Free and Non-free Multiplicities on the $A_3$ Arrangement

TL;DR

This paper classifies all free and non-free multiplicities on the $A_3$ arrangement, revealing that free cases belong to two known families, using a novel homological obstruction linked to multivariate spline theory.

Contribution

It provides a complete classification of free multiplicities on the $A_3$ arrangement with a new homological obstruction method.

Findings

All free multiplicities on $A_3$ are in two known families.

Introduces a new homological obstruction to determine freeness.

Connects freeness classification to multivariate spline theory.

Abstract

We give a complete classification of free and non-free multiplicities on the $A_3$ braid arrangement. Namely, we show that all free multiplicities on $A_3$ fall into two families that have been identified by Abe-Terao-Wakefield (2007) and Abe-Nuida-Numata (2009). The main tool is a new homological obstruction to freeness derived via a connection to multivariate spline theory.