Characteristic varieties of arrangements

Details Characteristic varieties of arrangements Characteristic varieties of arrangements

1998-01-11

arxiv.org/abs/math/9801048v3

Mathematical Proceedings of the Cambridge Philosophical Society 127 (1999), no. 1, 33-53

Mathematics

Algebraic Geometry

LaTeX2e, 20 pages. A reference to Libgober's recent work in math.AG/9801070 is added. Several points are clarified, a new exam

Scientific paper

10.1017/S0305004199003576

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.

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