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Multiple addition, deletion and restriction theorems for hyperplane arrangements


Authors: Takuro Abe and Hiroaki Terao
Journal: Proc. Amer. Math. Soc. 147 (2019), 4835-4845
MSC (2010): Primary 32S22; Secondary 52C35
DOI: https://doi.org/10.1090/proc/14592
Published electronically: June 10, 2019
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Abstract: In the study of free arrangements, the most useful result to construct/check free arrangements is the addition-deletion theorem in [J. Fac. Sci. Univ. Tokyo 27 (1980), 293-320]. Recently, the multiple version of the addition theorem was proved in [J. Eur. Math. Soc. 18 (2016), 1339-1348], called the multiple addition theorem (MAT), to prove the ideal-free theorem. The aim of this article is to give the deletion version of MAT, the multiple deletion theorem (MDT). Also, we can generalize MAT to get MAT2 from the viewpoint of our new proof. Moreover, we introduce the restriction version, a multiple restriction theorem (MRT). Applications of MAT2, including the combinatorial freeness of the extended Catalan arrangements, are given.


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Additional Information

Takuro Abe
Affiliation: Institute of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan
Email: abe@imi.kyushu-u.ac.jp

Hiroaki Terao
Affiliation: Department of Mathematics, Hokkaido University, Hokkaido 060-0808, Japan
Email: terao@math.sci.hokudai.ac.jp

DOI: https://doi.org/10.1090/proc/14592
Keywords: Hyperplane arrangements, free arrangements, addition-deletion and restriction theorems, Catalan arrangements
Received by editor(s): January 22, 2018
Received by editor(s) in revised form: February 10, 2019
Published electronically: June 10, 2019
Additional Notes: The first author was partially supported by KAKENHI, JSPS Grant-in-Aid for Scientific Research (B) 16H03924, and Grant-in-Aid for Exploratory Research 16K13744.
Communicated by: Patricia Hersh
Article copyright: © Copyright 2019 American Mathematical Society