Braid monodromy of complex line arrangements

Abstract

Let V be the complex vector space C^l, \mathscr{A} an arrangement in V, i.e. a finite family of hyperplanes in V In [11], Moishezon associated to any algebraic plane curve \mathscr{C} of degree n a braid monodromy homomorphism θ F_s→B(n), where F_s is a free group, B(n) is the Artin braid group. In this paper, we will determine the braid monodromy for the case when \mathscr{C} is an arrangement \mathscr{A} of complex lines in C², using the notion of labyrinth of an arrangement. As a corollary we get the braid monodromy presentation for the fundamental group of the complement to the arrangement.