Inductive Freeness of Ziegler's Canonical Multiderivations

TL;DR

This paper extends Ziegler's theorem by proving that inductive freeness of a hyperplane arrangement implies the inductive freeness of its restriction with multiplicity, with additional results on deletions and other freeness notions.

Contribution

It establishes that inductive freeness is preserved under restrictions with multiplicity, generalizing Ziegler's theorem and exploring related freeness concepts.

Findings

Inductive freeness of arrangements implies inductive freeness of their restricted multiarrangements.

If a deletion is free and the restriction is inductively free, then the restriction with multiplicity is inductively free.

Counterparts are shown for additive and recursive freeness notions.

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 $\kappa$ 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.