2009/03/23 by Nero Budur, Budur, Nero, Morihiko Saito +1 · 3 citations
Computer Science · Mathematics · #32S22 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Computational Geometry and Mesh Generation #FOS: Mathematics #Mathematics and Applications #math.AG #msc:32S22
paper · pdf · doi:10.48550/arxiv.0903.3839
30 pages, corrections are made after the suggestions of the referee
openalex publication_date 2009/03/23 · arxiv created 2009/08/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In an earlier version of this paper written by the second named author, we showed that the jumping coefficients of a hyperplane arrangement depend only on the combinatorial data of the arrangement as conjectured by Mustata. For this we proved a similar assertion on the spectrum. After this first proof was written, the first named author found a more conceptual proof using the Hirzebruch-Riemann-Roch theorem where the assertion on the jumping numbers was proved without reducing to that for the spectrum. In this paper we improve these methods and show that the jumping numbers and the spectrum are calculable in low dimensions without using a computer. In the reduced case we show that these depend only on fewer combinatorial data, and give completely explicit combinatorial formulas for the jumping coefficients and (part of) the spectrum in the case the ambient dimension is 3 or 4. We also give an analogue of Mustata's formula for the spectrum.