Inductive and Recursive Freeness of Localizations of Multiarrangements

TL;DR

This paper extends the understanding of free multiarrangements by proving that inductive and recursive freeness are preserved under localizations and product constructions, and explores the implications for canonical classes.

Contribution

It introduces the extension of localizations to inductive and recursive freeness and demonstrates their compatibility with product constructions in multiarrangements.

Findings

Recursive free multiarrangements are compatible with product construction.

Some canonical free multiarrangements are not inductively free.

The class of inductively and recursively free multiarrangements is closed under localizations.

Abstract

The class of free multiarrangements is known to be closed under taking localizations. We extend this result to the stronger notions of inductive and recursive freeness. As an application, we prove that recursively free multiarrangements are compatible with the product construction for multiarrangements. In addition, we show how our results can be used to derive that some canonical classes of free multiarrangements are not inductively free.