Jumping coefficients and spectrum of a hyperplane arrangement

By Nero Budur, Morihiko Saito. 2009

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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 Mustaţǎ. 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 Mustaţǎ’s formula for the spectrum.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7ba2790f9e44dfc15bf15e04a9054c9c59b2f06d