Characteristic varieties of arrangements
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 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.
Cite
@article{arxiv.math/9801048,
title = {Characteristic varieties of arrangements},
author = {Daniel C. Cohen and Alexander I. Suciu},
journal= {arXiv preprint arXiv:math/9801048},
year = {2007}
}
Comments
LaTeX2e, 20 pages. A reference to Libgober's recent work in math.AG/9801070 is added. Several points are clarified, a new example is included