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Characteristic varieties of arrangements

Published online by Cambridge University Press:  01 July 1999

DANIEL C. COHEN  and
ALEXANDER I. SUCIU
DANIEL C. COHEN
Affiliation:
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803; e-mail: [email protected], http://math.lsu.edu/˜cohen
ALEXANDER I. SUCIU
Affiliation:
Department of Mathematics, Northeastern University, Boston, MA 02115; e-mail: [email protected], http://www.math.neu.edu/˜suciu
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Abstract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 127 , Issue 1 , July 1999 , pp. 33 - 53
Copyright
The Cambridge Philosophical Society 1999

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