Boundary behaviour of Hurwitz schemes
Boundary behaviour of Hurwitz schemes
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Hurwitz 方案的边界行为
DOI：
10.1007/978-1-4612-4264-2_7
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发表时间：
1995
期刊：
影响因子：
--
通讯作者：
T. Ekedahl
中科院分区：
文献类型：
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作者：
T. Ekedahl
本文的目的是得到一种简便的计算射影线的覆盖退化的拓扑类型的方法。这直接导致了Hurwitz计划的理论。Hurwitz格式通常被认为是具有指定次数和局部行为的曲线到P1的映射的模空间。我们宁愿考虑映射的伽罗瓦船体，从而考虑曲线与有限群的作用，只有平凡的不变量1-上同调。如果该组然后提供了一个传递置换表示，一个曲线与映射到P 1是获得和本地的行为是很容易描述的（共轭类）稳定子群的点。现在，如果曲线和群作用在一个族中变化，则曲线族在一般有限基扩张之后具有到适当基的稳定扩张，并且群作用自动扩张。如果我们观察边界点，我们将得到群G在稳定曲线上的作用。这张注记实际上是用从商曲线的一种基本群到G的同态来描述这种作用，这种描述完全类似于众所周知的光滑曲线的情况。一个直接紧化的赫尔维茨计划是由哈里斯和芒福德在[哈穆]，很明显，人们可以提取一个描述可能的拓扑类型从他们的建设。
The purpose of this paper is to get a convenient way of computing the topological types of degenerations of covers of the projective line. This leads immediately to the theory of Hurwitz schemes. Hurwitz schemes are normally thought of as the moduli spaces of maps of curves to P 1 with prescribed degree and local behaviour. We will rather consider the Galois hull of the mapping and thus consider curves together with an action of a finite group with only trivial invariants on 1-cohomology. If the group then is provided with a transitive permutation representation, a curve with a map toP 1 is obtained and the local behaviour is easily described in terms of the (conjugacy classes of) stabiliser subgroups of points. Now, if the curve and the group action varies in a family the family of curves has, after a generically finite base extension, a stable extension to a proper base and the group action automatically extends. If we look at boundary points we will then get an action of our group G on a stable curve. What this note actually does is to provide a description of such actions in terms of homomorphisms from a kind of fundamental group of the quotient curve into G, a description which is completely analogous to the well known case of a smooth curve. A direct compactification of the Hurwitz scheme was provided by Harris and Mumford in [Ha-Mu] and it is clear that one could extract a description of the possible topological types from their construction.