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A Characterization of High Order Freeness for Product Arrangements and Answers to Holm’s Questions

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Abstract

An m-free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m-free imply (m + 1)-free for any arrangement? (2)Are all arrangements m-free for m large enough? In this paper, we characterize m-freeness for product arrangements, while we prove that all localizations of an m-free arrangement are m-free. From these results, we give answers to Holm’s questions.

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Acknowledgments

The authors would like to thank Noriyuki Abe and Mutsumi Saito for useful discussions about the proof of Claim 4.10. The first author is partially supported by JSPS Grant-in-Aid for Scientific Research (B) 16H03924, and Grant-in-Aid for Exploratory Research 16K13744. The second author is supported by JSPS Grant-in-Aid for Young Scientists (B) 16K17582.

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Authors and Affiliations

  1. Institute of Mathematics for Industry, Kyushu University, Fukuoka, 819-0395, Japan

    Takuro Abe

  2. Department of Mathematics, Nagoya Institute of Technology, Aichi, 466-8555, Japan

    Norihiro Nakashima

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  1. Takuro Abe
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  2. Norihiro Nakashima
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Correspondence to Norihiro Nakashima.

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Presented by: Vyjayanthi Chari

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Abe, T., Nakashima, N. A Characterization of High Order Freeness for Product Arrangements and Answers to Holm’s Questions. Algebr Represent Theor 24, 585–599 (2021). https://doi.org/10.1007/s10468-020-09961-1

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  • DOI: https://doi.org/10.1007/s10468-020-09961-1

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