Nets in P-2 and Alexander Duality

A net in P-2 is a configuration of lines 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 = r. In the context of line arrangements in P-2, applying Alexander duality to the resulting ideal yields insight into the combinatorial structure of nets.