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.