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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Totally free arrangements of hyperplanes


Authors: Takuro Abe, Hiroaki Terao and Masahiko Yoshinaga
Journal: Proc. Amer. Math. Soc. 137 (2009), 1405-1410
MSC (2000): Primary 32S22
Published electronically: November 5, 2008
MathSciNet review: 2465666
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Abstract: A central arrangement $ \mathcal{A}$ of hyperplanes in an $ \ell$-dimensional vector space $ V$ is said to be totally free if a multiarrangement $ (\mathcal{A}, m)$ is free for any multiplicity $ m : \mathcal{A}\rightarrow \Z_{> 0}$. It has been known that $ \mathcal{A}$ is totally free whenever $ \ell \le 2$. In this article, we will prove that there does not exist any totally free arrangement other than the obvious ones, that is, a product of one-dimensional arrangements and two-dimensional ones.


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Additional Information

Takuro Abe
Affiliation: Department of Mathematics, Hokkaido University, Kita-10, Nishi-8, Kita-Ku, Sapporo, 060-0810, Japan
Address at time of publication: Department of Mathematics, Kyoto University, Kitashirakawa-Oiwake-Cho, Sakyo-Ku, Kyoto, 606-8502, Japan
Email: abetaku@math.kyoto-u.ac.jp

Hiroaki Terao
Affiliation: Department of Mathematics, Hokkaido University, Kita-10, Nishi-8, Kita-Ku, Sapporo, 060-0810, Japan
Email: terao@math.sci.hokudai.ac.jp

Masahiko Yoshinaga
Affiliation: Department of Mathematics, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe, 657-8501, Japan
Email: myoshina@math.kobe-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09755-4
PII: S 0002-9939(08)09755-4
Received by editor(s): May 16, 2008
Published electronically: November 5, 2008
Communicated by: Martin Lorenz
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.