Nets in $$\mathbb {P}^2$$ and Alexander Duality

Abstract

A net in $$\mathbb {P}^2$$ is a configuration of lines $$\mathcal {A}$$ and points X satisfying certain incidence properties. Nets appear in a variety of settings, ranging from quasigroups to combinatorial design to classification of Kac–Moody algebras to cohomology jump loci of hyperplane arrangements. For a matroid M and rank r, we associate a monomial ideal (a monomial variant of the Orlik–Solomon ideal) to the set of flats of M of rank $$\le r$$ . In the context of line arrangements in $$\mathbb {P}^2$$ , applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets.