Characteristic varieties of arrangements

The kth 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-k(1)(A), one of the cohomology support loci of the Orlik-Solomon algebra. Using work of Arapura [1], we conclude that all irreducible components of V-k(A) which pass through the identity element of (C*)(n) are combinatorially determined, and that R-k(1)(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.