Braid Monodromy and Topology of Plane Curves

In this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve. Introduction Our purpose in this paper is to relate the topological embedding of algebraic curves to a refinement of a well-known invariant of curves such as braid monodromy. Roughly speaking, braid monodromy is defined for a triple (C , L , P), where C ⊂ P2 is a curve, L ⊂ P2 is a line not contained in C , and P ∈ L , as follows. Let us consider homogeneous coordinates [x : y : z] such that P = [0 : 1 : 0], L = {z = 0}, and C = { f (x, y, z) = 0}. Let d be the y-degree of C ; that is, d = degy( f (x, y, 1)). The pencil H of lines passing through P (and different from L) is parametrized by x ∈ C. By the theorem of continuity of roots, H determines a representation of a free group F on the braid group on d strings, which is called a braid monodromy of the triple (C , L , P). The free group F corresponds to the fundamental group of an r -punctured complex line, where the punctures come from the nongeneric elements of H with respect to C . The classical definition of braid monodromy refers to generic choices of L and P , for example, P / ∈ C and L transversal to C . In this work we allow certain nongeneric choices. Braid monodromy is a strong invariant of plane curves. It is fair to say that the main ideas that lead to this invariant have already been used in the classic works of O. Zariski [13] and E. van Kampen [6] to find the fundamental group of the complement of a curve. The first explicit definition of braid monodromy was made by DUKE MATHEMATICAL JOURNAL Vol. 118, No. 2, c © 2003 Received 17 May 2001. Revision received 14 May 2002. 2000 Mathematics Subject Classification. Primary 14D05, 14H30; Secondary 14H50, 20F36. Artal’s work partially supported by Dirección General de Enseñanza Superior grant number PB1997-0284-C0202 and Dirección General de Ciencia y Tecnologı́a grant number BFM-1488-CO2-02. Carmona’s work partially supported by Dirección General de Enseñanza Superior grant number PB1997-0284C02-02 and Dirección General de Ciencia y Tecnologı́a grant number BFM-1488-CO2-01. Cogolludo’s work partially supported by Dirección General de Enseñanza Superior grant number PB1997-0284C02-01 and Dirección General de Ciencia y Tecnologı́a grant number BFM-1488-CO2-02.