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arxiv: 2606.20421 · v2 · pith:PJQU7H6Pnew · submitted 2026-06-18 · 🧮 math.AG · math.AC· math.CO

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

Alexandru Dimca , Piotr Pokora This is my paper

Pith reviewed 2026-06-26 15:16 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.CO
keywords Ziegler pairsline arrangementsintersection latticeMilnor algebraJacobian relationfree resolutionexponent data
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The pith

The intersection lattice determines the exponent data for line arrangements with fewer than 9 lines, while six Ziegler pairs of 10-line arrangements share the lattice, Jacobian degree, and Milnor Hilbert function but differ in their minimal

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

This paper investigates Ziegler pairs of line arrangements from numerical and homological viewpoints. It first shows that the intersection lattice fixes the exponent data for every arrangement of fewer than nine lines. The authors then exhibit six distinct Ziegler pairs at ten lines. Each pair consists of arrangements that agree exactly on the intersection lattice, the minimal degree of a Jacobian relation, and the Hilbert function of the Milnor algebra, yet differ in their minimal graded free resolutions. A sympathetic reader cares because the examples mark the smallest degree at which the lattice no longer determines all the listed algebraic data.

Core claim

For arrangements of d<9 lines the intersection lattice determines the exponent data considered here. Six distinct Ziegler pairs with d=10 are listed. In particular, higher-degree examples are constructed 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.

What carries the argument

Ziegler pairs of line arrangements, which share an intersection lattice and certain numerical and homological data but are distinguished by their minimal graded free resolutions of the Milnor algebra.

If this is right

  • The intersection lattice determines the exponent data for every arrangement of d<9 lines.
  • Six distinct Ziegler pairs exist among arrangements of exactly ten lines.
  • Each such pair shares the intersection lattice, the minimal degree of a Jacobian relation, and the Hilbert function of the Milnor algebra.
  • The pairs are separated only by the structure of their minimal graded free resolutions.

Where Pith is reading between the lines

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

  • The number of such Ziegler pairs is likely to increase for degrees larger than 10.
  • Computing the full minimal resolution rather than only the Hilbert function becomes necessary for distinguishing arrangements once the number of lines reaches 10.
  • The transition from lattice-determined behavior to the existence of non-isomorphic pairs occurs between degrees 9 and 10.

Load-bearing premise

The six listed ten-line pairs can be shown to match exactly on intersection lattice, minimal Jacobian degree, and Milnor algebra Hilbert function while their minimal graded free resolutions differ.

What would settle it

A direct computation of the minimal graded free resolutions for any one of the six claimed pairs that shows the resolutions coincide would falsify the existence of that Ziegler pair.

read the original abstract

We study Ziegler pairs of line arrangements from both numerical and homological perspectives. First, we show that for arrangements of $d<9$ lines the intersection lattice determines the exponent data considered here. Then we list six distinct Ziegler pair with $d=10$. 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.

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

1 major / 1 minor

Summary. The paper studies Ziegler pairs of line arrangements. It proves that for d < 9 the intersection lattice determines the exponent data. It then constructs six distinct Ziegler pairs with d = 10 that share the same intersection lattice, the same minimal degree of a Jacobian relation, and the same Hilbert function of the Milnor algebra, yet have non-isomorphic minimal graded free resolutions.

Significance. If the explicit constructions and homological verifications hold, the result establishes a sharp transition from non-existence (d < 9) to abundance (six examples at d = 10) of Ziegler pairs. The examples demonstrate that minimal graded free resolutions can distinguish arrangements agreeing on lattice, Jacobian degree, and Milnor Hilbert function, supplying concrete higher-degree instances of this phenomenon.

major comments (1)
  1. [Section presenting the d=10 examples] The central claim that the six d=10 examples form Ziegler pairs rests on verification that their minimal graded free resolutions differ while the intersection lattice, minimal Jacobian degree, and Milnor-algebra Hilbert function coincide. The manuscript must supply the explicit Betti tables (or syzygy matrices) or a reproducible computational script (e.g., Macaulay2 output) that isolates the difference in graded ranks or shifts; without this data the homological distinction cannot be independently confirmed.
minor comments (1)
  1. [Abstract] The abstract contains the typographical error "six distinct Ziegler pair" (should be "pairs").

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the constructive suggestion regarding the presentation of the d=10 examples. We address the major comment below.

read point-by-point responses

  1. Referee: [Section presenting the d=10 examples] The central claim that the six d=10 examples form Ziegler pairs rests on verification that their minimal graded free resolutions differ while the intersection lattice, minimal Jacobian degree, and Milnor-algebra Hilbert function coincide. The manuscript must supply the explicit Betti tables (or syzygy matrices) or a reproducible computational script (e.g., Macaulay2 output) that isolates the difference in graded ranks or shifts; without this data the homological distinction cannot be independently confirmed.

    Authors: We agree that the manuscript would benefit from explicit data to allow independent verification of the differences in the minimal graded free resolutions. In the revised version we will add the Betti tables for each of the six arrangements. These tables will display the graded ranks and shifts that distinguish the resolutions while confirming that the intersection lattices, minimal Jacobian degrees, and Milnor-algebra Hilbert functions remain identical across the pairs. revision: yes

Circularity Check

0 steps flagged

No circularity: results rest on explicit lattice determinations and constructions

full rationale

The paper claims a determination result for d<9 (intersection lattice fixes exponent data) followed by explicit construction of six d=10 Ziegler pairs that match on lattice, Jacobian degree, and Milnor Hilbert function but differ in minimal free resolutions. No quoted equations or steps reduce a claimed prediction to a fitted input by construction, invoke load-bearing self-citations, smuggle ansatzes, or rename known results. The derivation chain is self-contained via direct combinatorial and homological comparisons rather than any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract alone; the paper appears to work within standard algebraic geometry of arrangements without introducing new postulated objects.

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discussion (0)

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Reference graph

Works this paper leans on

22 extracted references · 5 canonical work pages · 1 internal anchor

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