Inductive Freeness of Ziegler's Canonical Multiderivations

Abstract

Let $\mathcal{A}$ be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction ${\mathcal{A}}^{\prime \prime }$ of $\mathcal{A}$ to any hyperplane endowed with the natural multiplicity $\kappa$ is then a free multiarrangement $({\mathcal{A}}^{\prime \prime },\kappa )$. 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 $({\mathcal{A}}^{\prime \prime },\kappa )$. In a related result we derive that if a deletion ${\mathcal{A}}^{\prime }$ of $\mathcal{A}$ is free and the corresponding restriction ${\mathcal{A}}^{\prime \prime }$ is inductively free, then so is $({\mathcal{A}}^{\prime \prime },\kappa )$ -- irrespective of the freeness of $\mathcal{A}$. In addition, we show counterparts of the latter kind for additive and recursive freeness.

20 pages. v2: Expanded Remark 1.4; added Examples 1.5 and 1.15; v3: updated bibliographic information, to appear in Discrete & Computational Geometry. arXiv admin note: text overlap with arXiv:1705.02767