Proceedings of the American Mathematical Society

Author(s): Takuro Abe; Hiroaki Terao; Masahiko Yoshinaga
Journal: Proc. Amer. Math. Soc. 137 (2009), 1405-1410.
MSC (2000): Primary 32S22
Posted: November 5, 2008
Retrieve article in: PDF

Abstract: A central arrangement $\mathcal{A}$ of hyperplanes in an $\ell$ -dimensional vector space

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.

1.: T. Abe, Free and non-free multiplicity on the deleted arrangement. Proc. Japan Acad. Ser. A 83 (2007), no. 7, 99-103. MR 2361419
2.: T. Abe, K. Nuida and Y. Numata, Bicolor-eliminable graphs and free multiplicities on the braid arrangement. arXiv:0712.4110.
3.: T. Abe, H. Terao and M. Wakefield, The characteristic polynomial of a multiarrangement. Adv. in Math. 215 (2007), 825-838. MR 2355609
4.: T. Abe, H. Terao and M. Wakefield, The Euler multiplicity and addition-deletion theorems for multiarrangements. J. London Math. Soc. 77 (2008), no. 2, 335-348. MR 2400395
5.: T. Abe and M. Yoshinaga, Coxeter multiarrangements with quasi-constant multiplicities. arXiv:0708.3228.
6.: P. Orlik and H. Terao, Arrangements of hyperplanes. Grundlehren der Mathematischen Wissenschaften, 300. Springer-Verlag, Berlin, 1992. MR 1217488 (94e:52014)
7.: J. G. Oxley, Matroid Theory. Oxford University Press, New York, 1992. MR 1207587 (94d:05033)
8.: W. T. Tutte, Lectures on matroids. J. Res. Nat. Bur. Standards Sect. B 69B (1965), 1-47. MR 0179781 (31:4023)
9.: M. Yoshinaga, Characterization of a free arrangement and conjecture of Edelman and Reiner. Invent. Math. 157 (2004), no. 2, 449-454. MR 2077250 (2005d:52044)
10.: M. Yoshinaga, On the freeness of 3-arrangements. Bull. London. Math. Soc. 37 (2005), no. 1, 126-134. MR 2105827 (2005i:52030)
11.: M. Yoshinaga, On the extendability of free multiarrangements. arXiv:0710.5044.
12.: G. M. Ziegler, Multiarrangements of hyperplanes and their freeness. Singularities (Iowa City, IA, 1986), 345-359, Contemp. Math., 90, Amer. Math. Soc., Providence, RI, 1989. MR 1000610 (90e:32015)

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32S22

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: 10.1090/S0002-9939-08-09755-4
PII: S 0002-9939(08)09755-4
Received by editor(s): May 16, 2008
Posted: November 5, 2008
Communicated by: Martin Lorenz
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6826 (e) ISSN 0002-9939 (p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS