Inductive Freeness of Ziegler's Canonical Multiderivations

TL;DR

This work proves that inductive freeness of a simple hyperplane arrangement ${\mathscr A}$ passes to its Ziegler restriction $({\mathscr A}'', \kappa)$, establishing the analogue of Ziegler’s freeness result for the stronger property of inductive freeness. The authors develop free filtrations of multiplicities, analyze Euler and Ziegler multiplicities along chains, and extend the core results to additive and recursive freeness, with broad consequences for Coxeter, supersolvable, and ideal-type arrangements. They also explore limits of these phenomena, showing, via counterexamples, that converses to the main theorem do not hold in general and that divisional freeness or concentrated multiplicities yield nuanced behavior. The techniques provide a robust framework for understanding freeness in multiarrangements and highlight the noncombinatorial nature of freeness in this context, while yielding practical criteria and applications in key classes of arrangements.

Abstract

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 $κ$ is then a free multiarrangement $(\mathcal A'',κ)$. 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 $(\mathcal A'',κ)$. 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 $(\mathcal A'',κ)$ -- irrespective of the freeness of $\mathcal A$. In addition, we show counterparts of the latter kind for additive and recursive freeness.

Theorems & Definitions (64)

• Definition 1.1
• Theorem 1.2: ziegler:multiarrangements
• Theorem 1.3
• Remark 1.4
• Example 1.5
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• Theorem 1.9
• Corollary 1.10