Inductive Freeness of Ziegler’s Canonical Multiderivations

Abstract

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