Critical points and resonance of hyperplane arrangements

TL;DR

This paper explores the relationship between the critical points of a master function associated with hyperplane arrangements, the geometry of the critical set, and resonance conditions, revealing bounds on the critical set’s codimension.

Contribution

It establishes a link between resonance in certain dimensions and the codimension of the critical set for specific classes of arrangements, including free and rank 3 arrangements.

Findings

Resonance in dimension p implies the critical set has codimension at most p.

The results apply to all free arrangements.

The findings include all rank 3 arrangements.

Abstract

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