For d<9 line arrangements the intersection lattice determines the exponent data; six Ziegler pairs with d=10 share the same lattice, Jacobian degree and Milnor algebra Hilbert function but have different minimal graded free resolutions.
On the Jacobian algebras of Ziegler pairs of plane arrangements
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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 $\mathbb{P}^2$.
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On Ziegler pairs of line arrangements: from non-existence to abundance
For d<9 line arrangements the intersection lattice determines the exponent data; six Ziegler pairs with d=10 share the same lattice, Jacobian degree and Milnor algebra Hilbert function but have different minimal graded free resolutions.