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Journal of the London Mathematical Society Advance Access published online on February 7, 2008
Journal of the London Mathematical Society, doi:10.1112/jlms/jdm110
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© 2008 London Mathematical Society

The Euler multiplicity and addition–deletion theorems for multiarrangements

Takuro Abe, Hiroaki Terao and Max Wakefield

Department of Mathematics
Hokkaido University
Sapporo 060-0810
Japan
abetaku{at}math.sci.hokudai.ac.jp
wakefield{at}math.sci.hokudai.ac.jp

The addition–deletion theorems for hyperplane arrangements, which were originally shown by Terao [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980) 293–320.], provide useful ways to construct examples of free arrangements. In this article, we prove addition–deletion theorems for multiarrangements. A key to the generalization is the definition of a new multiplicity, called the Euler multiplicity, of a restricted multiarrangement. We compute the Euler multiplicities in many cases. Then we apply the addition–deletion theorems to various arrangements, including supersolvable arrangements and the Coxeter arrangement of type A3, to construct free and non-free multiarrangements.

2000 Mathematics Subject Classification 32S22 (primary), 52C35 (secondary).

The first author is supported by the 21st Century COE Program ‘Mathematics of Nonlinear Structures via Singularities’, Hokkaido University. The second author is supported in part by the Japan Society for the Promotion of Science. The third author is supported by NSF grant # 0600893 and the NSF Japan program.

Received December 31, 2006; revised September 14, 2007;
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