Strong Ziegler Pairs in Arrangement Theory

• Strong Ziegler pairs are arrangements or curves with identical combinatorial data but distinct minimal free resolutions of their Jacobian or Milnor algebras.
• They are detected using invariants like mdr, graded Betti tables, and Hilbert functions, which signal non-combinatorial discrepancies.
• Concrete examples in line, plane, and hyperplane arrangements illustrate how topologically similar configurations can differ significantly in their algebraic structure.

Searching arXiv for the cited works to ground the article in the current literature. Strong Ziegler pairs are a class of combinatorially indistinguishable plane-curve or hyperplane-arrangement pairs that are separated by Jacobian or Milnor-algebra invariants. In the formulation for line arrangements, a strong Ziegler pair consists of two arrangements with isomorphic intersection lattices whose Milnor algebras have different minimal free resolutions; in the formulation for reduced plane curves, the pair has equivalent combinatorics but distinct graded syzygy modules $\mathrm{AR}(B)$, equivalently non-isomorphic graded Milnor algebras. For plane arrangements in $\mathbb{P}^3$, the phrase itself is not introduced, but the homological framework of Ziegler pairs, together with the properties (HP), (HF), (MDR), and (SPEC), provides a precise way to measure how strong the non-combinatorial discrepancy is (Kühne et al., 2024, Bannai et al., 10 Sep 2025, Dimca et al., 28 Apr 2026).

1. Definitions and terminological scope

The modern literature uses a hierarchy of notions centered on how much combinatorial data are fixed and how much Jacobian data are allowed to vary. For line arrangements in $\mathbb{P}^2_\mathbb{C}$, the intersection lattice $L(\mathcal{L})$ is the lattice of flats of the underlying rank-$3$ matroid, while the weak combinatorics is

$W(\mathcal{L})=(d;t_2,\dots,t_d),$

with $t_r$ the number of points where exactly $r$ lines meet (Kühne et al., 2024).

Notion	Fixed data	Distinguishing invariant
Weak Ziegler pair	Same weak combinatorics	Different $\mathrm{mdr}$
Ziegler pair	Same intersection lattice / matroid	Different $\mathrm{mdr}$
Strong Ziegler pair	Same intersection lattice or equivalent combinatorics	Different minimal free resolutions or distinct graded $\mathbb{P}^3$0 / Milnor algebras

In the 2024 treatment of line arrangements, a Ziegler pair is defined by the condition $\mathbb{P}^3$1 together with $\mathbb{P}^3$2, and a strong Ziegler pair is defined by the stronger condition that the minimal free resolutions of the Milnor algebras are different (Kühne et al., 2024). In the 2025 treatment of reduced plane curves, the definition is phrased in terms of equivalent combinatorics and non-isomorphic graded syzygy modules

$\mathbb{P}^3$3

with the same distinction detected through the graded structure of the Milnor algebra $\mathbb{P}^3$4 (Bannai et al., 10 Sep 2025).

This terminological variation is substantive. In the line-arrangement setting, “strong” is stronger than a change in the minimal degree of a Jacobian relation; in the plane-curve setting, it is keyed directly to the non-isomorphism class of $\mathbb{P}^3$5 or $\mathbb{P}^3$6. For plane arrangements in $\mathbb{P}^3$7, the phrase “strong Ziegler pair” is absent, but the refined comparison of graded Betti data, Hilbert function, Hilbert polynomial, and specialization behavior supplies the same conceptual role (Dimca et al., 28 Apr 2026).

2. Jacobian algebras, syzygies, and homological detectors

The common algebraic object is the Jacobian, or Milnor, algebra. For a homogeneous polynomial $\mathbb{P}^3$8 defining a reduced plane curve $\mathbb{P}^3$9, the Jacobian ideal is

$\mathbb{P}^2_\mathbb{C}$0

and the Milnor algebra is

$\mathbb{P}^2_\mathbb{C}$1

A minimal graded free resolution of $\mathbb{P}^2_\mathbb{C}$2 has the form

$\mathbb{P}^2_\mathbb{C}$3

For an $\mathbb{P}^2_\mathbb{C}$4-syzygy curve, the same resolution is written as

$\mathbb{P}^2_\mathbb{C}$5

and the exponents are $\mathbb{P}^2_\mathbb{C}$6 with

$\mathbb{P}^2_\mathbb{C}$7

(Kühne et al., 2024).

For reduced plane curves, the first syzygy module is

$\mathbb{P}^2_\mathbb{C}$8

and strong Ziegler behavior is detected by the graded $\mathbb{P}^2_\mathbb{C}$9-module structure of $L(\mathcal{L})$0, equivalently by the graded algebra structure of $L(\mathcal{L})$1. In practice, the papers compare minimal graded free resolutions, because differing shifts or multiplicities force non-isomorphism of the Milnor algebras (Bannai et al., 10 Sep 2025).

For reduced surfaces in $L(\mathcal{L})$2, the ambient ring is $L(\mathcal{L})$3, the Jacobian ideal is

$L(\mathcal{L})$4

and the Jacobian algebra is again $L(\mathcal{L})$5. The minimal free resolution of $L(\mathcal{L})$6 is written as

$L(\mathcal{L})$7

while the corresponding resolution of the module of Jacobian derivations $L(\mathcal{L})$8 is

$L(\mathcal{L})$9

The graded Betti data are encoded by the increasing sequences

$3$0

and these sequences are the principal homological detectors of Ziegler behavior in $3$1 (Dimca et al., 28 Apr 2026).

The point of the strong notion is therefore not merely that two arrangements fail to be combinatorially rigid, but that the failure persists at the level of higher syzygies. In the strongest examples now available, the difference is visible simultaneously in the number of syzygies, their degrees, and the asymptotic behavior of the Hilbert function.

3. Hierarchies of strength

The strongest explicit hierarchy is given for line arrangements and hyperplane arrangements. In the line-arrangement framework, weak Ziegler pairs hold only weak combinatorics fixed, Ziegler pairs hold the full intersection lattice fixed but require a change in $3$2, and strong Ziegler pairs require a change in the full minimal free resolution. The difference is substantial: a strong Ziegler pair may have the same $3$3 and still fail to have the same Betti table (Kühne et al., 2024).

For plane arrangements in $3$4, the 2026 framework refines this further by attaching auxiliary properties to a Ziegler pair. The pair satisfies (HP) when the Hilbert polynomials of $3$5 and $3$6 coincide; it satisfies (HF) when the full Hilbert functions coincide; it satisfies (MDR) when the smallest syzygy degree agrees,

$3$7

and it satisfies (SPEC$3$8) or (SPEC) when one arrangement is obtained from the other inside a flat family with constant lattice, with (SPEC) additionally requiring constant graded Betti numbers along the non-special fibers (Dimca et al., 28 Apr 2026).

These refinements show that “strength” has several axes. One may compare two pairs with the same lattice and different Betti tables, but one pair may still satisfy (HF), while another may already differ in Hilbert polynomial. The elliptic matroid examples $3$9 and $W(\mathcal{L})=(d;t_2,\dots,t_d),$0 are subtle in exactly this sense: their Betti tables differ, but the Hilbert functions coincide. By contrast, the cone over Ziegler’s classical line pair in $W(\mathcal{L})=(d;t_2,\dots,t_d),$1 has different Betti data, different Hilbert function, and different Hilbert polynomial (Dimca et al., 28 Apr 2026).

A plausible implication is that the term “strong” is best understood as contextual rather than absolute. In $W(\mathcal{L})=(d;t_2,\dots,t_d),$2, the 2024 definition fixes the criterion at “different minimal free resolutions.” In $W(\mathcal{L})=(d;t_2,\dots,t_d),$3, where finer asymptotic invariants are available, the same language naturally extends to pairs that additionally fail (HF) or (HP).

4. Canonical line-arrangement examples

The classical background is Ziegler’s $W(\mathcal{L})=(d;t_2,\dots,t_d),$4-line example: two real line arrangements with $W(\mathcal{L})=(d;t_2,\dots,t_d),$5 triple points and $W(\mathcal{L})=(d;t_2,\dots,t_d),$6 double points, identical intersection lattices, and a geometric distinction in that in one arrangement the $W(\mathcal{L})=(d;t_2,\dots,t_d),$7 triple points lie on a conic while in the other only $W(\mathcal{L})=(d;t_2,\dots,t_d),$8 do. In the 2024 formulation, this is a Ziegler pair because the line-conic incidence changes $W(\mathcal{L})=(d;t_2,\dots,t_d),$9 (Kühne et al., 2024).

The first new family in that paper comes from a rank-$t_r$0 matroid $t_r$1 on $t_r$2 elements with weak combinatorics

$t_r$3

Its realization space decomposes into two $t_r$4-dimensional components,

$t_r$5

meeting along a $t_r$6-dimensional singular locus that is a smooth conic. For generic realizations on $t_r$7 or $t_r$8, the Milnor algebra has resolution

$t_r$9

so the arrangement is $r$0-syzygy with exponents $r$1 and $r$2. On the singular locus, the resolution changes to

$r$3

with exponents $r$4 and $r$5. This yields a genuine Ziegler pair with the same matroid but different minimal Jacobian relation degree (Kühne et al., 2024).

The decisive strong example is the matroid $r$6 on $r$7 elements with weak combinatorics

$r$8

Its realization space has three one-dimensional components and a zero-dimensional singular locus. On the component defined by $r$9 with generic $\mathrm{mdr}$0, the Milnor algebra has resolution

$\mathrm{mdr}$1

so the arrangement is $\mathrm{mdr}$2-syzygy with exponents $\mathrm{mdr}$3 and $\mathrm{mdr}$4. At the special points $\mathrm{mdr}$5, the resolution becomes

$\mathrm{mdr}$6

so the arrangement is $\mathrm{mdr}$7-syzygy with exponents $\mathrm{mdr}$8 and the same $\mathrm{mdr}$9. This is the characteristic strong Ziegler phenomenon: the minimal degree of a Jacobian relation remains fixed, but the full homological type changes from $\mathrm{mdr}$0-syzygy to $\mathrm{mdr}$1-syzygy (Kühne et al., 2024).

The same paper places these examples next to weaker contrasts. Its smallest counterexample to the Numerical Terao’s Conjecture consists of two $\mathrm{mdr}$2-line arrangements with

$\mathrm{mdr}$3

one free with exponents $\mathrm{mdr}$4 and one plus-one generated with exponents $\mathrm{mdr}$5. This is not presented as a strong Ziegler pair, because the hypothesis fixed there is weak combinatorics rather than a common matroid, but it supplies the surrounding principle that weak combinatorics do not control the homological structure of the Milnor algebra (Kühne et al., 2024).

5. Plane arrangements and conic–line arrangements

For plane arrangements in $\mathrm{mdr}$6, the 2026 paper studies Ziegler pairs through the Jacobian algebra and its graded Betti data. Its basic example starts from Ziegler’s classical line pair

$\mathrm{mdr}$7

in $\mathrm{mdr}$8, with

$\mathrm{mdr}$9

Passing to the cones

$\mathbb{P}^3$00

in $\mathbb{P}^3$01 gives

$\mathbb{P}^3$02

The Hilbert polynomials are

$\mathbb{P}^3$03

These arrangements have the same lattice but differ in Betti data, Hilbert function, and Hilbert polynomial. At the same time, their complements and Milnor fibrations are homotopy equivalent, because the central arrangements are lattice-isotopic. The contrast is therefore purely algebraic rather than topological (Dimca et al., 28 Apr 2026).

The same paper gives a non-cone pair of plane arrangements in $\mathbb{P}^3$04 with

$\mathbb{P}^3$05

and graded Betti data

$\mathbb{P}^3$06

$\mathbb{P}^3$07

Here $\mathbb{P}^3$08, so (MDR) holds even though the rest of the resolution differs substantially. This example shows that strong homological divergence need not come from coning and need not coincide with a change in the smallest syzygy degree (Dimca et al., 28 Apr 2026).

A different strand appears in conic–line arrangements. The 2025 paper on degree $\mathbb{P}^3$09 and $\mathbb{P}^3$10 conic–line arrangements defines a strong Ziegler pair for reduced plane curves by equivalent combinatorics together with distinct graded modules $\mathbb{P}^3$11 and $\mathbb{P}^3$12, equivalently non-isomorphic graded Milnor algebras (Bannai et al., 10 Sep 2025). Its first benchmark is the classical Zariski pair of sextics with six cusps. For

$\mathbb{P}^3$13

and an explicit sextic $\mathbb{P}^3$14 with six cusps not lying on a conic, the minimal resolutions are

$\mathbb{P}^3$15

and

$\mathbb{P}^3$16

The Zariski pair is therefore also a strong Ziegler pair (Bannai et al., 10 Sep 2025).

The same paper proves the existence of strong Ziegler pairs for degree-$\mathbb{P}^3$17 conic–line arrangements with combinatorics

$\mathbb{P}^3$18

and of a degree-$\mathbb{P}^3$19 example obtained by adding a further bitangent line to a degree-$\mathbb{P}^3$20 configuration (Bannai et al., 10 Sep 2025). For type $\mathbb{P}^3$21, however, the situation is deliberately different: there are Zariski pairs whose Milnor algebras all have the same minimal resolution

$\mathbb{P}^3$22

so the topology can differ without producing a strong Ziegler pair at the level of minimal free resolutions (Bannai et al., 10 Sep 2025). This isolates an important boundary: Zariski-pair behavior and strong Ziegler behavior are related, but not equivalent.

6. Constructions, realization spaces, and broader significance

A central mechanism behind strong Ziegler phenomena is the geometry of realization spaces. In the 2024 line-arrangement paper, the decisive examples come from matroids with singular realization spaces. For $\mathbb{P}^3$23, the singular locus is a smooth conic; for $\mathbb{P}^3$24, it is zero-dimensional. The homological type is generic on the smooth components and jumps at the singular locus. The authors explicitly suggest that hidden collinearities among intersection points, present at singular realizations but not encoded by the matroid, are a possible explanation for the change in Jacobian syzygies (Kühne et al., 2024).

Another systematic mechanism is coning and generic hyperplane addition. For reduced curves $\mathbb{P}^3$25 with $\mathbb{P}^3$26, the cone $\mathbb{P}^3$27 has a new degree-$\mathbb{P}^3$28 derivation

$\mathbb{P}^3$29

and if

$\mathbb{P}^3$30

is the resolution of $\mathbb{P}^3$31, then the resolution of $\mathbb{P}^3$32 is

$\mathbb{P}^3$33

This is the algebraic explanation for why coning can convert line-arrangement Ziegler pairs into plane-arrangement examples with more pronounced Hilbert-polynomial separation (Dimca et al., 28 Apr 2026).

The 2025 addition theorem for hyperplane arrangements pushes this idea much further. It defines a Ziegler pair in arbitrary dimension as a pair of central arrangements with isomorphic intersection lattices but different minimal free resolutions of their modules of logarithmic derivations $\mathbb{P}^3$34. Starting from a rank-$\mathbb{P}^3$35 Ziegler pair, one may add a combinatorially generic hyperplane in $\mathbb{P}^3$36, or repeatedly cone and then add a generic hyperplane in higher dimension, obtaining irreducible Ziegler pairs of arbitrarily large size and arbitrarily large dimension (Abe et al., 23 Sep 2025). A plausible implication is that these addition and coning theorems furnish a systematic source of higher-dimensional strong examples whenever “strong” is interpreted as a controlled divergence of derivation-module or Jacobian-module resolutions.

The resulting picture is that strong Ziegler pairs are a precise expression of non-combinatoriality in arrangement theory. Depending on context, the fixed datum may be weak combinatorics, full intersection lattice, or the broader combinatorics of a plane curve; the separating datum may be $\mathbb{P}^3$37, the full graded Betti table, the Hilbert function, the Hilbert polynomial, or the graded module structure of $\mathbb{P}^3$38 or $\mathbb{P}^3$39. The strongest known examples show that one may preserve the lattice, and sometimes even preserve $\mathbb{P}^3$40 or the topology of the complement, while changing the full homological structure of the Jacobian or derivation module (Kühne et al., 2024, Dimca et al., 28 Apr 2026, Bannai et al., 10 Sep 2025).

A separate use of the name “Ziegler” occurs in representation theory, where the Ziegler spectrum $\mathbb{P}^3$41 is the topological space of indecomposable pure-injectives and exact structures are classified by closed subsets of an open subset of that spectrum (Sauter, 2 Jun 2025). This is a different construction from strong Ziegler pairs of arrangements, but the coexistence of the two usages explains why papers are careful about context.