Totally free arrangements of hyperplanes

Abe, Takuro; Terao, Hiroaki; Yoshinaga, Masahiko

doi:10.1090/S0002-9939-08-09755-4

Totally free arrangements of hyperplanes
HTML articles powered by AMS MathViewer

by Takuro Abe, Hiroaki Terao and Masahiko Yoshinaga

Proc. Amer. Math. Soc. 137 (2009), 1405-1410

DOI: https://doi.org/10.1090/S0002-9939-08-09755-4

Published electronically: November 5, 2008

PDF | Request permission

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 $m : \mathcal {A}\rightarrow \mathbb {Z} _{> 0}$. 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.

References

Takuro Abe, Free and non-free multiplicity on the deleted $A_3$ arrangement, Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 7, 99–103. MR 2361419
T. Abe, K. Nuida and Y. Numata, Bicolor-eliminable graphs and free multiplicities on the braid arrangement. arXiv:0712.4110.
Takuro Abe, Hiroaki Terao, and Max Wakefield, The characteristic polynomial of a multiarrangement, Adv. Math. 215 (2007), no. 2, 825–838. MR 2355609, DOI 10.1016/j.aim.2007.04.019
Takuro Abe, Hiroaki Terao, and Max Wakefield, The Euler multiplicity and addition-deletion theorems for multiarrangements, J. Lond. Math. Soc. (2) 77 (2008), no. 2, 335–348. MR 2400395, DOI 10.1112/jlms/jdm110
T. Abe and M. Yoshinaga, Coxeter multiarrangements with quasi-constant multiplicities. arXiv:0708.3228.
Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488, DOI 10.1007/978-3-662-02772-1
James G. Oxley, Matroid theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. MR 1207587
W. T. Tutte, Lectures on matroids, J. Res. Nat. Bur. Standards Sect. B 69B (1965), 1–47. MR 179781, DOI 10.6028/jres.069B.001
Masahiko Yoshinaga, Characterization of a free arrangement and conjecture of Edelman and Reiner, Invent. Math. 157 (2004), no. 2, 449–454. MR 2077250, DOI 10.1007/s00222-004-0359-2
Masahiko Yoshinaga, On the freeness of 3-arrangements, Bull. London Math. Soc. 37 (2005), no. 1, 126–134. MR 2105827, DOI 10.1112/S0024609304003704
M. Yoshinaga, On the extendability of free multiarrangements. arXiv:0710.5044.
Günter M. Ziegler, Multiarrangements of hyperplanes and their freeness, Singularities (Iowa City, IA, 1986) Contemp. Math., vol. 90, Amer. Math. Soc., Providence, RI, 1989, pp. 345–359. MR 1000610, DOI 10.1090/conm/090/1000610