Characteristic varieties of arrangements
The kth Fitting ideal of the Alexander invariant B of an arrangement [script A] of n complex hyperplanes defines a characteristic subvariety, Vk([script A]), of the algebraic torus ([open face C]*)n. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of Vk([script A]). For any arrangement [script A], we show that the tangent cone at the identity of this variety coincides with [script R]1k(A), one of the cohomology support loci of the Orlik–Solomon algebra. Using work of Arapura , we conclude that all irreducible components of Vk([script A]) which pass through the identity element of ([open face C]*)n are combinatorially determined, and that [script R]1k(A) is the union of a subspace arrangement in [open face C]n, thereby resolving a conjecture of Falk . We use these results to study the reflection arrangements associated to monomial groups.(Received January 20 1998)
(Revised August 17 1998)
1 Partially supported by grant LEQSF(1996-99)-RD-A-04 from the Louisiana Board of Regents.
2 Partially supported by NSF grant DMS–9504833.