关于圆锥线排列的无共轭基本群的结构
On the structure of conjugation-free fundamental groups of conic-line arrangements
摘要：
超平面排列的补的基本群在研究相应的超平面排列中起着重要的作用。特别地，对于超平面排列的大族，这个基本群，同构于线排列的补的基本群，具有一些显著的性质：要么它是自由群和自由阿尔贝群的直和，要么它具有无共轭的几何表示。本文首先对以下关键引理给出了完整的证明：如果我们画一条新的直线只通过一个实直线排列的一个交点，而该实直线排列的基本群是无共轭的，那么新的直线排列的基本群也是无共轭的。其次，我们将这个引理推广到圆锥线排列的情形。此外，我们证明，一旦图相关的圆锥线安排（定义略有不同，比相应的图的线安排）没有循环，那么它的补的基本群有一个共轭自由几何表示，此外可以写为一个直和的自由群和自由阿尔贝群。证明了如果图由一个圈组成，且二次曲线不通过圈顶点对应的所有重点，则基本群也具有无共轭几何表示.作为结论，我们通过定义无共轭图的概念，扩展了具有无共轭几何表示的实数线排列族（对于它们的基本群）。我们也将这一概念推广到某些家庭的圆锥线安排。
英文摘要：
The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkable properties: either it is a direct sum of free groups and a free abelian group, or it has a conjugation-free geometric presentation. In this paper, we first give a complete proof to the following key lemma: if we draw a new line through only one intersection point of a given real line arrangement whose fundamental group is conjugation-free, then the fundamental group of the new arrangement is also conjugation-free. Second, we generalize this lemma to the case of conic-line arrangements. Moreover, we prove that once the graph associated to conic-line arrangements (defined slightly different than the corresponding graph for line arrangements) has no cycles, then the fundamental group of its complement has a conjugation-free geometric presentation and in addition can be written as a direct sum of free groups and a free abelian group. Also, we show that if the graph consists of one cycle, and the conic does not pass through all the multiple points corresponding to the vertices of the cycle, then the fundamental group has a conjugation-free geometric presentation as well. For conclusion, we extend the family of real line arrangements having a conjugation-free geometric presentation (for their fundamental group) by defining the notion of a conjugation-free graph. We also extend this notion to certain families of conic-line arrangements.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
