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

This paper presents a counterexample to the Generalized Terao Conjecture by constructing two arrangements of nine lines in the complex projective plane with identical intersection lattices but distinct minimal degrees of Jacobian relations.

Original authors: Alexandru Dimca, Piotr Pokora

Published 2026-07-03

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CC BY 4.0

Original authors: Alexandru Dimca, Piotr Pokora

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect designing two different skyscrapers. In the world of mathematics, these skyscrapers are called "line arrangements" (specifically, nine lines drawn on a flat, infinite canvas called the complex projective plane).

The authors of this paper, Alexandru Dimca and Piotr Pokora, have built two very special skyscrapers, let's call them Building A and Building B.

The Setup: Two Buildings, Same Blueprint?

Here is the twist: If you look at the blueprint (which mathematicians call the "intersection lattice"), Building A and Building B look exactly the same.

They both have 9 floors (lines).
They both have exactly the same pattern of where the lines cross each other.
They both have one spot where four lines meet, seven spots where three lines meet, and nine spots where two lines meet.

If you were a detective looking only at the blueprint, you would swear these two buildings are identical twins.

The Discovery: A Hidden Difference

However, the authors discovered that despite having the same blueprint, the buildings have a hidden structural difference. They measured something called the "minimal degree of Jacobian relations" (let's call this the "Stability Score").

Think of the Stability Score as a measure of how much "wiggle room" or "flexibility" the structure has before it starts to wobble.

Building A has a Stability Score of 4.
Building B has a Stability Score of 5.

Even though the blueprints are identical, the internal physics of the buildings are different. One is slightly more rigid than the other.

The Big Claim: Breaking a Rule

For a long time, mathematicians believed in a rule called the Generalized Terao Conjecture. This rule was like a promise that said:

"If a building is flexible enough (specifically, if its Stability Score is less than half its total height), then you can predict its Stability Score just by looking at the blueprint."

In our case:

The buildings are 9 stories high.
Half of 9 is 4.5.
Building A has a score of 4, which is less than 4.5. So, it fits the "flexible" category.

According to the old rule, if you saw Building A's blueprint, you should be able to say, "Ah, this blueprint always produces a Stability Score of 4."

But the authors proved this rule wrong.
They showed that you can take that exact same blueprint and build a second version (Building B) that has a Stability Score of 5.

The Analogy: The Cookie Cutter

Imagine you have a cookie cutter shaped like a star.

The Conjecture said: "If you use this star cutter, the dough will always stretch exactly 4 inches before tearing."
The Counterexample says: "Actually, if you use this exact same star cutter, you can make one cookie that stretches 4 inches, and another cookie (made from the same dough, same cutter) that stretches 5 inches."

The shape of the cookie (the blueprint) doesn't tell you everything about how the dough behaves (the mathematical properties).

Why Does This Matter?

The authors didn't just say "we found a difference." They built the actual mathematical equations for both buildings to prove it exists.

They wrote down the exact recipe (equations) for Building A and Building B.
They ran the numbers (linear algebra) to prove Building A's score is 4 and Building B's is 5.
They showed that while the "blueprint" is identical, the "homological type" (the deep structural DNA of the building) is different. One is a "plus-one generated" type, and the other is a "type 2B" type.

The Conclusion

This paper is a "nine-line counterexample." It uses a specific case of nine lines to shatter a long-held belief in the mathematical community. It proves that combinatorics (counting and patterns) is not always enough to predict the deep algebraic properties of these geometric shapes.

In short: Same map, different territory. The old rule that "the map tells you everything about the terrain" has been proven false for this specific type of mathematical landscape.

Technical Summary: A Nine-Line Counterexample to a Conjecture on the Minimal Degree of Jacobian Relations

Problem Statement
The paper addresses the combinatorial determination of algebraic invariants associated with line arrangements in the complex projective plane $\mathbb{P}^2$ . Specifically, it investigates the Generalized Terao Conjecture (formulated by the first author in [4]), which posits that for a line arrangement $A$ of degree $d$ , if the minimal degree of a Jacobian relation, denoted $mdr(f)$, satisfies the strict inequality $mdr(f) < d/2$, then $mdr(f)$ is determined solely by the combinatorics of the arrangement (i.e., its intersection lattice).

While classical examples by Ziegler and Yuzvinsky demonstrate that arrangements with isomorphic intersection lattices can exhibit different Jacobian syzygy behaviors, these examples typically involve arrangements with only double and triple points. The conjecture suggests that the range $mdr(f) < d/2$ imposes a rigidity that forces combinatorial determination. The authors seek to test the validity of this conjecture in the "borderline" case of arrangements with higher multiplicity points.

Methodology
The authors construct two distinct arrangements of nine lines, denoted $A$ (defined by polynomial $f$ ) and $B$ (defined by polynomial $g$ ), over the field of rational numbers $\mathbb{Q}$ .

Combinatorial Verification: Using linear algebra (determinant calculations of line vectors), the authors verify that $A$ and $B$ possess isomorphic intersection lattices. Both arrangements share the same weak combinatorics: one quadruple point ( $n_4=1$ ), seven triple points ( $n_3=7$ ), and nine double points ( $n_2=9$ ), yielding a total Tjurina number $\tau = 46$ . This distinguishes them from the classical Ziegler–Yuzvinsky pair, which has different combinatorics.
Computational Algebra: The minimal degree of Jacobian relations, $mdr(f)$, is defined as the smallest integer $q$ such that the kernel of the linear map $\Phi_{f,q}: S^3_q \to S_{q+d-1}$ is non-trivial. The authors compute the ranks of these maps for $q = 0, \dots, 7$ for both polynomials $f$ and $g$ .
Homological Analysis: Beyond computing $mdr$, the authors determine the minimal graded free resolutions of the modules of logarithmic derivations $D_0(f)$ and $D_0(g)$ (equivalently, the Milnor algebras $M(f)$ and $M(g)$ ). They classify the arrangements according to the homological type hierarchy introduced in [2], specifically looking at the "type" $t(C) = d_1 + d_2 + 1 - d$ , where $d_i$ are the degrees of minimal generators.

Key Results
The computations yield the following definitive results:

Different Minimal Degrees: For arrangement $A$ , $mdr(f) = 4$. For arrangement $B$ , $mdr(g) = 5$.
Combinatorial Identity: Despite $mdr(f) \neq mdr(g)$ , the intersection lattices of $A$ and $B$ are isomorphic.
Violation of the Conjecture: Since the degree of the arrangements is $d=9$ , the condition $mdr(f) < d/2 $(i.e.,$ 4 < 4.5$) is satisfied for arrangement $A$ . However, arrangement $B$ shares the same combinatorics but has $mdr(g) = 5$. This directly contradicts Conjecture 1.1 (Conjecture 3.5 in [4]).
Homological Distinction: The arrangements differ significantly in their homological structure:
- Arrangement $A$ is of Type 1 (plus-one generated) with exponents $(4, 5, 6)$ .
- Arrangement $B$ is of Type 2B (four-syzygy curve) with exponents $(5, 5, 5, 5)$ .
- Consequently, their Milnor algebras have distinct minimal graded free resolutions and different Hilbert functions, although they stabilize to the same Tjurina number.
Freeness Defect: The authors note that the freeness defect $\nu$ is identical for both arrangements ( $\nu(A) = \nu(B) = 2$ ). Thus, the counterexample specifically targets the combinatoriality of $mdr$ in the specified range, rather than the freeness defect.

Significance and Claims
The paper claims to provide the first explicit counterexample to the Generalized Terao Conjecture in the range $mdr(f) < d/2$. The significance lies in:

Refuting Rigidity: Demonstrating that even in the range where $mdr(f)$ is expected to be combinatorially rigid (strictly less than half the degree), the invariant is not determined by the intersection lattice alone.
Novelty of Example: The example is distinct from the classical Ziegler–Yuzvinsky pair, utilizing a configuration with a quadruple point and a specific weak combinatorics $(9, 7, 1)$ , representing a "borderline" nine-line case.
Homological Sharpness: The counterexample is robust, as the two arrangements differ not only in the numerical invariant $mdr$ but also in their entire homological type and minimal resolutions.

The authors conclude that the conjecture is false as stated, highlighting that the transition range $d_1 \geq (d-2)/2$ allows for non-combinatorial behavior of $mdr$.