On the Jacobian algebras of Ziegler pairs of plane arrangements

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 $\mathbb{P}^2$.

• The paper demonstrates that Ziegler pairs with identical intersection lattices can have distinct graded Betti numbers and Hilbert invariants in their Jacobian algebras.
• It employs coning constructions and explicit computational examples to reveal that classical combinatorial data fail to predict key algebraic resolutions.
• The study connects moduli space analysis, elliptic matroids, and hierarchies of free arrangements, offering new tools for classifying algebraic pathologies.

Jacobian Algebras and Ziegler Pairs of Plane Arrangements

Introduction and Problem Setting

This paper provides a systematic analysis of Jacobian (Milnor) algebras associated to Ziegler pairs of plane arrangements in projective 3-space. A Ziegler pair comprises two arrangements $:f=0$ and $':f'=0$ in $\mathbb{P}^3$ with isomorphic intersection lattices but distinct graded Betti numbers in the minimal resolutions of their Jacobian algebras. The primary motivation is to elucidate the subtle relationship between the combinatorics of an arrangement (encoded by its intersection lattice) and the algebraic invariants provided by its Jacobian algebra and associated resolutions.

Contrary to prior expectations from the case of line arrangements in $\mathbb{P}^2$, the authors demonstrate that in $\mathbb{P}^3$, neither the Hilbert polynomial, the Hilbert function, the minimal degree of a Jacobian syzygy, nor the finer structure of the minimal resolution is determined by the intersection lattice or the topology of the complement.

Main Definitions and Properties

The paper formalizes several algebraic and geometric properties for Ziegler pairs $(\mathcal{A},\mathcal{A}')$:

• (HP): The Jacobian algebras $M(f)$ and $M(f')$ share the same Hilbert polynomial.
• (HF): $M(f)$ and $M(f')$ possess identical Hilbert functions.
• (MDR): The minimal degree of a first Jacobian syzygy (mdr) coincides for both.
• (SPEC) / (SPEC$':f'=0$0): One arrangement is a (possibly nontrivial) specialization of the other, possibly with constant Betti data after the specialization process.

The authors provide explicit constructions and counterexamples for pairs satisfying some, but not all, of these criteria.

Coning Construction and Transfer of Phenomena

A major technical contribution is the "cone" operation. By taking cones over Ziegler pairs of line arrangements in $':f'=0$1, they transfer non-determinacy phenomena to higher dimensions. Specifically, if a pair of line arrangements share the intersection lattice but differ in their Hilbert functions or minimal resolutions, then their cones have isomorphic plane arrangement lattices in $':f'=0$2 but, crucially, their Jacobian algebras differ in Hilbert polynomial and Betti data.

Further, a general recursive result is established: $':f'=0$3-fold cones of isomorphic arrangements in $':f'=0$4 yield isomorphic intersection lattices in $':f'=0$5, but again with potentially distinct algebraic invariants.

Explicit Examples and Graded Betti Data

Several explicit, computationally detailed examples are provided. The most instructive case involves arrangements $':f'=0$6 and $':f'=0$7 in $':f'=0$8 whose intersection lattices are isomorphic but whose Jacobian algebras yield

$':f'=0$9

demonstrating sharp failure of combinatorial determination. Their minimal graded Betti data differ, and the discrepancy is rooted in the associated pair of line arrangements in $\mathbb{P}^3$0, for which full Hilbert series and syzygy structure are computed.

Another set of examples—constructed via generic hyperplane sections of higher-dimensional cones—explicitly realizes Ziegler pairs not interpretable as cones, thereby demonstrating that the phenomenon is not an artifact of the coning process.

Moduli Spaces and Rigidity

The paper analyzes the realization spaces (moduli) of arrangements with a given intersection lattice. For one nontrivial class, the realization space is a dense open subset of $\mathbb{P}^3$1. By explicit parameterization, it is shown that the locus where the Betti data change is typically of positive dimension, and that nontrivial families of arrangements with identical lattices but different algebraic types exist even inside high-dimensional moduli spaces.

Elliptic Matroids and Modular Interpretation

Beyond ad hoc examples, the authors examine a family of so-called elliptic matroids $\mathbb{P}^3$2 (for $\mathbb{P}^3$3), whose realizations correspond (up to projective equivalence) to moduli of point arrangements on plane cubic curves. For particular $\mathbb{P}^3$4, such as $\mathbb{P}^3$5 or $\mathbb{P}^3$6, the realization space is isomorphic to the modular curve $\mathbb{P}^3$7 (see "Modular curves $\mathbb{P}^3$8 as moduli spaces of point arrangements and applications" (Borisov et al., 2024)). In these families, despite the strong number-theoretic structure, the Betti data of Jacobian algebras can still vary in families with constant combinatorics.

Tameness and Hierarchies of Free/Nearly Free Arrangements

For cones over plane curves, the authors relate the module-theoretic property of tameness (in the sense of logarithmic vector fields) to that of the underlying curve, showing precise transfer of freeness, nearly-free, and strictly plus-one-generated status, following the context of the new hierarchy for plane curves ("A new hierarchy for complex plane curves" (Dimca et al., 13 Jan 2026, Dimca et al., 10 Feb 2026)).

This establishes that, at least for cones, the homological structure of the Jacobian algebra is tightly controlled by that of the base curve, and the cone operation does not introduce novel algebraic pathologies in this hierarchy.

Theoretical and Practical Implications

These findings definitively answer several open questions concerning the extent to which combinatorial data (intersection lattice, topology of the complement, Milnor fiber) determine the algebraic resolutions associated with arrangements in $\mathbb{P}^3$9. The answer is negative in a strong sense: even within lattice-isotopic families and topologically equivalent pairs, the Hilbert polynomial, Hilbert function, and the full set of graded Betti numbers can and do vary.

The theoretical implication is that computational invariants of singular hypersurfaces cannot, in general, be predicted from the intersection lattice or the topology of the complement, necessitating more sophisticated geometric and algebraic methods for distinguishing arrangements. This impacts the study of characteristic varieties, resonance varieties, and related invariants in the theory of arrangements, in both combinatorial and algebro-geometric settings.

On the practical side, the explicit parameterizations and algorithmic approach provided here, coupled with symbolic computation methods (e.g., via OSCAR), facilitate systematic enumeration and classification of arrangement types, offering benchmarks for further explorations in computational algebraic geometry and combinatorial topology.

Conclusion

This work rigorously establishes and illustrates that, for plane arrangements in $\mathbb{P}^2$0, the Jacobian algebra and its associated graded Betti numbers are not determined by combinatorial or topological data derived from intersection lattices, even for arrangements that are lattice-isotopic and have homeomorphic complements. The paper provides new tools for constructing and classifying such pathologies, draws connections to the moduli of elliptic curve arrangements and modular curves, and elucidates the precise transfer of homological properties under coning. These results prompt a re-evaluation of which invariants are combinatorially determined and open directions for the interaction of arrangement theory, algebraic geometry, and singularity theory.