Braid Monodromy of Algebraic Curves

Cogolludo-Agustín, José Ignacio

Annales mathématiques Blaise Pascal, Tome 18 (2011) no. 1, pp. 141-209.

These are the notes from a one-week course on Braid Monodromy of Algebraic Curves given at the Université de Pau et des Pays de l’Adour during the Première 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.

Classification : 32S50, 14D05, 14H30, 14H50, 32S05, 57M10
Mots clés : Fundamental group, algebraic variety, quasi-projective group, pencil of hypersurfaces

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