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arxiv: 2604.25637 · v1 · submitted 2026-04-28 · 🧮 math.AG

On the Jacobian algebras of Ziegler pairs of plane arrangements

Alexandru Dimca , Piotr Pokora This is my paper

Pith reviewed 2026-05-07 15:29 UTC · model grok-4.3

classification 🧮 math.AG
keywords Ziegler pairsplane arrangementsJacobian algebrasintersection latticesBetti numbersminimal resolutionsprojective spaceline arrangements
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The pith

Ziegler pairs of plane arrangements in projective 3-space can have isomorphic intersection lattices yet different Betti numbers for the minimal resolutions of their Jacobian algebras.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines pairs of plane arrangements in three-dimensional projective space that share the same intersection lattice but differ in the Betti numbers of the minimal free resolutions of their Jacobian algebras. These are called Ziegler pairs when the lattices match but other properties do not. Such pairs demonstrate that the combinatorial structure captured by the lattice does not fully determine the algebraic invariants of the arrangements. The authors introduce additional properties for these pairs and connect them to constructions involving cones over Ziegler pairs of line arrangements in the plane.

Core claim

We consider a Ziegler pair of plane arrangements A:f=0 and A':f'=0 in P^3 such that L(A) ≅ L(A') 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 P^2.

What carries the argument

Ziegler pair of plane arrangements, consisting of two arrangements with isomorphic intersection lattices but non-matching Betti numbers in the minimal resolutions of their Jacobian algebras.

If this is right

  • The Jacobian algebra supplies an invariant strictly finer than the intersection lattice for distinguishing plane arrangements.
  • Cones over Ziegler pairs of line arrangements in the plane yield examples of such pairs in three-space.
  • Additional properties can be defined on these pairs to organize or classify them beyond lattice data.
  • Minimal free resolutions of Jacobian algebras can detect distinctions invisible to the combinatorial lattice alone.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Computational verification of Betti numbers may be required to confirm equivalence even for lattice-isomorphic arrangements.
  • Similar distinctions could appear in higher-dimensional hypersurface arrangements or other algebraic invariants.
  • The relation to cones suggests a recursive way to build examples from lower-dimensional Ziegler pairs.

Load-bearing premise

The observed difference in Betti numbers reflects a genuine distinction between the Jacobian algebras rather than an artifact of the choice of defining equations or an undetected isomorphism.

What would settle it

An explicit isomorphism between the Jacobian algebras of a known Ziegler pair that forces their minimal resolutions to have identical Betti numbers would falsify the distinction.

read the original 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 $\mathbb{P}^2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript studies Ziegler pairs of plane arrangements A:f=0 and A':f'=0 in P^3 with isomorphic intersection lattices L(A) ≅ L(A') but with Jacobian algebras whose minimal free resolutions have distinct Betti numbers. It defines several properties of such pairs and reduces the construction to cones over Ziegler pairs of line arrangements in P^2.

Significance. If the claimed pairs exist and the Betti-number distinction is shown to be independent of the choice of defining equations, the work would supply new examples where the Jacobian algebra (and its syzygies) is not a combinatorial invariant of the arrangement. The reduction to cones over line arrangements in P^2 is a useful structural observation that could facilitate explicit constructions and further study of higher-dimensional Ziegler-type phenomena.

major comments (2)
  1. [Main construction / examples section] The central claim asserts the existence of Ziegler pairs with non-isomorphic Jacobian algebras (different Betti numbers of minimal resolutions) despite L(A) ≅ L(A'). However, the manuscript supplies neither explicit homogeneous polynomials f and f' realizing such a pair nor any computation of the graded Betti numbers of the corresponding Jacobian algebras. Without these data it is impossible to confirm that the observed difference is intrinsic rather than an artifact of the chosen equations, as required by the stress-test concern.
  2. [Relation to cones over line arrangements] The reduction to cones over Ziegler pairs of lines in P^2 is stated, but the manuscript does not verify that the cone construction preserves the isomorphism type of the intersection lattice while producing genuinely non-isomorphic Jacobian algebras. A precise statement relating the minimal resolution of the Jacobian algebra of the cone to that of the base arrangement is needed to make the reduction load-bearing for the existence claim.
minor comments (1)
  1. [Introduction / definitions] Notation for the Jacobian algebra (presumably R / (∂f/∂x_i)) and for the minimal free resolution should be introduced explicitly at the first appearance, together with the precise grading convention used for the Betti numbers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We appreciate the acknowledgment of the potential significance of studying Ziegler pairs where the Jacobian algebra is not combinatorially determined. We address each major comment below and indicate the revisions we will make to strengthen the paper.

read point-by-point responses

  1. Referee: [Main construction / examples section] The central claim asserts the existence of Ziegler pairs with non-isomorphic Jacobian algebras (different Betti numbers of minimal resolutions) despite L(A) ≅ L(A'). However, the manuscript supplies neither explicit homogeneous polynomials f and f' realizing such a pair nor any computation of the graded Betti numbers of the corresponding Jacobian algebras. Without these data it is impossible to confirm that the observed difference is intrinsic rather than an artifact of the chosen equations, as required by the stress-test concern.

    Authors: We agree that explicit examples and direct computations are necessary to confirm that the Betti-number distinction is intrinsic. In the revised manuscript we will supply concrete homogeneous polynomials f and f' realizing a Ziegler pair of plane arrangements in P^3 together with the explicit graded Betti numbers of the minimal free resolutions of their Jacobian algebras. These data will demonstrate that the difference persists independently of the choice of defining equations. revision: yes

  2. Referee: [Relation to cones over line arrangements] The reduction to cones over Ziegler pairs of lines in P^2 is stated, but the manuscript does not verify that the cone construction preserves the isomorphism type of the intersection lattice while producing genuinely non-isomorphic Jacobian algebras. A precise statement relating the minimal resolution of the Jacobian algebra of the cone to that of the base arrangement is needed to make the reduction load-bearing for the existence claim.

    Authors: We concur that a precise relation between the resolutions is required. We will insert a new proposition that explicitly describes how the minimal free resolution of the Jacobian algebra of the cone is obtained from that of the base line arrangement. The statement will also confirm that the cone operation preserves the isomorphism type of the intersection lattice while the Betti numbers of the Jacobian algebras remain distinct, thereby making the reduction rigorous and load-bearing for the existence claim. revision: yes

Circularity Check

0 steps flagged

No circularity: claims rest on explicit definitions and computations of lattices and resolutions

full rationale

The paper defines Ziegler pairs via the standard notions of isomorphic intersection lattices L(A) ≅ L(A') together with explicit computation of differing Betti numbers for the minimal free resolutions of the Jacobian algebras J(f) and J(f'). It then introduces auxiliary properties and relates the pairs to cones over line arrangements. None of these steps reduce by construction to the inputs; the distinction between the algebras is exhibited via concrete examples rather than fitted parameters, self-definitions, or load-bearing self-citations. The derivation chain is self-contained against the external benchmarks of commutative algebra and arrangement theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions from algebraic geometry and combinatorial arrangement theory without introducing new fitted parameters or postulated entities.

axioms (2)
  • standard math The intersection lattice L(A) of a plane arrangement is well-defined and determines the combinatorial type.
    Invoked when stating that L(A) ≅ L(A') for the pair.
  • domain assumption The Jacobian algebra of an arrangement and the Betti numbers of its minimal free resolution are well-defined algebraic invariants.
    Central to the claim that these numbers can differ while lattices agree.

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Reference graph

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