Inductive Freeness of Ziegler's Canonical Multiderivations

Torsten Hoge, Gerhard Roehrle

Published 2022-04-20Version 1

Let $\mathcal A$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction $\mathcal A"$ of $\mathcal A$ to any hyperplane endowed with the natural multiplicity $\kappa$ is then a free multiarrangement $(\mathcal A",\kappa)$. The aim of this paper is to prove an analogue of Ziegler's theorem for the stronger notion of inductive freeness: if $\mathcal A$ is inductively free, then so is the multiarrangement $(\mathcal A",\kappa)$. In a related result we derive that if a deletion $\mathcal A'$ of $\mathcal A$ is free and the corresponding restriction $\mathcal A"$ is inductively free, then so is $(\mathcal A",\kappa)$ -- irrespective of the freeness of $\mathcal A$. In addition, we show counterparts of the latter kind for additive and recursive freeness.