The Euler multiplicity and addition–deletion theorems for multiarrangements

Abstract

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.