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Mathematical Proceedings of the Cambridge Philosophical Society (1999), 127: 33-53 Cambridge University Press
doi:10.1017/S0305004199003576
Research Article

Characteristic varieties of arrangements

DANIEL C. COHEN a1 fn1 and ALEXANDER I. SUCIU a2 fn2
a1 Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803; e-mail: cohen@math.lsu.edu, http://math.lsu.edu/˜cohen
a2 Department of Mathematics, Northeastern University, Boston, MA 02115; e-mail: alexsuciu@neu.edu, http://www.math.neu.edu/˜suciu

Abstract

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 [1], 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 [11]. We use these results to study the reflection arrangements associated to monomial groups.

(Received January 20 1998)
(Revised August 17 1998)


fn1 Partially supported by grant LEQSF(1996-99)-RD-A-04 from the Louisiana Board of Regents.

fn2 Partially supported by NSF grant DMS–9504833.


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