:structure sheaf
3128:
3127:
3102:
3097:
3096:
3077:
3076:
3049:
3015:
2977:
2976:
2954:
2953:
2924:
2923:
2894:
2893:
2881:
2834:singular points
2818:
2806:
2772:
2771:
2749:
2744:
2738:
2714:
2713:
2690:
2675:
2670:
2669:
2627:
2620:
2608:
2607:
2581:
2576:
2575:
2545:
2544:
2497:
2496:
2473:
2468:
2467:
2435:
2434:
2431:
2378:
2377:
2346:
2345:
2342:trigonal curves
2310:
2309:
2278:
2277:
2254:
2243:
2242:
2217:
2216:
2185:
2184:
2165:
2164:
2145:
2144:
2125:
2124:
2101:
2100:
2069:
2068:
2065:
2063:
2030:
2011:
1988:
1983:
1982:
1963:
1962:
1932:
1919:
1914:
1913:
1879:
1878:
1857:
1852:
1851:
1822:
1821:
1820:. For the case
1798:
1781:
1780:
1760:
1759:
1738:
1737:
1730:
1704:
1699:
1698:
1671:
1652:
1623:
1622:
1603:
1602:
1583:
1582:
1560:
1559:
1556:
1526:
1515:
1502:
1489:
1476:
1475:
1469:
1452:
1451:
1426:
1425:
1406:
1405:
1384:
1371:
1358:
1345:
1340:
1339:
1320:
1319:
1291:
1287:
1282:
1281:
1236:
1219:
1218:
1182:
1177:
1176:
1127:
1110:
1109:
1108:whose sections
1086:
1081:
1080:
1048:
1047:
1044:
1039:
1007:
1006:
948:
947:
924:
923:
882:
881:
854:
853:
818:
817:
798:
797:
768:
767:
730:
729:
705:
655:
654:
625:
624:
605:
604:
585:
584:
561:
560:
531:
530:
494:
493:
474:
473:
443:singular points
423:
422:
400:
399:
374:
373:
345:
344:
322:
321:
302:
301:
282:
281:
235:
234:
185:
184:
183:, two divisors
165:
164:
161:
104:
90:
89:
21:
12:
11:
5:
5914:
5912:
5904:
5903:
5893:
5892:
5888:
5887:
5872:Lazarsfeld, R.
5869:
5850:Hartshorne, R.
5847:
5841:
5820:
5817:
5816:
5793:
5778:
5745:
5704:
5684:
5667:
5666:
5664:
5661:
5660:
5659:
5654:
5649:
5642:
5639:
5624:
5619:
5614:
5611:
5606:
5602:
5598:
5595:
5592:
5587:
5583:
5579:
5559:
5556:
5553:
5533:
5530:
5527:
5520:
5515:
5507:
5495:
5490:
5482:
5475:
5469:
5463:
5457:
5454:
5451:
5448:
5443:
5437:
5412:
5407:
5385:
5373:
5370:
5357:
5352:
5347:
5344:
5340:
5336:
5333:
5330:
5325:
5322:
5318:
5314:
5311:
5308:
5303:
5298:
5293:
5290:
5286:
5265:
5245:
5223:
5201:
5198:
5195:
5192:
5189:
5172:
5171:
5151:
5149:
5138:
5135:
5127:
5126:
5115:
5112:
5107:
5103:
5099:
5095:
5091:
5088:
5085:
5082:
5079:
5076:
5050:
5047:
5024:
5021:
5018:
5015:
5004:
5003:
4992:
4989:
4986:
4983:
4978:
4974:
4970:
4966:
4962:
4956:
4953:
4947:
4944:
4915:
4891:
4885:
4877:
4872:
4869:
4866:
4862:
4858:
4855:
4850:
4847:
4843:
4835:
4829:
4822:
4818:
4813:
4807:
4799:
4795:
4791:
4788:
4785:
4782:
4779:
4752:
4749:
4720:
4714:
4696:
4695:
4684:
4681:
4676:
4672:
4668:
4664:
4660:
4657:
4654:
4651:
4620:
4609:
4608:
4597:
4594:
4591:
4586:
4582:
4578:
4574:
4570:
4567:
4562:
4557:
4553:
4549:
4545:
4541:
4538:
4535:
4532:
4527:
4522:
4518:
4514:
4510:
4506:
4503:
4500:
4497:
4466:
4463:
4460:
4457:
4452:
4448:
4425:
4419:
4411:
4406:
4403:
4400:
4396:
4392:
4389:
4384:
4381:
4377:
4369:
4363:
4356:
4352:
4347:
4341:
4333:
4329:
4325:
4322:
4319:
4316:
4313:
4289:
4286:
4281:
4275:
4267:
4263:
4259:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4190:
4187:
4173:
4167:
4164:
4161:
4158:
4155:
4150:
4145:
4141:
4137:
4134:
4131:
4126:
4122:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4094:
4085:
4082:
4077:
4072:
4044:
4041:
4038:
4016:
4011:
3987:
3982:
3977:
3974:
3969:
3965:
3961:
3956:
3952:
3948:
3928:
3923:
3919:
3915:
3912:
3907:
3903:
3878:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3849:
3844:
3840:
3836:
3833:
3830:
3825:
3821:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3793:
3784:
3779:
3752:
3730:
3727:
3724:
3721:
3716:
3711:
3706:
3701:
3696:
3693:
3690:
3687:
3684:
3681:
3659:
3654:
3649:
3644:
3639:
3636:
3621:
3618:
3592:
3589:
3586:
3581:
3559:
3555:
3551:
3547:
3543:
3540:
3537:
3517:
3497:
3494:
3491:
3486:
3464:
3443:
3439:
3435:
3411:
3408:
3405:
3402:
3398:
3394:
3390:
3369:
3346:
3343:
3319:
3315:
3311:
3290:
3270:
3250:
3230:
3209:
3205:
3201:
3173:
3144:
3140:
3136:
3105:
3084:
3073:
3072:
3052:
3048:
3045:
3039:
3035:
3031:
3027:
3018:
3013:
3009:
3006:
3002:
2998:
2994:
2990:
2987:
2984:
2961:
2940:
2936:
2932:
2907:
2904:
2901:
2880:
2877:
2873:Henri Poincaré
2817:
2814:
2805:
2804:Other examples
2802:
2785:
2782:
2779:
2748:
2745:
2740:Main article:
2737:
2734:
2721:
2699:
2694:
2689:
2684:
2679:
2666:
2665:
2654:
2651:
2645:
2640:
2636:
2633:
2630:
2624:
2618:
2615:
2590:
2585:
2563:
2560:
2557:
2553:
2532:
2529:
2526:
2523:
2518:
2513:
2510:
2507:
2504:
2482:
2477:
2455:
2452:
2449:
2444:
2430:
2427:
2414:
2411:
2408:
2405:
2402:
2398:
2394:
2391:
2388:
2385:
2363:
2358:
2354:
2327:
2322:
2318:
2295:
2290:
2286:
2263:
2258:
2253:
2250:
2230:
2227:
2224:
2202:
2197:
2193:
2172:
2163:and dimension
2152:
2132:
2110:
2086:
2081:
2077:
2064:
2061:
2058:
2045:
2042:
2037:
2033:
2029:
2026:
2023:
2018:
2014:
2010:
2006:
2002:
1997:
1992:
1970:
1950:
1947:
1944:
1939:
1935:
1931:
1926:
1922:
1901:
1898:
1895:
1892:
1889:
1886:
1864:
1860:
1835:
1832:
1829:
1807:
1802:
1797:
1794:
1791:
1788:
1767:
1745:
1729:
1726:
1711:
1707:
1686:
1683:
1678:
1674:
1670:
1667:
1664:
1659:
1655:
1651:
1647:
1643:
1639:
1635:
1631:
1610:
1590:
1567:
1555:
1552:
1538:
1532:
1529:
1522:
1518:
1514:
1509:
1505:
1501:
1496:
1492:
1488:
1483:
1479:
1472:
1468:
1465:
1462:
1459:
1436:
1433:
1413:
1391:
1387:
1383:
1378:
1374:
1370:
1365:
1361:
1357:
1352:
1348:
1327:
1306:
1302:
1298:
1294:
1290:
1269:
1266:
1263:
1260:
1255:
1250:
1245:
1240:
1235:
1232:
1229:
1226:
1206:
1203:
1200:
1197:
1194:
1189:
1185:
1160:
1157:
1154:
1151:
1146:
1141:
1136:
1131:
1126:
1123:
1120:
1117:
1095:
1090:
1068:
1065:
1062:
1057:
1043:
1040:
1038:
1035:
1014:
983:
980:
977:
974:
971:
968:
963:
958:
955:
933:
911:
908:
903:
898:
895:
892:
889:
863:
834:
830:
826:
805:
784:
780:
776:
755:
752:
749:
746:
743:
740:
737:
717:
712:
708:
703:
699:
696:
693:
690:
687:
684:
679:
674:
671:
668:
665:
662:
641:
637:
633:
612:
592:
570:
547:
543:
539:
518:
515:
512:
504:
501:
481:
465:); see below.
430:
407:
387:
384:
381:
361:
358:
355:
352:
342:function field
329:
309:
289:
275:
274:
260:
257:
254:
251:
248:
245:
242:
215:
212:
209:
201:
198:
195:
192:
172:
160:
157:
120:
115:
109:
103:
100:
97:
13:
10:
9:
6:
4:
3:
2:
5913:
5902:
5899:
5898:
5896:
5885:
5884:3-540-22533-1
5881:
5877:
5873:
5870:
5867:
5866:0-387-90244-9
5863:
5859:
5855:
5851:
5848:
5844:
5842:0-471-05059-8
5838:
5834:
5830:
5826:
5822:
5821:
5812:
5808:
5804:
5797:
5794:
5789:
5785:
5781:
5775:
5771:
5767:
5763:
5759:
5755:
5749:
5746:
5741:
5737:
5732:
5727:
5723:
5719:
5715:
5708:
5705:
5699:
5697:
5695:
5693:
5691:
5689:
5685:
5681:
5677:
5672:
5669:
5662:
5658:
5655:
5653:
5650:
5648:
5645:
5644:
5640:
5638:
5622:
5612:
5604:
5600:
5596:
5593:
5590:
5585:
5581:
5557:
5554:
5551:
5528:
5518:
5493:
5473:
5467:
5455:
5449:
5441:
5410:
5383:
5371:
5369:
5345:
5342:
5331:
5323:
5320:
5316:
5309:
5291:
5288:
5284:
5276:, defined as
5263:
5243:
5199:
5193:
5190:
5187:
5178:
5168:
5159:
5155:
5152:This section
5150:
5147:
5143:
5142:
5136:
5134:
5132:
5113:
5105:
5101:
5086:
5083:
5080:
5077:
5074:
5067:
5066:
5065:
5048:
5045:
5022:
5019:
5016:
5013:
4990:
4987:
4984:
4976:
4972:
4954:
4951:
4945:
4942:
4935:
4934:
4933:
4931:
4889:
4870:
4867:
4864:
4860:
4848:
4845:
4841:
4833:
4820:
4811:
4797:
4793:
4789:
4780:
4777:
4769:
4750:
4747:
4736:
4718:
4701:
4682:
4674:
4670:
4655:
4652:
4649:
4642:
4641:
4640:
4638:
4634:
4618:
4595:
4592:
4584:
4580:
4568:
4560:
4555:
4551:
4539:
4533:
4530:
4525:
4520:
4516:
4501:
4498:
4495:
4488:
4487:
4486:
4484:
4481:, there is a
4480:
4479:relative Proj
4464:
4461:
4458:
4455:
4450:
4446:
4423:
4404:
4401:
4398:
4394:
4382:
4379:
4375:
4367:
4354:
4345:
4331:
4327:
4323:
4314:
4311:
4303:
4287:
4279:
4265:
4261:
4257:
4249:
4230:
4227:
4224:
4215:
4212:
4204:
4200:
4195:
4188:
4185:
4171:
4162:
4159:
4156:
4143:
4139:
4135:
4132:
4129:
4124:
4120:
4110:
4107:
4104:
4098:
4092:
4083:
4056:
4042:
4039:
4036:
4014:
3985:
3975:
3967:
3963:
3959:
3954:
3950:
3926:
3921:
3917:
3913:
3910:
3905:
3901:
3890:
3876:
3867:
3864:
3861:
3858:
3855:
3842:
3838:
3834:
3831:
3828:
3823:
3819:
3809:
3806:
3803:
3797:
3791:
3782:
3766:
3722:
3709:
3704:
3688:
3685:
3682:
3679:
3657:
3637:
3634:
3627:
3624:Consider the
3619:
3616:
3611:
3608:
3606:
3587:
3549:
3538:
3535:
3515:
3492:
3462:
3437:
3423:
3409:
3406:
3403:
3400:
3392:
3367:
3341:
3313:
3288:
3268:
3248:
3228:
3203:
3190:
3185:
3171:
3163:
3159:
3138:
3125:
3103:
3082:
3050:
3046:
3043:
3033:
3025:
3016:
3011:
3007:
2996:
2985:
2982:
2975:
2974:
2973:
2959:
2934:
2920:
2905:
2902:
2899:
2890:
2886:
2878:
2876:
2874:
2870:
2866:
2862:
2858:
2853:
2851:
2847:
2843:
2839:
2838:Weil divisors
2835:
2831:
2827:
2823:
2815:
2813:
2811:
2803:
2801:
2799:
2783:
2777:
2769:
2768:normal bundle
2764:
2762:
2758:
2754:
2746:
2743:
2735:
2733:
2719:
2697:
2682:
2652:
2649:
2638:
2634:
2631:
2628:
2616:
2613:
2606:
2605:
2604:
2588:
2558:
2524:
2505:
2502:
2480:
2450:
2428:
2426:
2412:
2409:
2406:
2400:
2396:
2392:
2386:
2383:
2361:
2356:
2352:
2343:
2325:
2320:
2316:
2293:
2288:
2284:
2261:
2248:
2228:
2225:
2222:
2200:
2195:
2191:
2170:
2150:
2130:
2084:
2079:
2075:
2059:
2057:
2035:
2031:
2027:
2024:
2016:
2012:
2000:
1995:
1968:
1948:
1945:
1937:
1933:
1924:
1920:
1899:
1896:
1893:
1890:
1887:
1884:
1862:
1858:
1849:
1833:
1830:
1827:
1805:
1792:
1789:
1786:
1779:
1765:
1743:
1735:
1727:
1725:
1709:
1705:
1676:
1672:
1668:
1665:
1657:
1653:
1641:
1633:
1608:
1588:
1581:
1565:
1553:
1550:
1536:
1530:
1527:
1520:
1516:
1512:
1507:
1503:
1499:
1494:
1490:
1486:
1481:
1477:
1470:
1466:
1463:
1460:
1457:
1448:
1434:
1431:
1411:
1389:
1385:
1381:
1376:
1372:
1368:
1363:
1359:
1355:
1350:
1346:
1325:
1304:
1300:
1296:
1292:
1288:
1261:
1248:
1243:
1227:
1224:
1201:
1195:
1192:
1187:
1183:
1174:
1152:
1139:
1134:
1118:
1115:
1093:
1063:
1041:
1036:
1034:
1032:
1028:
1012:
1004:
1000:
995:
981:
978:
975:
972:
969:
966:
956:
953:
909:
896:
893:
879:
852:
851:linear system
847:
828:
803:
778:
750:
744:
741:
738:
735:
715:
710:
706:
701:
691:
685:
672:
669:
635:
610:
590:
541:
513:
502:
499:
479:
471:
466:
464:
460:
456:
452:
451:Weil divisors
448:
444:
428:
421:Note that if
419:
405:
382:
356:
350:
343:
327:
307:
287:
280:
255:
249:
246:
243:
240:
233:
232:
231:
229:
210:
199:
196:
193:
190:
170:
158:
156:
154:
149:
147:
143:
139:
138:
132:
113:
101:
98:
88:
84:
81:on a general
80:
77:
73:
69:
65:
61:
60:linear system
56:
54:
50:
46:
38:
34:
30:
25:
19:
5875:
5853:
5832:
5825:P. Griffiths
5805:. Springer.
5802:
5796:
5761:
5748:
5721:
5717:
5707:
5679:
5671:
5396:embedded in
5375:
5368:(page 158).
5179:
5175:
5162:
5158:adding to it
5153:
5130:
5128:
5005:
4929:
4767:
4734:
4699:
4697:
4636:
4610:
4301:
4247:
4202:
4198:
4196:
4192:
4058:
3939:for a fixed
3892:
3768:
3623:
3609:
3424:
3186:
3184:should be).
3123:
3074:
2921:
2884:
2882:
2868:
2864:
2854:
2819:
2807:
2765:
2760:
2756:
2755:for a curve
2752:
2750:
2667:
2432:
2066:
1731:
1557:
1450:
1045:
1030:
996:
877:
850:
848:
469:
467:
420:
276:
227:
162:
152:
150:
145:
141:
135:
133:
87:ringed space
78:
71:
59:
57:
48:
42:
35:, as in the
28:
5724:: 163–176.
5165:August 2019
2123:on a curve
1736:is a curve
999:line bundle
159:Definitions
153:Kodaira map
5663:References
3613:See also:
3124:base locus
2885:base locus
2879:Base locus
1621:, denoted
85:or even a
5829:J. Harris
5740:0001-5962
5613:∈
5594:⋯
5555:∈
5474:⊗
5346:∈
5321:−
5289:−
5197:↪
5106:∗
5090:→
5084:−
5049:~
5023:≃
5017:−
4985:×
4977:∗
4961:↪
4955:~
4876:∞
4861:⨁
4857:→
4846:−
4821:⊗
4794:⊗
4781:
4751:~
4675:∗
4659:→
4619:≃
4593:×
4585:∗
4561:∗
4540:≃
4531:⊗
4526:∗
4505:↪
4462:×
4410:∞
4395:⨁
4391:→
4380:−
4355:⊗
4328:⊗
4315:
4285:→
4262:⊗
4219:Γ
4216:⊂
4133:…
3976:∈
3832:…
3692:Γ
3689:∈
3648:→
3539:
3407:≥
3401:⋅
3345:~
3047:
3026:∈
3012:⋂
2986:
2781:↪
2688:→
2650:−
2509:Γ
2387:≥
2252:→
2032:ω
1891:−
1796:→
1706:ω
1673:ω
1231:Γ
1228:∈
1122:Γ
1119:∈
979:−
973:
957:
888:Γ
711:∗
686:∖
664:Γ
503:∈
200:∈
5895:Category
5831:(1994).
5760:(2011).
5641:See also
2840:(in the
1778:morphism
1037:Examples
76:divisors
5788:2807457
5682:, 21.3.
3620:Example
3189:nefness
2889:variety
2861:Zariski
1981:map to
1756:with a
1171:define
372:. Here
340:of the
144:, or a
66:in the
5882:
5864:
5839:
5786:
5776:
5738:
5680:EGA IV
5006:Since
4904:where
4611:where
3241:, and
3075:where
3060:
2603:where
1758:degre
559:. Let
453:: see
262:
137:pencil
83:scheme
3741:, so
3281:. If
2466:over
2241:-map
1580:genus
5880:ISBN
5862:ISBN
5837:ISBN
5774:ISBN
5736:ISSN
4205:and
4197:Let
4088:Proj
4055:, so
3787:Proj
3083:Supp
3044:Supp
2883:The
2808:The
2376:for
1912:and
441:has
226:are
140:, a
47:, a
5807:doi
5766:doi
5726:doi
5722:132
5256:to
5236:on
5160:.
4778:Sym
4312:Sym
3508:on
3164:of
3126:of
3108:eff
3055:eff
3021:eff
2770:to
1877:is
1578:of
1079:on
1001:or
970:dim
954:dim
880:of
766:of
507:Div
472:on
461:or
300:on
230:if
204:Div
146:web
142:net
74:of
62:of
43:In
5897::
5874:,
5856:,
5852:,
5827:;
5784:MR
5782:.
5772:.
5734:.
5720:.
5716:.
5687:^
5637:.
4485::
4067:Bl
3536:Bl
3172:Bl
3008::=
2983:Bl
2852:.
2763:.
2732:.
2425:.
2067:A
2056:.
1724:.
994:.
849:A
468:A
449:,
418:.
155:.
131:.
27:A
5886:.
5868:.
5845:.
5813:.
5809::
5790:.
5768::
5742:.
5728::
5623:r
5618:P
5610:]
5605:r
5601:x
5597::
5591::
5586:0
5582:x
5578:[
5558:X
5552:x
5532:)
5529:1
5526:(
5519:r
5514:P
5506:O
5494:r
5489:P
5481:O
5468:X
5462:O
5456:=
5453:)
5450:1
5447:(
5442:X
5436:O
5411:r
5406:P
5384:X
5356:}
5351:d
5343:D
5339:|
5335:)
5332:D
5329:(
5324:1
5317:f
5313:{
5310:=
5307:)
5302:d
5297:(
5292:1
5285:f
5264:Y
5244:X
5222:d
5200:X
5194:Y
5191::
5188:f
5167:)
5163:(
5131:V
5114:.
5111:)
5102:V
5098:(
5094:P
5087:B
5081:X
5078::
5075:f
5046:X
5020:B
5014:X
4991:.
4988:X
4982:)
4973:V
4969:(
4965:P
4952:X
4946::
4943:i
4930:B
4914:I
4890:n
4884:I
4871:0
4868:=
4865:n
4854:)
4849:1
4842:L
4834:X
4828:O
4817:)
4812:X
4806:O
4798:k
4790:V
4787:(
4784:(
4768:B
4748:X
4735:X
4719:X
4713:O
4700:V
4683:.
4680:)
4671:V
4667:(
4663:P
4656:X
4653::
4650:f
4637:i
4596:X
4590:)
4581:V
4577:(
4573:P
4569:=
4566:)
4556:X
4552:V
4548:(
4544:P
4537:)
4534:L
4521:X
4517:V
4513:(
4509:P
4502:X
4499::
4496:i
4465:X
4459:V
4456:=
4451:X
4447:V
4424:X
4418:O
4405:0
4402:=
4399:n
4388:)
4383:1
4376:L
4368:X
4362:O
4351:)
4346:X
4340:O
4332:k
4324:V
4321:(
4318:(
4302:k
4288:L
4280:X
4274:O
4266:k
4258:V
4248:V
4234:)
4231:L
4228:,
4225:X
4222:(
4213:V
4203:X
4199:L
4172:)
4166:)
4163:g
4160:,
4157:f
4154:(
4149:]
4144:n
4140:x
4136:,
4130:,
4125:0
4121:x
4117:[
4114:]
4111:t
4108:,
4105:s
4102:[
4099:k
4093:(
4084:=
4081:)
4076:X
4071:(
4043:g
4040:,
4037:f
4015:n
4010:P
3986:1
3981:P
3973:]
3968:0
3964:t
3960::
3955:0
3951:s
3947:[
3927:g
3922:0
3918:t
3914:+
3911:f
3906:0
3902:s
3877:)
3871:)
3868:g
3865:t
3862:+
3859:f
3856:s
3853:(
3848:]
3843:n
3839:x
3835:,
3829:,
3824:0
3820:x
3816:[
3813:]
3810:t
3807:,
3804:s
3801:[
3798:k
3792:(
3783:=
3778:X
3751:X
3729:)
3726:)
3723:d
3720:(
3715:O
3710:,
3705:n
3700:P
3695:(
3686:g
3683:,
3680:f
3658:1
3653:P
3643:X
3638::
3635:p
3591:)
3588:D
3585:(
3580:O
3558:)
3554:|
3550:D
3546:|
3542:(
3516:X
3496:)
3493:D
3490:(
3485:O
3463:X
3442:|
3438:D
3434:|
3410:0
3404:C
3397:|
3393:D
3389:|
3368:C
3342:D
3318:|
3314:D
3310:|
3289:C
3269:X
3249:C
3229:X
3208:|
3204:D
3200:|
3143:|
3139:D
3135:|
3104:D
3051:D
3038:|
3034:D
3030:|
3017:D
3005:)
3001:|
2997:D
2993:|
2989:(
2960:X
2939:|
2935:D
2931:|
2906:a
2903:=
2900:x
2784:Y
2778:C
2761:C
2757:C
2753:Y
2720:k
2698:N
2693:P
2683:k
2678:P
2653:1
2644:)
2639:n
2635:d
2632:+
2629:n
2623:(
2617:=
2614:N
2589:N
2584:P
2562:)
2559:V
2556:(
2552:P
2531:)
2528:)
2525:d
2522:(
2517:O
2512:(
2506:=
2503:V
2481:n
2476:P
2454:)
2451:d
2448:(
2443:O
2413:1
2410:+
2407:g
2404:)
2401:2
2397:/
2393:1
2390:(
2384:d
2362:d
2357:1
2353:g
2326:3
2321:1
2317:g
2294:1
2289:2
2285:g
2262:1
2257:P
2249:C
2229:1
2226::
2223:2
2201:1
2196:2
2192:g
2171:r
2151:d
2131:C
2109:d
2085:r
2080:d
2076:g
2062:d
2060:g
2044:)
2041:)
2036:C
2028:,
2025:C
2022:(
2017:0
2013:H
2009:(
2005:P
2001:=
1996:1
1991:P
1969:2
1949:2
1946:=
1943:)
1938:C
1934:K
1930:(
1925:0
1921:h
1900:2
1897:=
1894:2
1888:g
1885:2
1863:C
1859:K
1834:2
1831:=
1828:g
1806:1
1801:P
1793:C
1790::
1787:f
1766:2
1744:C
1710:C
1685:)
1682:)
1677:C
1669:,
1666:C
1663:(
1658:0
1654:H
1650:(
1646:P
1642:=
1638:|
1634:K
1630:|
1609:K
1589:g
1566:C
1537:)
1531:y
1528:x
1521:2
1517:w
1513:+
1508:2
1504:z
1500:+
1495:2
1491:y
1487:+
1482:2
1478:x
1471:(
1467:+
1464:E
1461:=
1458:D
1435:y
1432:x
1412:E
1390:2
1386:w
1382:+
1377:2
1373:z
1369:+
1364:2
1360:y
1356:+
1351:2
1347:x
1326:D
1305:)
1301:s
1297:/
1293:t
1289:(
1268:)
1265:)
1262:2
1259:(
1254:O
1249:,
1244:3
1239:P
1234:(
1225:t
1205:)
1202:s
1199:(
1196:Z
1193:=
1188:s
1184:D
1159:)
1156:)
1153:2
1150:(
1145:O
1140:,
1135:3
1130:P
1125:(
1116:s
1094:3
1089:P
1067:)
1064:2
1061:(
1056:O
1025:(
1013:D
982:1
976:W
967:=
962:d
932:d
910:.
907:)
902:L
897:,
894:X
891:(
878:W
862:d
833:|
829:D
825:|
804:f
783:|
779:D
775:|
754:)
751:f
748:(
745:+
742:D
739:=
736:E
716:,
707:k
702:/
698:)
695:}
692:0
689:{
683:)
678:L
673:,
670:X
667:(
661:(
640:|
636:D
632:|
611:X
591:D
569:L
546:|
542:D
538:|
517:)
514:X
511:(
500:D
480:X
429:X
406:f
386:)
383:f
380:(
360:)
357:X
354:(
351:k
328:f
308:X
288:f
259:)
256:f
253:(
250:+
247:D
244:=
241:E
214:)
211:X
208:(
197:E
194:,
191:D
171:X
119:)
114:X
108:O
102:,
99:X
96:(
79:D
39:.
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