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arXiv · 2208.01557

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

Nancy Abdallah·Hal Schenck
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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.

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Nancy Abdallah, Hal Schenck. 2022-08-02. Nets in $\mathbb P^2$ and Alexander Duality. https://arxiv.org/abs/2208.01557

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