TY - JOUR
T1 - Deletion theorem and combinatorics of hyperplane arrangements
AU - Abe, Takuro
N1 - Funding Information:
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.
Publisher Copyright:
© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2019/2/8
Y1 - 2019/2/8
N2 - 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.
AB - 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.
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U2 - 10.1007/s00208-018-1713-9
DO - 10.1007/s00208-018-1713-9
M3 - Article
AN - SCOPUS:85048668730
SN - 0025-5831
VL - 373
SP - 581
EP - 595
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 1-2
ER -