Extendability of the $B_2$-arrangement

TL;DR

This paper investigates the extendability of free multiarrangements, showing that unlike the type A_2 case, certain multiplicities in type B_2 do not admit free extensions, impacting higher rank cases.

Contribution

It demonstrates that for Coxeter arrangement of type B_2, infinitely many multiplicities lack free extensions, revealing limitations in extendability.

Findings

No free extension exists for certain multiplicities in B_2 arrangements.

Contrast with type A_2 where free extensions always exist.

Implications for free extension existence in higher rank arrangements.

Abstract

Let $(\mathscr{A},m)$ be a free multiarrangement, and let $\mathscr{E}$ be an extension of $(\mathscr{A},m)$. It is well known that if $\mathscr{A}$ is the Coxeter arrangement of type $A_2$, then a free extension of $(\mathscr{A},m)$ always exists. In this work, we demonstrate that if $\mathscr{A}$ is the Coxeter arrangement of type $B_2$, there exist infinitely many multiplicities for which no free extension of $(\mathscr{A},m)$ exists. This result has immediate consequences for the existence of free extensions in higher rank.