Jumping Numbers of Hyperplane Arrangements

Abstract

Saito [8] 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.

Key Words:

• Arrangements
• Multiplier ideals
• Spectrum

2000 Mathematics Subject Classification:

• 32S22

ACKNOWLEDGMENTS

We would like to thank Saito, to whom we are indebted for the proof of Lemma 3.2, for sharing with us the results of his preprint [8] which was the inspiration behind this article, and for many comments. We also thank M. Mustaţă, M. Schulze, and Z. Teitler for useful discussions. The author was supported by the NSF grant DMS-0700360.

Notes

Communicated by L. Ein.