Inductive Freeness of Ziegler’s Canonical Multiderivations

Abstract

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