A nine-line counterexample to a conjecture on the minimal degree of Jacobian relations

Alexandru Dimca; Piotr Pokora

arxiv: 2607.01985 · v1 · pith:AVYPFSEPnew · submitted 2026-07-02 · 🧮 math.AG · math.AC· math.CO

A nine-line counterexample to a conjecture on the minimal degree of Jacobian relations

Alexandru Dimca , Piotr Pokora This is my paper

Pith reviewed 2026-07-03 06:03 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.CO

keywords line arrangementsJacobian relationsTerao conjectureintersection latticesminimal degreeweak combinatoricsprojective plane

0 comments

The pith

Two nine-line arrangements in the plane have isomorphic intersection lattices but different minimal degrees of Jacobian relations, giving a counterexample to the generalized Terao conjecture.

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

The paper constructs two explicit arrangements of nine lines in the complex projective plane. Both arrangements share the same weak combinatorics consisting of nine double points, seven triple points, and one quadruple point, which implies that their intersection lattices are isomorphic. The defining polynomials f and g of these arrangements nevertheless satisfy mdr(f) equals 4 and mdr(g) equals 5. For total degree d equals 9 this yields mdr(f) less than d over 2, which directly violates the prediction of the generalized Terao conjecture that isomorphic lattices should force the minimal degree to be at least half the degree.

Core claim

The authors exhibit two explicit nine-line arrangements f and g in the complex projective plane whose intersection lattices are isomorphic, yet mdr(f) = 4 while mdr(g) = 5. Since d = 9, this yields mdr(f) < d/2, furnishing a counterexample to the Generalized Terao Conjecture.

What carries the argument

The minimal degree of Jacobian relations (mdr) of the defining polynomial of a line arrangement, which is the lowest degree of a non-trivial syzygy among the three partial derivatives.

If this is right

Isomorphic intersection lattices do not determine the minimal degree of Jacobian relations for line arrangements of degree 9.
The generalized Terao conjecture fails already for arrangements whose weak combinatorics is (9,7,1).
The classical Ziegler-Yuzvinsky pair is not the only counterexample up to weak combinatorics.
The value of mdr can differ by 1 even when the intersection lattice is held fixed.

Where Pith is reading between the lines

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

Explicit equations for the two arrangements allow independent verification of the mdr values by computer algebra.
The result suggests that additional combinatorial or geometric invariants beyond the intersection lattice may be needed to control mdr.
Similar discrepancies might appear for arrangements of other degrees that share the same lattice but differ in some finer invariant.

Load-bearing premise

The two constructed nine-line arrangements really do have isomorphic intersection lattices.

What would settle it

A computation of the syzygy module for the partial derivatives of the first polynomial that finds no relation in degree 4 would show that mdr(f) is not 4.

read the original abstract

We construct two arrangements of nine lines in the complex projective plane with isomorphic intersection lattices but with different minimal degrees of Jacobian relations. The common weak combinatorics is \[ (n_2,n_3,n_4)=(9,7,1), \] so the example is not the classical Ziegler-Yuzvinsky pair, whose weak combinatorics is $(n_{2},n_{3}) = (18,6)$. For the two defining equations $f$ and $g$ we prove \[ {\rm mdr}(f)=4,\qquad {\rm mdr}(g)=5. \] Since the degree is $d=9$, the first equality gives ${\rm mdr}(f)<d/2$. Hence the pair gives a counterexample to the Generalized Terao Conjecture.

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.

Desk Editor's Note private letter to a colleague

The paper claims two nine-line arrangements share an intersection lattice but have mdr 4 and 5, countering the generalized Terao conjecture, yet the lattice claim rests only on shared multiplicity counts.

read the letter

Two nine-line arrangements are presented with the same weak combinatorics but different minimal degrees of Jacobian relations, which would give a counterexample to the generalized Terao conjecture since one mdr is 4, below d/2 for d=9.

The work identifies a new multiplicity vector (9,7,1) distinct from the older Ziegler-Yuzvinsky case. The authors state they construct explicit polynomials f and g and prove the mdr values are 4 and 5 respectively.

The central soft spot is the assertion that the intersection lattices are isomorphic. The abstract ties this only to the shared counts of points of each multiplicity. Those counts do not fix the lattice; different incidence structures can produce the same numbers without the lattices matching. The stress-test note is on target here. A direct check of the matroid or the full set of lines through each point would be needed to secure the claim. Without that, the counterexample does not yet hold.

The mdr calculations are asserted as proved, but the abstract supplies no equations or steps, so the verification strength is not visible from the summary.

This is for specialists tracking Terao-type questions in arrangement theory. A reader focused on whether combinatorial data controls algebraic invariants would want to examine the example if the lattice step is solid.

It deserves peer review. The topic is live and a genuine counterexample would matter, so referees should be asked to verify the lattice isomorphism and the Jacobian relation computations.

Referee Report

1 major / 1 minor

Summary. The manuscript constructs two nine-line arrangements in the complex projective plane that realize the same weak combinatorics (n₂,n₃,n₄)=(9,7,1). It asserts that these arrangements have isomorphic intersection lattices, computes mdr(f)=4 for one defining polynomial and mdr(g)=5 for the other, and concludes that the pair furnishes a counterexample to the Generalized Terao Conjecture because mdr(f)<d/2 with d=9.

Significance. If the lattice isomorphism and the mdr computations are verified, the result would separate the minimal degree of Jacobian relations from the intersection lattice for arrangements of fixed degree, providing a concrete counterexample to a conjecture in combinatorial algebraic geometry.

major comments (1)

[Abstract] Abstract and introduction: the claim that the two arrangements have isomorphic intersection lattices rests solely on the shared weak combinatorics (n₂,n₃,n₄)=(9,7,1). Weak combinatorics records only point multiplicities and supplies no incidence data; distinct non-isomorphic lattices can realize the same multiplicity vector. The manuscript must supply an explicit verification of lattice isomorphism (e.g., via matroid comparison, incidence matrix, or direct computation of the intersection poset) for the central claim to hold.

minor comments (1)

[Abstract] The abstract states that mdr(f)=4 and mdr(g)=5 are proved but does not indicate where the explicit polynomials or the Jacobian relation computations appear; a brief pointer to the relevant section or equation would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit verification of the lattice isomorphism. We address the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract and introduction: the claim that the two arrangements have isomorphic intersection lattices rests solely on the shared weak combinatorics (n₂,n₃,n₄)=(9,7,1). Weak combinatorics records only point multiplicities and supplies no incidence data; distinct non-isomorphic lattices can realize the same multiplicity vector. The manuscript must supply an explicit verification of lattice isomorphism (e.g., via matroid comparison, incidence matrix, or direct computation of the intersection poset) for the central claim to hold.

Authors: We agree that shared weak combinatorics is insufficient to guarantee isomorphic intersection lattices, as distinct lattices can share the same multiplicity vector. In the revised manuscript we will add an explicit verification of the lattice isomorphism, for instance by exhibiting the incidence matrices of the two arrangements or by direct computation and comparison of their intersection posets. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit construction of distinct mdr values on claimed isomorphic lattices.

full rationale

The paper constructs two explicit nine-line arrangements, states their common weak combinatorics (n2,n3,n4)=(9,7,1), asserts the lattices are isomorphic, and proves mdr(f)=4, mdr(g)=5 directly for the defining equations. No derivation step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation chain; the counterexample rests on the geometric objects and their computed invariants rather than any renaming or ansatz smuggling. This is the normal case of a self-contained explicit example.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result relies on standard facts from algebraic geometry over the complex numbers and the definition of the minimal degree of Jacobian relations; no free parameters or new entities are introduced in the abstract.

axioms (1)

standard math The minimal degree of Jacobian relations is well-defined for a homogeneous polynomial defining a line arrangement in CP^2.
Invoked implicitly when stating mdr(f) and mdr(g).

pith-pipeline@v0.9.1-grok · 5668 in / 1144 out tokens · 25131 ms · 2026-07-03T06:03:18.018859+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

8 extracted references · 1 canonical work pages

[1]

Abe,Plus-one generated and next to free arrangements of hyperplanes, Int

T. Abe,Plus-one generated and next to free arrangements of hyperplanes, Int. Math. Res. Not. IMRN2021, no. 12, 9233–9261
[2]

T. Abe, A. Dimca and P. Pokora,A new hierarchy for complex plane curves, Canad. Math. Bull., published online 2025, 1–25.doi:10.4153/S0008439525101422

work page doi:10.4153/s0008439525101422 2025
[3]

Dimca,Hyperplane arrangements: an introduction, Universitext, Springer, Cham, 2017

A. Dimca,Hyperplane arrangements: an introduction, Universitext, Springer, Cham, 2017

2017
[4]

Dimca,On free curves and related open problems, Rev

A. Dimca,On free curves and related open problems, Rev. Roumaine Math. Pures Appl.69(2024), no. 2, 129–150

2024
[5]

Dimca and G

A. Dimca and G. Sticlaru,From Pascal’s theorem to the geometry of Ziegler’s line arrangements, J. Algebraic Combin.60(2024), 991–1009

2024
[6]

Dimca and G

A. Dimca and G. Sticlaru,Plus-one generated curves, Briançon-type polynomials and eigenscheme ideals, Result. Math.80(2)(2025), Paper No. 51, 22 p

2025
[7]

Yuzvinsky,The first two obstructions to the freeness of arrangements, Trans

S. Yuzvinsky,The first two obstructions to the freeness of arrangements, Trans. Amer. Math. Soc.335(1993), no. 1, 231–244

1993
[8]

G. M. Ziegler,Combinatorial construction of logarithmic differential forms, Adv. Math.76(1989), 116–154. Université Côte d’Azur, CNRS, LJAD, France and Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania Email address:Alexandru.DIMCA@univ-cotedazur.fr Department of Mathematics, UKEN Krakow, Podchora ¸żych 2, PL-30-084 Kra...

1989

[1] [1]

Abe,Plus-one generated and next to free arrangements of hyperplanes, Int

T. Abe,Plus-one generated and next to free arrangements of hyperplanes, Int. Math. Res. Not. IMRN2021, no. 12, 9233–9261

[2] [2]

T. Abe, A. Dimca and P. Pokora,A new hierarchy for complex plane curves, Canad. Math. Bull., published online 2025, 1–25.doi:10.4153/S0008439525101422

work page doi:10.4153/s0008439525101422 2025

[3] [3]

Dimca,Hyperplane arrangements: an introduction, Universitext, Springer, Cham, 2017

A. Dimca,Hyperplane arrangements: an introduction, Universitext, Springer, Cham, 2017

2017

[4] [4]

Dimca,On free curves and related open problems, Rev

A. Dimca,On free curves and related open problems, Rev. Roumaine Math. Pures Appl.69(2024), no. 2, 129–150

2024

[5] [5]

Dimca and G

A. Dimca and G. Sticlaru,From Pascal’s theorem to the geometry of Ziegler’s line arrangements, J. Algebraic Combin.60(2024), 991–1009

2024

[6] [6]

Dimca and G

A. Dimca and G. Sticlaru,Plus-one generated curves, Briançon-type polynomials and eigenscheme ideals, Result. Math.80(2)(2025), Paper No. 51, 22 p

2025

[7] [7]

Yuzvinsky,The first two obstructions to the freeness of arrangements, Trans

S. Yuzvinsky,The first two obstructions to the freeness of arrangements, Trans. Amer. Math. Soc.335(1993), no. 1, 231–244

1993

[8] [8]

G. M. Ziegler,Combinatorial construction of logarithmic differential forms, Adv. Math.76(1989), 116–154. Université Côte d’Azur, CNRS, LJAD, France and Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania Email address:Alexandru.DIMCA@univ-cotedazur.fr Department of Mathematics, UKEN Krakow, Podchora ¸żych 2, PL-30-084 Kra...

1989