Note on resonance varieties

TL;DR

This paper investigates the irreducibility of resonance varieties in graded rings, especially Orlik-Solomon algebras, providing conditions for irreducibility and exploring their relation to hyperplane arrangements and Betti numbers.

Contribution

It establishes criteria for the irreducibility of the first resonance variety in Orlik-Solomon algebras and examines the conjecture relating Betti numbers to resonance varieties.

Findings

First resonance variety is irreducible for stable monomial ideals.

Irreducibility of the first resonance variety characterized by 2-linear resolution.

Counterexample shows the conjecture does not hold universally.

Abstract

We study the irreducibility of resonance varieties of graded rings over an exterior algebra E with particular attention to Orlik-Solomon algebras. We prove that for a stable monomial ideal in E the first resonance variety is irreducible. If J is an Orlik- Solomon ideal of an essential central hyperplane arrangement, then we show that its first resonance variety is irreducible if and only if the subideal of J generated by all degree 2 elements has a 2-linear resolution. As an application we characterize those hyperplane arrangements of rank less than or equal to 3 where J is componentwise linear. Higher resonance varieties are also considered. We prove results supporting a conjecture of Schenck-Suciu relating the Betti numbers of the linear strand of J and its first resonance variety. A counter example is constructed that this conjecture is not true for arbitrary graded ideals.