The kth Fitting ideal of the Alexander invariant
B of an arrangement [Ascr ] of n complex hyperplanes defines a
characteristic subvariety, Vk([Ascr ]), of the algebraic torus
([Copf ]*)n. In the combinatorially determined case
where B decomposes as a direct sum
of local Alexander invariants, we obtain a complete description of
Vk([Ascr ]). For any
arrangement [Ascr ], we show that the tangent cone at the identity of this variety coincides
with [Rscr ]1k(A), one of the cohomology
support loci of the Orlik–Solomon algebra.
Using work of Arapura [1], we conclude that all irreducible
components of Vk([Ascr ]) which pass through the identity
element of ([Copf ]*)n are combinatorially determined,
and that [Rscr ]1k(A) is the union of a subspace
arrangement in [Copf ]n, thereby resolving a
conjecture of Falk [11]. We use these results to study the
reflection arrangements associated to monomial groups.
Characteristic varieties of hypersurface complements