On Ziegler pairs of line arrangements: from non-existence to abundance

Alexandru Dimca; Piotr Pokora

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arXiv · 2606.20421

On Ziegler pairs of line arrangements: from non-existence to abundance

Alexandru Dimca·Piotr Pokora

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Abstract

We study Ziegler pairs of line arrangements from both numerical and homological perspectives. We distinguish classical Ziegler pairs, for which the minimal degree of a Jacobian relation denoted by ${\rm mdr}$ differs, from the more general notion detected by distinct graded Betti data. First, we show that for arrangements of $d<9$ lines the intersection lattice determines ${\rm mdr}$, and hence classical Ziegler pairs do not occur in this range. Then we list several distinct Ziegler pairs with $d=10$ in the broader sense. In particular, we construct higher-degree examples with the same intersection lattice, the same minimal degree of a Jacobian relation, and the same Hilbert function of the Milnor algebra, but with different minimal graded free resolutions. Another rather surprising consequence is that being maximal Tjurina for a line arrangement is not determined by the combinatorics.

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Alexandru Dimca, Piotr Pokora. 2026-06-18. On Ziegler pairs of line arrangements: from non-existence to abundance. https://arxiv.org/abs/2606.20421

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