Free and non-free multiplicities on the A3 arrangement

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.

Inequalities for free multi-braid arrangements

“…In this note we prove that a multiplicity in the balanced cone is free if and only if it is a free ANN multiplicity. This partially generalizes the recent classification of all free multiplicities on the A 3 braid arrangement [8], which is joint work of the author with Francisco, Mermin, and Schweig. To state our result more concretely we shall associate to the multi-braid arrangement (A ℓ , m) an edge-labeled complete graph (K ℓ+1 , m).…”

“…Remark 6.8. Conjecture 6.7 is proved for the A 3 braid arrangement in [8]. Using Macaulay2 [10], we have verified Conjecture 6.7 for many multiplicities on the A 4 arrangement.…”

“…There are four such sums, two of the form 2s + t and two of the form 2t + s. So, applying Theorem 1.1, if m 3 s,t is in the balanced cone of multiplicities, it is free if and only if 6(s − t) 2 ≤ 4 · 3, or |s − t| ≤ 1. In fact, using the classification from [8], it follows that m 3 s,t is a free multiplicity if and only if |s − t| ≤ 1 (regardless of whether m 3 s,t is in the balanced cone or not). Example 1.4.…”

“…• or a σ-mountain. We assume G = H and show that DV (G) > q G ℓ in each of these cases, where ℓ is one less than the number of vertices of G. The inequality DV (G) > 3q G can easily be verified by hand for each of the twelve graphs on four vertices which are not signed-eliminable (see Table 1); this is also done in [8,Corollary 6.2]. If G is a σ-cycle, σ-mountain, or σ-hill on (ℓ + 1) vertices we will show that DV(G) and q G are given by the formulas:…”

A homological characterization for freeness of multi-arrangements

“…Chain complexes having very similar properties to D • (A, m) appear in the theory of algebraic splines [10,27]; applying techniques of Schenck and Stiller [25,28] yields our main result, stated below. Weaker versions of this statement have been proved recently and used to classify free multiplicities on several rank three arrangements [15,13,14]. For simple arrangements, the forward direction of the first statement in Theorem 1.1 follows from work of Brandt and Terao [12].…”