The braid monodromy of plane algebraic curves and hyperplane arrangements

Abstract

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 RandellArvola 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.Mathematics Subject Classification (1991). Primary 14H30, 20F36, 52B30; Secondary 05B35, 32S25, 57M05.

Torsion in Milnor fiber homology

“…The "braid monodromy" η : G(B) → P 2p may be determined using the techniques of [6], [7], and [3]. In fact, this map may be obtained by an appropriate modification of the calculation in [3, §2.2] of the braid monodromy of the full monomial arrangement defined by x 3 Q(A p ), which we now carry out.…”

Untitled

“…. , x n−1 via the Artin representation, see [8] for details and further references. In particular, We do not know whether (a) holds if C is replaced by a field K of positive characteristic.…”

On the genus of reducible surfaces and degenerations of surfaces

“…[3], [4], [6]). We expect that such degenerations will find even more applications, e.g., in the classification of surfaces with low invariants, in braid monodromy computations (see [5], [11], [12], [15]), in the birational classification of higher-dimensional varieties, etc.…”