Ziegler Pairs in Arrangement Theory

• Ziegler pairs are pairs of arrangements or plane curves sharing identical combinatorial structures while exhibiting different Jacobian invariants.
• They are classified into weak, classical, homological, and strong types based on variations in minimal Jacobian relations and graded free resolutions.
• The study of Ziegler pairs uncovers failures of combinatorial rigidity, impacts Terao-type conjectures, and extends to higher dimensions through coning and spectrum analysis.

Ziegler pairs are comparison objects used most prominently in the theory of arrangements and plane curves to isolate the gap between combinatorics and Jacobian-algebra data. In the arrangement-theoretic literature, the common core is that two objects have the same prescribed combinatorial structure—typically the same intersection lattice or the same weak combinatorics—but differ in syzygetic or homological invariants attached to the Jacobian ideal. Recent papers distinguish weak Ziegler pairs, Ziegler pairs, and strong Ziegler pairs, and extend the notion from line arrangements in $\mathbb{P}^2$ to conic-line arrangements, plane arrangements in $\mathbb{P}^3$, and higher-dimensional hyperplane arrangements (Lampa-Baczyńska et al., 2024, Dimca et al., 18 Jun 2026).

1. Core definitions and terminological variants

Recent papers use several related definitions rather than a single universal one. In each case, the invariant that is held fixed is combinatorial, while the invariant that is allowed to vary is homological.

Notion	Fixed data	Variable data
Weak Ziegler pair	Same weak combinatorics	Different $\operatorname{mdr}$
Ziegler pair (classical line-arrangement sense)	Isomorphic intersection lattices	Different $\operatorname{mdr}$
Ziegler pair (homological sense)	Isomorphic intersection lattices	Different graded Betti numbers or minimal graded free resolutions
Strong Ziegler pair	Same combinatorics or isomorphic intersection lattices	Non-isomorphic Milnor algebras or different minimal graded free resolutions

For reduced plane curves with smooth components, a weak Ziegler pair $(C_1,C_2)$ is defined by equality of weak combinatorics together with $$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$ (Lampa-Baczyńska et al., 2024). For line arrangements in $\mathbb{P}^2$, one paper recalls the classical definition: two arrangements form a Ziegler pair if their intersection lattices are isomorphic but their minimal degrees of Jacobian relations are different (Kühne et al., 2024). A later homological reformulation declares two line arrangements to be a Ziegler pair when the intersection lattices are isomorphic but the graded Betti numbers in the minimal graded free resolutions of their Jacobian algebras differ (Dimca et al., 18 Jun 2026). In plane-curve language, a strong Ziegler pair is a pair with equivalent combinatorics but non-isomorphic Milnor algebras, equivalently distinct graded $S$-modules of Jacobian syzygies (Bannai et al., 10 Sep 2025).

This definitional evolution reflects a shift from a numerical criterion based on $\operatorname{mdr}$ to a finer homological criterion based on the full minimal resolution. A plausible implication is that the phrase now names a hierarchy of failure modes for combinatorial determinacy rather than a single invariant.

2. Jacobian-algebra framework

Let $S=\mathbb{C}[x,y,z]$ for plane curves, or $\mathbb{P}^3$0 for plane arrangements in $\mathbb{P}^3$1. If $\mathbb{P}^3$2 is a reduced homogeneous polynomial defining a curve or arrangement, the Jacobian ideal is

$\mathbb{P}^3$3

in the plane-curve setting, and

$\mathbb{P}^3$4

for hypersurfaces in $\mathbb{P}^3$5. The Milnor, or Jacobian, algebra is

$\mathbb{P}^3$6

The module of Jacobian syzygies is

$\mathbb{P}^3$7

and the minimal degree of non-trivial Jacobian relations is

$\mathbb{P}^3$8

For $\mathbb{P}^3$9-syzygy curves, the first exponent $\operatorname{mdr}$0 satisfies $\operatorname{mdr}$1 (Lampa-Baczyńska et al., 2024).

In line-arrangement theory, the minimal graded free resolution of $\operatorname{mdr}$2 records the graded Betti numbers and the degree sequence of minimal generators of $\operatorname{mdr}$3. Those degrees are often referred to as exponents, with $\operatorname{mdr}$4 (Dimca et al., 18 Jun 2026). In the plane-arrangement setting in $\operatorname{mdr}$5, the minimal graded free resolution of $\operatorname{mdr}$6 has three nontrivial homological steps, and the corresponding degree sequences $\operatorname{mdr}$7, $\operatorname{mdr}$8, and $\operatorname{mdr}$9 determine the graded Betti numbers (Dimca et al., 28 Apr 2026).

The relevance of these objects to Ziegler pairs is direct. If two arrangements have the same combinatorics but different $\operatorname{mdr}$0, different degree sequences, different Hilbert functions, or different minimal resolutions of $\operatorname{mdr}$1, then the Jacobian algebra detects geometry not encoded in the combinatorial data. The stronger the invariant that differs, the stronger the corresponding Ziegler phenomenon.

3. Historical development in line arrangements

The classical point of departure is the 9-line example associated with Ziegler and Yuzvinsky: two line arrangements with the same intersection lattice but different minimal degrees of Jacobian relations,

$\operatorname{mdr}$2

Geometrically, one realization has six triple points on a smooth conic, while the other does not (Dimca et al., 18 Jun 2026).

A systematic low-degree analysis later showed that for arrangements of $\operatorname{mdr}$3 lines, the intersection lattice determines the exponent data considered there, so no nontrivial Ziegler pairs occur in that range (Dimca et al., 18 Jun 2026). This produces a threshold phenomenon: rigidity for $\operatorname{mdr}$4, non-rigidity beginning at $\operatorname{mdr}$5.

Beyond that threshold, the phenomenon becomes abundant. The same paper lists six distinct Ziegler pairs with $\operatorname{mdr}$6, including examples with identical $\operatorname{mdr}$7 and identical Hilbert functions but different minimal free resolutions. Its 11-line family is especially sharp: on an open set of a two-parameter family, the generic member has degree sequence

$\operatorname{mdr}$8

while a special member at

$\operatorname{mdr}$9

has degree sequence

$(C_1,C_2)$0

The two arrangements have the same intersection lattice, the same Tjurina number $(C_1,C_2)$1, the same Hilbert series, and the same $(C_1,C_2)$2, but different minimal graded free resolutions (Dimca et al., 18 Jun 2026).

This progression from the original 9-line example to families with property (HF) and property (MDR) shows that combinatorial equivalence can fail to determine not only the first Jacobian syzygy but also higher syzygetic structure. A plausible implication is that the minimal resolution of the Jacobian algebra is strictly finer than the standard numerical invariants traditionally used in arrangement theory.

4. Weak combinatorics, Terao-type conjectures, and realization spaces

For conic-line arrangements with ordinary singularities, weak combinatorics is encoded by

$(C_1,C_2)$3

where $(C_1,C_2)$4 is the number of lines, $(C_1,C_2)$5 the number of smooth conics, and $(C_1,C_2)$6 the number of $(C_1,C_2)$7-fold intersection points (Lampa-Baczyńska et al., 2024). In that setting, a weak Ziegler pair fixes this vector but allows $(C_1,C_2)$8 to vary. A foundational example consists of two conic-line arrangements over $(C_1,C_2)$9 with weak combinatorics

$$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$0

both nearly free, but with

$$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$1

and exponent triples

$$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$2

This is described as the first known weak Ziegler pair in that class and the first such example where both curves are nearly free (Lampa-Baczyńska et al., 2024).

The same tension appears in Terao-type questions. The Numerical Terao’s Conjecture asks whether weak combinatorics determines freeness under quasi-homogeneous hypotheses (Lampa-Baczyńska et al., 2024). For line arrangements, a smallest known counterexample uses $$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$3 lines: two arrangements with the same weak-combinatorics

$$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$4

one free with exponents $$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$5 and one plus-one generated with exponents $$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$6 (Kühne et al., 2024).

Singular matroid realization spaces provide another mechanism for producing Ziegler phenomena. Using rank-3 matroids on 12 elements with singular realization spaces, one paper constructs a Ziegler pair of degree $$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$7 line arrangements with weak-combinatorics

$$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$8

where the generic realization has $$\operatorname{mdr}(C_1)\neq \operatorname{mdr}(C_2)$$9 and a realization on the singular locus has $\mathbb{P}^2$0. The same paper constructs a strong Ziegler pair with weak-combinatorics

$\mathbb{P}^2$1

where one arrangement is 5-syzygy with exponents $\mathbb{P}^2$2 and the other is 3-syzygy with exponents $\mathbb{P}^2$3 (Kühne et al., 2024).

These examples show that weak combinatorics can fail already at the level of $\mathbb{P}^2$4, while full intersection-lattice data can fail at the level of the entire minimal resolution. They also indicate that singularities in realization spaces are a natural source of Ziegler behavior.

5. Cones, plane arrangements, and higher-dimensional proliferation

The passage from line arrangements in $\mathbb{P}^2$5 to plane arrangements in $\mathbb{P}^2$6 is governed by coning. If $\mathbb{P}^2$7 defines a reduced curve $\mathbb{P}^2$8, the cone is the surface

$\mathbb{P}^2$9

in $S$0 (Dimca et al., 28 Apr 2026). Coning preserves combinatorics in the sense that line arrangements with isomorphic intersection lattices yield plane arrangements with isomorphic intersection lattices. It also interacts sharply with Jacobian algebras: for certain line Ziegler pairs satisfying $S$1 and failing $S$2, the corresponding plane arrangements have different Hilbert polynomials. In a central example, the two cones have

$S$3

The same paper also gives a plane-arrangement Ziegler pair not arising as a cone, with Hilbert polynomials

$S$4

It follows that for plane arrangements in $S$5, the Hilbert polynomial $S$6 is determined neither by the intersection lattice nor by the topology of the complement or Milnor fibration (Dimca et al., 28 Apr 2026).

A parallel development concerns arbitrary dimension. Starting from a Ziegler pair in $S$7, one can add a combinatorially generic hyperplane. If

$S$8

then for $S$9 one has

$\operatorname{mdr}$0

under the genericity hypothesis. This addition theorem implies that adding a generic hyperplane to both members of a Ziegler pair produces a new Ziegler pair. After combining this with successive coning, one obtains the first known families of Ziegler pairs in arbitrary dimension and size, including irreducible examples in dimensions $\operatorname{mdr}$1 (Abe et al., 23 Sep 2025).

The cumulative effect of these constructions is to transform Ziegler pairs from isolated low-dimensional anomalies into a stable mechanism for generating higher-dimensional counterexamples to naive combinatorial rigidity.

6. Conic-line arrangements, Zariski pairs, and strong Ziegler behavior

The relation between Zariski pairs and strong Ziegler pairs is especially visible for conic-line arrangements. A Zariski pair is a pair of reduced plane curves with the same combinatorics but different embedded topology, meaning that no homeomorphism of $\operatorname{mdr}$2 carries one curve to the other. A strong Ziegler pair requires the same combinatorics but non-isomorphic Milnor algebras, equivalently distinct graded modules $\operatorname{mdr}$3 of Jacobian syzygies (Bannai et al., 10 Sep 2025).

A basic irreducible example is Zariski’s sextic pair: two sextics with six cusps, one with all six cusps on a conic and one without such a conic. Their complements have different fundamental groups,

$\operatorname{mdr}$4

and their Milnor algebras have different minimal graded free resolutions, so the pair is simultaneously a Zariski pair and a strong Ziegler pair (Bannai et al., 10 Sep 2025).

For degree $\operatorname{mdr}$5 conic-line arrangements, the paper identifies four combinatorics—$\operatorname{mdr}$6, $\operatorname{mdr}$7, $\operatorname{mdr}$8, and $\operatorname{mdr}$9—for which known Zariski pairs are also strong Ziegler pairs. By contrast, the degree $S=\mathbb{C}[x,y,z]$0 combinatorics $S=\mathbb{C}[x,y,z]$1 yields Zariski pairs whose Milnor algebras all have the same computed minimal resolution

$S=\mathbb{C}[x,y,z]$2

so the Zariski-pair property does not automatically imply strong Ziegler behavior (Bannai et al., 10 Sep 2025).

In degree $S=\mathbb{C}[x,y,z]$3, the same paper studies a Zariski triple $S=\mathbb{C}[x,y,z]$4 of conic-line arrangements, distinguished topologically by splitting type. The Milnor algebra of $S=\mathbb{C}[x,y,z]$5 has resolution

$S=\mathbb{C}[x,y,z]$6

whereas both $S=\mathbb{C}[x,y,z]$7 and $S=\mathbb{C}[x,y,z]$8 have

$S=\mathbb{C}[x,y,z]$9

Hence $\mathbb{P}^3$00 and $\mathbb{P}^3$01 are strong Ziegler pairs (Bannai et al., 10 Sep 2025).

These examples clarify a common misconception. Zariski pairs and strong Ziegler pairs measure different failures of combinatorial rigidity—one topological, one homological—and overlap only in special families.

7. Adjacent uses around the Ziegler spectrum

Outside arrangement theory, pair constructions related to the Ziegler spectrum occur in a different sense. In the theory of torsion pairs for finite-dimensional algebras and artinian rings, pairs

$\mathbb{P}^3$02

with $\mathbb{P}^3$03 a closed rigid subset of the Ziegler spectrum and $\mathbb{P}^3$04 a set of indecomposable injectives, parametrize torsion pairs in $\mathbb{P}^3$05. The paper formalizes these as cosilting pairs and proves bijections between torsion pairs, cosilting torsion pairs, 2-term cosilting complexes, and such pairs $\mathbb{P}^3$06 (Hügel et al., 2024).

A second nearby development concerns generalized Ziegler spectra for enriched ringoids and schemes. There, one has left and right Ziegler spectra linked by an Auslander–Gruson–Jensen duality on coherent objects, and for a left coherent enriched ringoid there is a closed embedding

$\mathbb{P}^3$07

from the injective spectrum into the Ziegler spectrum (Garkusha, 2024).

These constructions are mathematically adjacent to arrangement-theoretic Ziegler pairs but conceptually distinct. In arrangement theory, a Ziegler pair compares two combinatorially equivalent arrangements. In Ziegler-spectrum theory, the relevant pairings compare spectra, injectives, definable subcategories, or cosilting data. The shared name points to a common historical source, but the technical meanings belong to different branches of mathematics.