Critical Points and Resonance of Hyperplane Arrangements

Abstract

Abstract If is a master function corresponding to a hyperplane arrangement 𝒜 and a collection of weights ⋋, we investigate the relationship between the critical set of , the variety defined by the vanishing of the one-form ⩊⋋ = d log , and the resonance of ⋋. For arrangements satisfying certain conditions, we show that if ⋋ is resonant in dimension p, then the critical set of has codimension at most p. These include all free arrangements and all rank 3 arrangements.