Tame coverings and fundamental groups of algebraic varieties: Part I: Branch loci with normal crossings; applications: theorems of Zariski and Picard
Tame coverings and fundamental groups of algebraic varieties: Part I: Branch loci with normal crossings; applications: theorems of Zariski and Picard
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代数簇的驯服覆盖和基本群：第一部分：具有正常交叉的分支轨迹；
DOI：
10.2307/2372850
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发表时间：
1959
影响因子：
1.7
通讯作者：
S. Abhyankar
中科院分区：
文献类型：
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作者：
S. Abhyankar
导论.本文研究任意基域上代数簇V减去子簇W的基本群，经典情形被归为特殊情形。这将通过首先研究v的包含W中的分支基因座的有限代数覆盖来完成。在这里的导言中，我们将只大致描述情况，并指出一些结果。V上的有限伽罗瓦群是V-同构覆盖的所有驯服（定义见第2节）有限伽罗瓦覆盖的有限伽罗瓦群，它们被确定为含有W中的分支轨迹，形成一个特殊类型的逆系统（V-W），我们称之为群塔。称群G是群塔xr的弱母群，如果G可拓扑化使得G的所有连续有限同态像的群塔同构于7 r;如果γ r同构于G的所有有限同态像的群塔（即如果G被看作是一个离散群），则称G是7 r的一个父群。（或稍弱：可能存在一个弱生成的弱父群）是抽象的类似的声明（第16节）证明了在经典情形下拓扑基本群sr 1（VW）是双生成的;因此，如果存在这样一个弱生成的父（或弱父）群，我们称它为V-W的一个驯服的基本父（或弱父）群。本文的一个主要结果（第12节）是，如果V是非奇异的单连通的，如果W只有法向交叉，如果W的不可约分支在大于1维的线性系统中移动，则表示这些不可约分支的个数。
Introduction. In this paper, we shall study the fundamental group of an algebraic variety V minus a subvariety W over an arbitrary ground field, the classical case being subsumed as a special case. This will be done via first studying finite alegbraic coverings of v with branch loci contained in W. Here in the introduction, we shall only approximately describe the situation and indicate some of the results. The finite galois groups over V of all tame (for definition see Section 2) finite galois coverings of V-isomorphic coverings being identified-with branch loci contained in W form an inverse system &'(V-W) of a special kind which we shall call a group tower. A group G is said to be a weak parent group of a group tower xr if G can be topologized so that the group tower of all continuous finite homomorphic images of G is isomorphic to 7r; if 7r is isomorphic to the group tower of all finite homomorphic images of G (i.e. if G is regarded as a discrete group), then G is said to be a parent group of 7r.1 The possible existence of a finitely generated parent group (or somewhat weaker: the possible existence of a finitely generated weak parent group) of r'(V -W) is the abstract analogue of the statement (Section 16) that in the classical case the topological fundamental group sr1(VW) is finitely generated; and hence if such a finitely generated parent (respectively, weak parent) group exists, we shall call it a tame fundamental parent (respectively, weak parent) group of V -W. Now one main result of this paper (Section 12) is that if V is nonsingular and simply connected, if W has only normal crossings and if the irreducible components of W move in linear systems of dimension greater than one, then denoting the number of these