Braid Monodromy of Algebraic Curves
Braid Monodromy of Algebraic Curves
复制标题
代数曲线的辫状单峰
DOI：
10.5802/ambp.295
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
J. Cogolludo
中科院分区：
文献类型：
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作者：
J. Cogolludo
这些笔记来自于2009年10月在法国-西班牙高等学校的首次会议上，在圣保罗大学和阿杜地区大学举办的为期一周的关于代数曲线的编织单值性的课程：小维度的辫子和拓扑群。这是打算是一个介绍性的调查，通过它，我们希望我们可以简要地概述权力的概念monodromy作为一个共同领域的群论，代数几何和拓扑的射影曲线。主要的经典结果在§2中陈述，其中给出了计算射影平面曲线的补曲线的基本群的表示的Zapriki-van坎彭方法。在§1中，这些结果以基本概念如基本群、局部平凡纤维化、分支覆盖和非分支覆盖的回顾作为序言，并首次看到了单值性。描述的主要动机，导致数学家研究这些对象包括整个第一章。最后，§3将讨论直接应用编织单值性的其他工具和进一步结果。虽然不是所有的证据都包括在内，但我们确实提供了那些相关的原始或简化版本，因为它们展示了在这种情况下最常用的技术，并导致更好地理解本调查中讨论的主要概念。因此，这里没有什么是原创的，除了试图汇集不同的结果和观点。不用说，这不是第一次，也希望不是最后一次关于这一专题的调查。关于编织单值性的其他方法，我们参考以下写得很好的论文[73，20，6]。最后，我们要感谢主办方和裁判在撰写和修改这些笔记的过程中所表现出的耐心和理解。
These are the notes from a one-week course on Braid Monodromy of Algebraic Curves given at the Universite de Pau et des Pays de l'Adour during the Premiere Ecole Franco-Espagnole: Groupes de tresses et topologie en petite dimension in October 2009. This is intended to be an introductory survey through which we hope we can briefly outline the power of the concept monodromy as a common area for group theory, algebraic geometry, and topology of projective curves. The main classical results are stated in §2, where the Zariski-van Kampen method to compute a presentation for the fundamental group of the complement to projective plane curves is presented. In §1 these results are prefaced with a review of basic concepts like fundamental groups, locally trivial fibrations, branched and unbranched coverings and a first peek at monodromy. Descriptions of the main motivations that have lead mathematicians to study these objects are included throughout this first chapter. Finally, additional tools and further results that are direct applications of braid monodromy will be considered in §3. While not all proofs are included, we do provide either originals or simplified versions of those that are relevant in the sense that they exhibit the techniques that are most used in this context and lead to a better understanding of the main concepts discussed in this survey. Nothing here is hence original, other than an attempt to bring together different results and points of view. It goes without saying that this is not the first, and hopefully not the last, survey on the topic. For other approaches to braid monodromy we refer to the following beautifully-written papers [73, 20, 6]. We finally wish to thank the organizers and the referee for their patience and understanding in the process of writing and correcting these notes.