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

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.

• The paper presents a counterexample to the Generalized Terao Conjecture by exhibiting two nine-line arrangements with identical intersection lattices but different mdr values.
• The paper employs explicit construction and detailed syzygy computations to reveal that mdr is not strictly a combinatorial invariant.
• The paper refines the understanding of hyperplane arrangement invariants, urging a reexamination of the relationship between algebraic properties and combinatorial data.

Counterexample to the Generalized Terao Conjecture on Minimal Degree of Jacobian Relations

Introduction and Motivation

This work constructs a concrete counterexample to a strengthened version of Terao’s conjecture, specifically concerning the behavior of the minimal degree of Jacobian relations ($mdr$) for line arrangements in the complex projective plane. The minimal degree of a Jacobian relation, denoted $mdr(f)$ for a reduced plane curve $f=0$, is a fundamental algebraic invariant associated with the syzygy module $AR(f)$ of the Jacobian ideal. Conjecture 3.5 (“Generalized Terao Conjecture”) posited that for line arrangements, if $mdr(f) < d/2$ (where $d$ is the number of lines), then $mdr(f)$ is determined purely by the intersection lattice.

The authors present two explicit arrangements of nine lines, denoted by $f$ and $g$, with isomorphic intersection lattices, but with distinct $mdr$ values in the rigid range $mdr(f)$0. This construction directly falsifies the aforementioned conjecture.

Construction of the Counterexample

The arrangements $mdr(f)$1 and $mdr(f)$2 are defined by explicit equations specifying nine lines in $mdr(f)$3, with combinatorics characterized by

• $mdr(f)$4 double points,
• $mdr(f)$5 triple points,
• $mdr(f)$6 quadruple point.

Both arrangements share identical intersection lattices, verified via determinant calculus of the concurrent sets of lines. Therefore, the arrangements are combinatorially indistinguishable at the level of their intersection data. Importantly, these examples are not isomorphic to the classical Ziegler–Yuzvinsky pair, as their combinatorics explicitly exhibit a quadruple point, rather than only double and triple points.

Algebraic and Homological Analysis of $mdr(f)$7 and the Syzygy Module

Using explicit computation of the syzygy spaces $mdr(f)$8 for various degrees $mdr(f)$9 via linear algebra over $f=0$0, the authors determine:

• For $f=0$1 ($f=0$2).
• For $f=0$3.

Thus, despite $f=0$4 and $f=0$5 having isomorphic intersection lattices, their $f=0$6 invariants differ—contradicting the claim that $f=0$7 should be combinatorial in this range.

The minimal free resolutions of the Jacobian syzygy modules $f=0$8 and $f=0$9 further distinguish the arrangements homologically. For $AR(f)$0, $AR(f)$1 has generators in degrees $AR(f)$2 and a relation in degree $AR(f)$3 (“plus-one generated”, type 1). For $AR(f)$4, $AR(f)$5 has four generators in degree $AR(f)$6 with two relations in degree $AR(f)$7 (“type 2B”). This stratification corroborates the difference at the level of Betti tables and exponents.

Implications for the Generalized Terao Conjecture

The presented pair $AR(f)$8 directly disproves Conjecture 3.5: although $AR(f)$9 for $mdr(f) < d/2$0, $mdr(f) < d/2$1 is not determined by the intersection lattice, as $mdr(f) < d/2$2 provides a distinct value with the same lattice. Thus, combinatorial data encoded in the intersection lattice alone is insufficient to determine $mdr(f) < d/2$3 in this range.

Additionally, the analysis shows that the freeness defect $mdr(f) < d/2$4 (related to the total Tjurina number and $mdr(f) < d/2$5) remains invariant at $mdr(f) < d/2$6 for both arrangements, indicating that this counterexample is specific to $mdr(f) < d/2$7 and not to all invariants potentially conjectured to be combinatorial.

Theoretical and Practical Implications

This result sharpens the understanding of the limits of combinatorial invariance in the theory of hyperplane arrangements, particularly for the subtle behavior of logarithmic vector fields and their syzygy modules. The example falls within a narrow range—$mdr(f) < d/2$8—where the classical combinatorial-analytic correspondence fails, even though it is known to hold in more extreme ranges.

Practically, this indicates that algebraic invariants such as $mdr(f) < d/2$9 and the structure of the Milnor and Jacobian syzygy modules can encode geometric data invisible to the combinatorics of the intersection lattice. Therefore, the pursuit of combinatorial characterization of other freeness invariants in arrangements must account for such subtle algebraic phenomena.

Future work may involve classifying other such transition-range counterexamples, refining conjectures for arrangements with restricted intersection types, and understanding the implications for the module theory of logarithmic vector fields and Milnor algebras.

Conclusion

The paper establishes a concise counterexample to the Generalized Terao Conjecture regarding the minimal degree of Jacobian relations for line arrangements, demonstrating explicitly that $d$0 is not always a combinatorial invariant in the sub-$d$1 range. The example distinguishes not only $d$2 but also homological types of the associated syzygy modules, separating analytic invariants from combinatorial structure in a precise and computable way. This necessitates a reevaluation of conjectures linking the combinatorics of arrangements to their deeper algebraic invariants and calls for a more nuanced approach to the study of the relationship between intersection lattices and the algebraic structure of their defining equations.