On the Jacobian algebras of Ziegler pairs of plane arrangements

TL;DR

This paper studies pairs of plane arrangements with identical intersection lattices but different Jacobian algebra resolutions, revealing subtle algebraic differences not captured by combinatorial data.

Contribution

It introduces properties for Ziegler pairs of plane arrangements and links their algebraic invariants to those of line arrangements in the projective plane.

Findings

Pairs have isomorphic intersection lattices but different Betti numbers in Jacobian algebra resolutions.

Properties of Ziegler pairs are related to cones over line arrangements in P^2.

Reveals algebraic distinctions beyond combinatorial similarities.

Abstract

We consider a Ziegler pair of plane arrangements, that is two plane arrangements $\mathcal{A}:f=0$ and $\mathcal{A}':f'=0$ in the projective space $\mathbb{P}^3$, such that the intersection lattices $L(\mathcal{A})$ and $L(\mathcal{A}')$ are isomorphic, but the Betti numbers of the minimal resolutions of their Jacobian algebras are not the same. We introduce several properties for such pairs and relate them to cones over Ziegler pairs of line arrangements in $\mathbb{P}^2$.