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Braid monodromy of algebraic curves. (English) Zbl 1254.32043
The author gives a rather complete introductory course on the braid monodromy of algebraic curves.
The first section is a review of the basic tools necessary for the understanding of the subject: the fundamental groupoid and the Seifert-van Kampen theorem to compute it on the one side and branched and unbranched coverings of algebraic varieties on the other side. The problem of existence of maximal Galois branched coverings is given some attention. Eventually the monodromy action on a locally trivial fibration is introduced and, in the special case where the fibre is a disc without a finite set of points, it is interpreted as the action of the braid group.
The previous tools are used to construct from \(\mathbb{P}^2 \setminus \mathcal{C}\), where \(\mathcal{C}\) is an algebraic curve, a locally trivial fibration on \(\mathbb{P}^1\) minus a finite set of points and to prove the Zariski-van Kampen theorem, which allows to compute \(\pi_1(\mathbb{P}^2 \setminus \mathcal{C})\). Many examples of applications are given.
The third section contains further topics of the theory of braid monodromy. Notably a discussion of local and global monodromy, Hurwitz classes and wiring diagrams.
The whole article is completed with many figures that help to understand every concept explained.

MSC:
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14H30 Coverings of curves, fundamental group
14H50 Plane and space curves
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
57M10 Covering spaces and low-dimensional topology
32S05 Local complex singularities
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