On Universal derivations for multiarrangements

TL;DR

This paper develops criteria for universal derivations in multiarrangements, characterizes them for 2-multiarrangements, and provides the first example of a non-reflection arrangement with a universal derivation different from the Euler derivation.

Contribution

It introduces a criterion for universal derivations and characterizes them for 2-multiarrangements, including new examples beyond reflection arrangements.

Findings

Provided a criterion for universal derivations.

Characterized universal derivations for 2-multiarrangements.

Presented the first non-reflection arrangement with a distinct universal derivation.

Abstract

The study of universal derivations for arbitrary multiarrangements and multiplicity functions was initiated by Abe, R\"ohrle, Stump, and Yoshinaga in 2024 which focused on arrangements arising from (well-generated) reflection groups. In this paper we provide a criterion for determining whether a derivation is universal along with a characterization of universal derivations for arbitrary 2-multiarrangements. As an application we give descriptions of universal derivations for several multiarrangements, including the so-called deleted $A_3$ arrangement. This is the first known example of a non-reflection arrangement that admits a universal derivation distinct from the Euler derivation.