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

TL;DR

This paper explores the combinatorial and algebraic properties of nets in the projective plane, linking incidence structures with Alexander duality of associated monomial ideals to gain new insights.

Contribution

It introduces a novel approach connecting nets in projective planes with Alexander duality applied to monomial ideals derived from matroid flats.

Findings

Alexander duality reveals structural properties of nets

Monomial ideals encode incidence relations in line arrangements

New algebraic tools for studying combinatorial configurations

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.