Mathematische Annalen

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

Deletion theorem and combinatorics of hyperplane arrangements

  • Takuro AbeEmail author


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 



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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