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Commentarii Mathematici Helvetici

, Volume 72, Issue 2, pp 285–315 | Cite as

The braid monodromy of plane algebraic curves and hyperplane arrangements

  • D. C. Cohen
  • A. I. Suciu
Article

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 Bn. 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.

Keywords. Braid monodromy, plane curve, hyperplane arrangement, fundamental group, polynomial cover, braid group, wiring diagram, intersection lattice. 

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Copyright information

© Birkhäuser Verlag, Basel 1997

Authors and Affiliations

  • D. C. Cohen
    • 1
  • A. I. Suciu
    • 2
  1. 1.Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803 USA, e-mail: cohen@math.lsu.eduUSA
  2. 2.Department of Mathematics, Northeastern University, Boston, MA 02115 USA, e-mail: alexsuciu@neu.eduUSA

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