The Braid Monodromy of Plane Algebraic Curves and Hyperplane Arrangements

Abstract

To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin's braid group B_n. Using Hansen's polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of the braid monodromy of an arrangement of complex affine hyperplanes, by means of an associated ``braided wiring diagram.'' The ensuing presentation of the fundamental group of the complement is shown to be Tietze-I equivalent to the Randell-Arvola presentation. Work of Libgober then implies that the complement of a line arrangement is homotopy equivalent to the 2-complex modeled on either of these presentations. Finally, we prove that the braid monodromy of a line arrangement determines the intersection lattice. Examples of Falk then show that the braid monodromy carries more information than the group of the complement, thereby answering a question of Libgober.

27 pages with 7 figures, author-supplied DVI file available at ftp://ftp.math.neu.edu/Pub/faculty/Suciu_Alex/papers/bmono.dvi AMSTeX v 2.1, pictex, edge-vertex-graphs

References in corpus (1)

• On the fundamental group of the complement of a complex hyperplane arrangement

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• Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
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• Line arrangements and direct sums of free groups
• On the connectivity of the realization spaces of line arrangements
• Parallel connections and bundles of arrangements
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• Cohomology rings and nilpotent quotients of real and complex arrangements
• Moduli Spaces of Arrangements of 10 Projective Lines with Quadruple Points
• On the homotopy type of the complement of an arrangement that is a 2-generic section of the parallel connection of an arrangement and a pencil of lines
• On graphic arrangement groups
• The homology groups of the Milnor fiber associated to a central arrangement of hyperplanes in $\CC^3$
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• Braid Monodromy Computation of Real Singular Curves
• Conjugation-free geometric presentations of fundamental groups of arrangements
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