On the Jacobian algebras of Ziegler pairs of plane arrangements

We consider a Ziegler pair of plane arrangements, that is two plane arrangements $\mathcal{A}:f=0$ and ${\mathcal{A}}^{\mathrm{\prime }}:f^{\mathrm{\prime }}=0$ in the projective space ${\mathbb{P}}^3$, such that the intersection lattices $L(\mathcal{A})$ and $L({\mathcal{A}}^{\mathrm{\prime }})$ 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 ${\mathbb{P}}^2$.