  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 complex
algebraic n-torus. 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 and Libgober, 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. We use these results to study the reflection arrangements
associated to monomial groups.
