Critical points and resonance of hyperplane arrangements

Abstract

If Φλ is a master function corresponding to a hyperplane arrangement A and a collection of weights λ, we investigate the relationship between the critical set of Φλ, the variety defined by the vanishing of the one-form ωλ = d log and Φλ, the resonance of λ. For arrangements satisfying certain conditions, we show that if λ is resonant in dimension p, then the critical set of Φλ has codimension at most p. These include all free arrangements and all rank 3 arrangements. © Canadian Mathematical Society 2011.