Jumping numbers of hyperplane arrangements

TL;DR

This paper provides a combinatorial formula for the jumping numbers of hyperplane arrangements, confirming their dependence solely on combinatorial data and offering new insights into related spectral invariants.

Contribution

It introduces a formula for jumping numbers based on combinatorics and offers a new proof of their dependence on combinatorial structure.

Findings

Derived a combinatorial formula for jumping numbers

Provided a new proof of their combinatorial dependence

Connected jumping numbers to Hodge spectrum and multiplicities

Abstract

M. Saito recently proved that the jumping numbers of a hyperplane arrangement depend only on the combinatorics of the arrangement. However, a formula in terms of the combinatorial data was still missing. In this note, we give a formula and a different proof of the fact that the jumping numbers of a hyperplane arrangement depend only on the combinatorics. We also give a combinatorial formula for part of the Hodge spectrum and for the inner jumping multiplicities.