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.

15pp