Abstract. The k th Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, V k (A), of the algebraic torus (C * ) n . In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of V k (A). For any arrangement A, we show that the tangent cone at the identity of this variety coincides with R 1 k (A), one of the cohomology support loci of the Orlik-Solomon algebra. Using work of Arapura [1] and Libgober [18], we conclude that all positive-dimensional components of V k (A) are combinatorially determined, and that R 1 k (A) is the union of a subspace arrangement in C n , thereby resolving a conjecture of Falk [11]. We use these results to study the reflection arrangements associated to monomial groups.

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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.

Abstract. We use stratified Morse theory to construct a complex to compute the cohomology of the complement of a hyperplane arrangement with coefficients in a complex rank one local system. The linearization of this complex is shown to be the Aomoto complex of the arrangement. Using this result, we establish the relationship between the cohomology support loci of the complement and the resonance varieties of the Orlik-Solomon algebra for any arrangement, and show that the latter are unions of subspace arrangements in general, resolving a conjecture of Falk. We also obtain lower bounds for the local system Betti numbers in terms of those of the Orlik-Solomon algebra, recovering a result of Libgober and Yuzvinsky. For certain local systems, our results provide new combinatorial upper bounds on the local system Betti numbers. These upper bounds enable us to prove that in non-resonant systems the cohomology is concentrated in the top dimension, without using resolution of singularities.