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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 tofreeness derived via a connection to multivariate spline theory.

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