Multiple fibers on rational elliptic surfaces
Multiple fibers on rational elliptic surfaces
复制标题
有理椭圆面上的多纤维
DOI：
10.1090/s0002-9947-1988-0936813-6
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发表时间：
1988
影响因子：
1.3
通讯作者：
W. Lang
中科院分区：
文献类型：
--
作者：
B. Harbourne;W. Lang
我们的主要结果，定理（0.1），分类的有理椭圆曲面上的多纤维代数闭域的任意特征。这样做的一个结果是，对于先前没有已知实例的几种科代拉纤维类型，存在加性类型的驯服多纤维的正特性。0.导论.本文研究了具有任意特征p的代数闭域k上的椭圆曲面f：X）B是完备光滑曲面X在完备光滑曲线B上的纤维化，使得类属纤维是亏格为1的光滑曲线。人们自然会想知道是什么样的特殊纤维产生的。我们在有理曲面的情况下回答了这个问题，作为结果，对于每个p > 0和除了Ib，B > 4以外的所有这类类型，都给出了加法型驯服多重纤维的例子（参见第5条）。据我们所知，迄今为止唯一公认的例子是多重数为2和科代拉型IoX，这是由于Katalina [Ka，L1]（尽管自从我们的结果获得以来，我们已经看到Katalina和Ueno [KU]的预印本给出了其他例子，主要是p 7和2，3和科代拉型，而不是Ib，B > 0）。任何特殊的纤维都是X上的有效因子，因此可以写为mF = mEnjCj，其中每条曲线Cj是约化的和不可约的，系数是g.c.d.的正整数。（{nj}）= 1。正整数m是光纤mF的倍数，当m > 1时，光纤mF被称为多重光纤。除数F = EnjCj的形式非常有限。这样一个除数是一个不可分解的曲线的典型类型[穆];拓扑分类的所有这些曲线已被科代拉[高]（参见。第5条）。我们把这种分类中出现的类别称为科代拉（纤维）类型。更一般地，科代拉型光纤mF是mT，其中T是F的类型。在代数闭域k上，已知曲面X上的重数为m的任何重纤维mF具有mT的形式，其中T是科代拉型，m可被（并且特征0等于）F在X中的法丛OX（F）s OF E Pic° F的阶，但这些条件在何种程度上足以使mT型多重纤维出现在某个椭圆曲面上尚不清楚。这里我们特别感兴趣的是加性类型的驯服多重纤维，这些纤维是多重数为m的纤维mF，其中Pic° F是基场k的加性群Ga，m等于Ga °f OX（F）s OF中的阶数。只有当k的特征p为正时（因为Ga只有在此时才有p挠），它们才是可能的，但人们对它们的存在知之甚少。编辑于1987年3月21日收到。1980年数学学科分类（1985年修订）。初级14 J25、14 M20、14 D99。
Our main result, Theorem (0.1), classifies multiple fibers on rational elliptic surfaces over algebraically closed fields of arbitrary characteristic. One result of this is the existence in positive characteristics of tame multiple fibers of additive type for several of the Kodaira fiber-types for which no examples were previously known. 0. Introduction. We work in this paper over an algebraically closed field k of arbitrary characteristic p. An elliptic surface f: X ) B is a fibration of a complete smooth surface X over a complete smooth curve B such that the generic fiber is a smooth curve of genus one. It is natural to wonder what kind of special fibers arise. We answer this in the case of rational surfaces, giving as a result examples of tame multiple fibers of additive type for every p > O and for all such types except Ib, b > 4 (cf. §5). To our knowledge, the only example heretofore recognized is of multiplicity 2 and Kodaira type IoX due to Katsura [Ka, L1] (although since our results were obtained we have seen a preprint of Katsura and Ueno [KU] giving other examples, mostly for p 7& 2, 3 and Kodaira type not Ib, b > 0). Any special fiber is an effective divisor on X and so can be written as mF = mEnjCj, where each curve Cj is reduced and irreducible and the coefficients are positive integers with g.c.d.({nj}) = 1. The positive integer m is the rnultiplicity of the fiber mF which is said to be a multiple fiber whenever m > 1. The form of the divisor F = EnjCj is very restricted. Such a divisor is an indecomposable curve of canonical type [Mu]; a topological classification of all such curves has been given by Kodaira [Ko] (cf. §5). We refer to the classes which arise in this classification as the Kodaira (fiber-)types. More generally, the Kodaira type of the fiber mF is mT, where T is the type of F. Over an algebraically closed field k it is known that any multiple fiber mF of multiplicity m on a surface X is of the form mT, where T is a Kodaira type and m is divisible by (and in characteristic 0 equal to) the order of the normal bundle OX(F) s OF E Pic° F of F in X, but the extent to which these conditions are sufficient for there to occur on some elliptic surface a multiple fiber of type mT is not known. Of particular interest to us here are the tame multiple fibers of additive type, these being fibers mF of multiplicity m where Pic° F is the additive group Ga of the ground field k and m equals the order in Ga °f OX(F) s OF. These are possible only if p, the characteristic of k, is positive (since Ga only then has ptorsion), but their occurrence is poorly understood. Received by the editors March 21, 1987. 1980 Mathematzcs Subject Classzficatzon (1985 Re"szon). Primary 14J25, 14M20, 14D99.