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Mathematische Annalen

, Volume 373, Issue 1–2, pp 581–595 | Cite as

Deletion theorem and combinatorics of hyperplane arrangements

  • Takuro AbeEmail author
Article
  • 79 Downloads

Abstract

We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give a sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. As a corollary, we prove that whether a free arrangement has a free filtration is also combinatorial. The proof is based on the result about a minimal set of generators of a logarithmic derivation module of a multiarrangement which satisfies the \(b_2\)-equality.

Mathematics Subject Classification

32S22 52S35 

Notes

Acknowledgements

The author is grateful to the anonymous referees for several comments and suggestions to this article. The author is partially supported by JSPS Grant-in-Aid for Scientific Research (B) JP16H03924, and Grant-in-Aid for Exploratory Research JP16K13744. We are grateful to Michael DiPasquale for informing an example in Remark 3.6.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics for IndustryKyushu UniversityFukuokaJapan

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