Fundamental groups of line arrangements: Enumerative aspects
Fundamental groups of line arrangements: Enumerative aspects
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线路排列的基本组：枚举方面
DOI：
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发表时间：
2000
期刊：
影响因子：
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通讯作者：
Alexander I. Suciu
中科院分区：
文献类型：
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作者：
Alexander I. Suciu
本文综述了复平面上直线排列的补数研究的一些新进展。我们调查的基本群体和有限覆盖这些补充，侧重于同调和枚举方面。本研究的统一框架是分层的基本组，G，上同调系数在秩1本地系统的跳跃轨迹的字符品种。计算这些“特征”变种上的某些扭点，可以得到关于补集的分支和非分支覆盖的同源性的信息，以及关于其基本群的低指数子群的数量的信息。最后，我们得出两个结论，表示G/G'（和，在某些情况下，G本身）在密切相关的“共振”品种的较低的中心系列的子。我们用一些详细的例子来说明讨论，其中一些揭示了新的现象。
This is a survey of some recent developments in the study of complements of line arrangements in the complex plane. We investigate the fundamental groups and finite covers of those complements, focusing on homological and enumerative aspects. The unifying framework for this study is the stratification of the character variety of the fundamental group, G, by the jumping loci for cohomology with coefficients in rank 1 local systems. Counting certain torsion points on these "characteristic" varieties yields information about the homology of branched and unbranched covers of the complement, as well as on the number of low-index subgroups of its fundamental group. We conclude with two conjectures, expressing the lower central series quotients of G/G' (and, in some cases, G itself) in terms of the closely related "resonance" varieties. We illustrate the discussion with a number of detailed examples, some of which reveal new phenomena.