On the extendability of free multiarrangements

TL;DR

This paper investigates the conditions under which free multiarrangements can be extended from lower to higher ranks, providing classifications, examples, and applications in hyperplane arrangement theory.

Contribution

It characterizes extendability of free multiarrangements for a special class and presents applications including non-free arrangements and interpolations between known arrangements.

Findings

Not all free multiarrangements are extendable.

Identifies conditions for extendability in a special class.

Provides examples of non-free arrangements and interpolations.

Abstract

A free multiarrangement of rank $k$ is defined to be extendable if it is obtained from a simple rank $(k+1)$ free arrangement by the natural restriction to a hyperplane (in the sense of Ziegler). Not all free multiarrangements are extendable. We will discuss extendability of free multiarrangements for a special class. We also give two applications. The first is to produce totally non-free arrangements. The second is to give interpolating free arrangements between extended Shi and Catalan arrangements.