1998/01/31 by Daniel C. Cohen, Alexander I. Suciu · 9 citations
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #math.AG #math.CO #math.GT #msc:14H30 #msc:14M12 #msc:20F36 #msc:52B30 #msc:57M05
paper · pdf · doi:10.1017/s0305004199003576
published as Mathematical Proceedings of the Cambridge Philosophical Society 127 (1999), no. 1, 33-53 · 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
arxiv created 1998/04/11 · openalex publication_date 1999/07/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The k th Fitting ideal of the Alexander invariant B of an arrangement [Ascr ] of n complex hyperplanes defines a characteristic subvariety, V k ([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 V k ([Ascr ]). For any arrangement [Ascr ], we show that the tangent cone at the identity of this variety coincides with [Rscr ] 1 k ( A ), one of the cohomology support loci of the Orlik–Solomon algebra. Using work of Arapura [ 1 ], we conclude that all irreducible components of V k ([Ascr ]) which pass through the identity element of ([Copf ]*) n are combinatorially determined, and that [Rscr ] 1 k ( 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.