Inductive Freeness of Ziegler’s Canonical Multiderivations

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