Strong Ziegler Pair Phenomenon

• Strong Ziegler pairs are pairs of plane configurations sharing the same incidence data but exhibiting distinct homological invariants, notably different minimal graded free resolutions of their Milnor algebras.
• They arise in studies of both reduced curves and line arrangements, where invariants like mdr, Hilbert functions, and graded Betti numbers are used to gauge subtle algebraic differences.
• The phenomenon underscores that combinatorial equivalence does not guarantee homological equivalence, providing deeper insights into singularity theory and the geometry of moduli.

A strong Ziegler pair is a pair of plane configurations in $\PP^2$ that share the same combinatorial data but are distinguished by the homological structure of their Jacobian, or Milnor, algebras. In recent work, the term has been used for both reduced plane curves and line arrangements, with closely related but not identical emphases: Bannai–Tokunaga use it for reduced curves with identical combinatorics and non-isomorphic graded Milnor algebras; Kühle–Luber–Pokora use it for line arrangements with the same matroid but different full minimal graded free resolutions; and Dimca–Pokora reserve “strong” for the especially rigid situation in which the intersection lattice, Hilbert function, and minimal degree of a Jacobian relation all agree, while higher Betti data still differ (Bannai et al., 10 Sep 2025, Kühne et al., 2024, Dimca et al., 18 Jun 2026).

1. Terminological scope and basic definitions

For a reduced plane curve $B \subset \PP^2$ defined by a homogeneous polynomial $f \in S=\C[x,y,z]$, the Jacobian ideal is

$J_f=\langle f_x,f_y,f_z\rangle,$

and the Milnor algebra is

$M(f)=S/J_f.$

This algebra, together with its graded syzygies and minimal free resolution, is the central invariant in the theory (Bannai et al., 10 Sep 2025).

A Zariski pair of reduced curves $B_1,B_2 \subset \PP^2$ is defined by two conditions: they have the same combinatorics, meaning the same incidence data among components and singularities, but there is no homeomorphism $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$. In Bannai–Tokunaga’s formulation, $(B_1,B_2)$ is a strong Ziegler pair if the curves share identical combinatorics but the graded $S$-modules $M(f_{B_1})$ and $B \subset \PP^2$0 are not isomorphic (Bannai et al., 10 Sep 2025).

For line arrangements $B \subset \PP^2$1, the combinatorics are usually encoded by the intersection lattice $B \subset \PP^2$2, equivalently the rank-$B \subset \PP^2$3 matroid $B \subset \PP^2$4. In the 2024 line-arrangement exposition, a Ziegler pair is a pair of arrangements with isomorphic intersection lattices but different $B \subset \PP^2$5, while a strong Ziegler pair requires that the entire minimal graded free resolutions of the Milnor algebras differ (Kühne et al., 2024). In the 2026 terminology, a Ziegler pair is already defined by differing graded Betti numbers of $B \subset \PP^2$6, and “strong” means that two further conditions hold: (HF), equality of Hilbert functions, and (MDR), equality of $B \subset \PP^2$7 (Dimca et al., 18 Jun 2026).

Source	Ambient objects	Criterion called “strong Ziegler pair”
(Bannai et al., 10 Sep 2025)	Reduced plane curves	Same combinatorics, non-isomorphic graded Milnor algebras
(Kühne et al., 2024)	Line arrangements	Same matroid, different full minimal graded free resolutions
(Dimca et al., 18 Jun 2026)	Line arrangements	A Ziegler pair satisfying (HF) and (MDR)

A plausible synthesis is that all three usages isolate the same underlying phenomenon: combinatorics alone does not determine the full Jacobian-syzygy package, but the exact threshold for calling the phenomenon “strong” varies across the literature.

2. Milnor algebras, syzygies, and homological detection

The relevant homological object is the module of Jacobian syzygies. For reduced curves in Bannai–Tokunaga’s setup,

$B \subset \PP^2$8

Two reduced curves form a strong Ziegler pair precisely when they admit the same intersection graph and the same singularity-type data on each stratum, but the graded syzygy modules $B \subset \PP^2$9 differ; equivalently, the minimal free resolutions of $f \in S=\C[x,y,z]$0 are not term-by-term equal (Bannai et al., 10 Sep 2025).

For line arrangements, the same philosophy is expressed in the language of exponents and minimal graded free resolutions. If $f \in S=\C[x,y,z]$1 and $f \in S=\C[x,y,z]$2, then $f \in S=\C[x,y,z]$3 is naturally graded, and one studies the degrees of the minimal homogeneous generators of $f \in S=\C[x,y,z]$4. The smallest such degree is

$f \in S=\C[x,y,z]$5

A free arrangement is $f \in S=\C[x,y,z]$6-syzygy, and more generally the Betti table of the minimal resolution records the full syzygetic complexity of $f \in S=\C[x,y,z]$7 (Kühne et al., 2024).

This homological viewpoint leads to a practical test. In Bannai–Tokunaga’s conic–line examples, the curves are constructed so that each pair has identical combinatorics, and then the minimal graded resolutions of

$f \in S=\C[x,y,z]$8

are computed explicitly, for instance with graded_free_resolution in Sage. Whenever the two resolutions differ in the number or shifts of summands $f \in S=\C[x,y,z]$9, the Milnor algebras cannot be isomorphic, and the pair is strong Ziegler (Bannai et al., 10 Sep 2025).

The 2026 line-arrangement work sharpens this by distinguishing several levels of coincidence. Conditions (HF) and (MDR) isolate cases where the Hilbert function and the first Jacobian-syzygy degree are fixed, so that only higher Betti numbers change. In that sense, the “strong” condition of (Dimca et al., 18 Jun 2026) is a refinement inside the already homological notion of Ziegler pair (Dimca et al., 18 Jun 2026).

3. Conic–line arrangements in degrees $J_f=\langle f_x,f_y,f_z\rangle,$0 and $J_f=\langle f_x,f_y,f_z\rangle,$1

The paper "Examples of strong Ziegler pairs of conic-line arrangements of degree 7 and 8" exhibits several Zariski pairs of reduced plane curves that are also strong Ziegler pairs, all among conic–line arrangements (Bannai et al., 10 Sep 2025).

In degree $J_f=\langle f_x,f_y,f_z\rangle,$2, four combinatorial types are treated. For combinatorics $J_f=\langle f_x,f_y,f_z\rangle,$3, the arrangement consists of two smooth conics $J_f=\langle f_x,f_y,f_z\rangle,$4 and three lines $J_f=\langle f_x,f_y,f_z\rangle,$5, with the specified incidence conditions that $J_f=\langle f_x,f_y,f_z\rangle,$6 meets $J_f=\langle f_x,f_y,f_z\rangle,$7 transversely, $J_f=\langle f_x,f_y,f_z\rangle,$8 is tangent to $J_f=\langle f_x,f_y,f_z\rangle,$9 once and tangent to $M(f)=S/J_f.$0, $M(f)=S/J_f.$1, and $M(f)=S/J_f.$2 is tangent to $M(f)=S/J_f.$3 and passes through $M(f)=S/J_f.$4. With

$M(f)=S/J_f.$5

$M(f)=S/J_f.$6

$M(f)=S/J_f.$7

where $M(f)=S/J_f.$8, one obtains

$M(f)=S/J_f.$9

Their minimal Milnor-algebra resolutions are

$B_1,B_2 \subset \PP^2$0

for $B_1,B_2 \subset \PP^2$1, and

$B_1,B_2 \subset \PP^2$2

for $B_1,B_2 \subset \PP^2$3. Since the graded resolutions differ, $B_1,B_2 \subset \PP^2$4 is a strong Ziegler pair (Bannai et al., 10 Sep 2025).

For combinatorics $B_1,B_2 \subset \PP^2$5, one keeps the same support $B_1,B_2 \subset \PP^2$6, but $B_1,B_2 \subset \PP^2$7 is tangent to $B_1,B_2 \subset \PP^2$8 at two points and meets each of $B_1,B_2 \subset \PP^2$9 outside $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$0, while $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$1 passes through one point on $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$2 and one on $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$3. Taking

$h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$4

and setting $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$5, $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$6, one gets the same two resolution shapes as above, hence another strong Ziegler pair (Bannai et al., 10 Sep 2025).

The combinatorics $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$7 case involves two transversal conics $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$8 and three lines $h\colon (\PP^2,B_1)\simeq (\PP^2,B_2)$9, with $(B_1,B_2)$0 bitangent. The resulting Zariski pair $(B_1,B_2)$1 again has Milnor resolutions of the same two types just displayed, and is therefore strong Ziegler. For combinatorics $(B_1,B_2)$2, with three conics $(B_1,B_2)$3 and one line $(B_1,B_2)$4 of type “inscribed conic plus tangent line,” the specific choices

$(B_1,B_2)$5

$(B_1,B_2)$6

yield $(B_1,B_2)$7 and $(B_1,B_2)$8; here the Milnor algebra resolutions differ in the shifts of the last syzygies, again proving the strong Ziegler property (Bannai et al., 10 Sep 2025).

Degree $(B_1,B_2)$9 is obtained by starting from degree $S$0 combinatorics $S$1 and adding a second bitangent line. For $S$2,

$S$3

has degree $S$4, and all such arrangements share the same incidence pattern: seven nodes, six tacnodes, and two ordinary triple points. A splitting-type invariant shows that $S$5 is a Zariski triple. Their Milnor-algebra resolutions are

$S$6

for $S$7, and

$S$8

for $S$9 and $M(f_{B_1})$0. Hence $M(f_{B_1})$1 and $M(f_{B_1})$2 are strong Ziegler pairs (Bannai et al., 10 Sep 2025).

4. Line arrangements, realization spaces, and the $M(f_{B_1})$3-line constructions

For line arrangements, the classical Ziegler phenomenon is formulated in terms of the intersection lattice or underlying matroid. The 2024 paper "On the numerical Terao's conjecture and Ziegler pairs for line arrangements" first distinguishes this from failures of the Numerical Terao Conjecture. Its $M(f_{B_1})$4-line examples $M(f_{B_1})$5 and $M(f_{B_1})$6 share the same weak-combinatorics $M(f_{B_1})$7, but their intersection lattices differ, so they do not form a Ziegler pair, even though $M(f_{B_1})$8 is free with exponents $M(f_{B_1})$9 and $B \subset \PP^2$00 is plus-one-generated with $B \subset \PP^2$01 (Kühne et al., 2024).

The genuine line-arrangement examples arise from singular matroid realization spaces. If $B \subset \PP^2$02 is a rank-$B \subset \PP^2$03 matroid realizable over $B \subset \PP^2$04, its realization space

$B \subset \PP^2$05

is an affine scheme, and Mnëv’s universality implies that every singularity type appears in some $B \subset \PP^2$06. The 2024 paper records that the smallest ground-set size admitting a truly singular realization space is $B \subset \PP^2$07, and exploits two explicit $B \subset \PP^2$08-matroids to produce Ziegler phenomena (Kühne et al., 2024).

For the first matroid $B \subset \PP^2$09,

$B \subset \PP^2$10

up to inverting a finite set of factors. Thus $B \subset \PP^2$11 has two irreducible components

$B \subset \PP^2$12

meeting along the smooth conic singular locus $B \subset \PP^2$13. Sampling realizations on $B \subset \PP^2$14, $B \subset \PP^2$15, and on $B \subset \PP^2$16 yields different Milnor resolutions. On each of $B \subset \PP^2$17 and $B \subset \PP^2$18,

$B \subset \PP^2$19

with exponents $B \subset \PP^2$20, while on the singular locus,

$B \subset \PP^2$21

with exponents $B \subset \PP^2$22. Any two arrangements sampled on distinct irreducible components share the same lattice but have $B \subset \PP^2$23 versus $B \subset \PP^2$24, hence form a classical Ziegler pair (Kühne et al., 2024).

For the second matroid $B \subset \PP^2$25,

$B \subset \PP^2$26

so the realization space has two one-dimensional components plus their finite intersection. On the general component $B \subset \PP^2$27, excluding the two special points, the Milnor algebra has resolution

$B \subset \PP^2$28

with exponents $B \subset \PP^2$29, whereas at the special points $B \subset \PP^2$30,

$B \subset \PP^2$31

with exponents $B \subset \PP^2$32. Because all these realizations have the same matroid but the full Betti table jumps, they form a strong Ziegler pair in the 2024 sense (Kühne et al., 2024).

5. Degree bounds, $B \subset \PP^2$33-line strong pairs, and higher-degree families

The 2026 paper "On Ziegler pairs of line arrangements: from non-existence to abundance" reorganizes the subject around numerical and homological invariants of line arrangements. It proves that for arrangements of $B \subset \PP^2$34 lines, the combinatorics already determine the exponent data considered there, so no Ziegler pair exists for $B \subset \PP^2$35 (Dimca et al., 18 Jun 2026).

The key numerical input is Theorem 2.3, together with classical bounds. If an arrangement has $B \subset \PP^2$36 lines and maximal multiplicity $B \subset \PP^2$37, then

$B \subset \PP^2$38

where $B \subset \PP^2$39 is the number of points of maximal multiplicity $B \subset \PP^2$40. If $B \subset \PP^2$41, then $B \subset \PP^2$42 exactly; more generally, if $B \subset \PP^2$43, then $B \subset \PP^2$44. By case-by-case analysis for $B \subset \PP^2$45, the paper concludes that no ordinary or strong Ziegler pair occurs below $B \subset \PP^2$46 (Dimca et al., 18 Jun 2026).

At $B \subset \PP^2$47, the situation changes. The paper lists six distinct Ziegler pairs with $B \subset \PP^2$48 lines known to date, but only one of them is strong in its stricter sense, namely the deletion example from an $B \subset \PP^2$49-line family. For the generic $B \subset \PP^2$50-line deletion $B \subset \PP^2$51 and the special one $B \subset \PP^2$52 at $B \subset \PP^2$53, both arrangements have $B \subset \PP^2$54, both have the same Hilbert function, and both have Hilbert-series numerator

$B \subset \PP^2$55

Their $B \subset \PP^2$56-degree sequences are

$B \subset \PP^2$57

and the minimal free resolutions of $B \subset \PP^2$58 are

$B \subset \PP^2$59

for $B \subset \PP^2$60, and

$B \subset \PP^2$61

for $B \subset \PP^2$62. The additional second syzygy in degree $B \subset \PP^2$63 changes the Betti table without changing either $B \subset \PP^2$64 or the Hilbert function; this is the prototypical strong Ziegler pair in the 2026 terminology (Dimca et al., 18 Jun 2026).

The same paper also gives a negative example of a nearby but weaker phenomenon. In the $B \subset \PP^2$65 orchard family, $B \subset \PP^2$66 and $B \subset \PP^2$67 share $B \subset \PP^2$68 and $B \subset \PP^2$69, and their $B \subset \PP^2$70-modules have different minimal free resolutions, but the pair fails (HF), since the Hilbert function already changes when the extra generator in degree $B \subset \PP^2$71 appears. Likewise, the triple-point extension of Ziegler’s $B \subset \PP^2$72-line pair satisfies (MDR) but not (HF) (Dimca et al., 18 Jun 2026).

Beyond $B \subset \PP^2$73, the 2026 paper constructs an $B \subset \PP^2$74-line family $B \subset \PP^2$75 on a nonvanishing locus $B \subset \PP^2$76. For a general point $B \subset \PP^2$77 in the allowed region $B \subset \PP^2$78, the degree sequence of $B \subset \PP^2$79 is $B \subset \PP^2$80, while at the special point $B \subset \PP^2$81 a rank drop in the multiplication map $B \subset \PP^2$82 changes the degree sequence to $B \subset \PP^2$83. Since both arrangements retain the same $B \subset \PP^2$84 and the same Hilbert function, deleting a suitable line yields an infinite one-parameter family of $B \subset \PP^2$85-line strong Ziegler pairs, and in fact infinitely many non-isomorphic strong Ziegler pairs in arbitrary large degree (Dimca et al., 18 Jun 2026).

6. Conceptual significance, exceptions, and open directions

The strong Ziegler-pair phenomenon establishes that Jacobian-syzygy data can distinguish configurations that combinatorics alone cannot separate. In the conic–line setting, the paper emphasizes that the subtle difference in the syzygy structure of the Jacobian ideal, as read off from the graded free resolution of $B \subset \PP^2$86, captures a finer invariant than just the embedded topology and yields a natural algebraic measure of “Zariski-type” inequivalence (Bannai et al., 10 Sep 2025).

One immediate consequence is that the relation between Zariski pairs and strong Ziegler pairs is nontrivial. Bannai–Tokunaga show that some examples of Zariski pairs are also strong Ziegler pairs, and state more broadly that all the classical Zariski pairs considered, including Zariski’s original sextic-cusp pair, turn out also to be strong Ziegler pairs. At the same time, there is an explicit exception in degree $B \subset \PP^2$87: the $B \subset \PP^2$88 configuration with a single bitangent line is a Zariski pair whose Milnor resolutions coincide, so it is not strong. Adding a second bitangent line restores the strong Ziegler property in degree $B \subset \PP^2$89 (Bannai et al., 10 Sep 2025).

For line arrangements, the principal misconception to avoid is that equal combinatorics should force equal homological data. The 2024 and 2026 papers show the contrary in progressively sharper forms. In 2024, singular realization spaces of $B \subset \PP^2$90-point matroids produced the first strong line-arrangement examples in that framework (Kühne et al., 2024). In 2026, the threshold for line-arrangement strong pairs, in the stricter (HF)+(MDR) sense, is pushed down to $B \subset \PP^2$91 lines, while non-existence below $B \subset \PP^2$92 lines is proved (Dimca et al., 18 Jun 2026).

The current research directions are also explicit. Bannai–Tokunaga suggest that Zariski tuples of conic–line arrangements are promising sources of strong Ziegler pairs and pose the broader problem of classifying all combinatorial types that admit non-isomorphic Milnor algebras (Bannai et al., 10 Sep 2025). The line-arrangement literature, by contrast, frames the subject as an abundance problem: once the small-$B \subset \PP^2$93 obstruction disappears and one can engineer a rank drop in the relevant multiplication map, strong pairs can be produced systematically in families (Dimca et al., 18 Jun 2026).

Taken together, these developments place strong Ziegler pairs at the intersection of singularity theory, arrangement theory, homological algebra, and the geometry of moduli. The common theme is that the full minimal free resolution of $B \subset \PP^2$94, or even the weaker datum of its higher Betti profile, can vary inside a fixed combinatorial type. That conclusion is now established for both conic–line arrangements and line arrangements, albeit with distinct terminological conventions and different thresholds for the first known examples (Bannai et al., 10 Sep 2025, Kühne et al., 2024, Dimca et al., 18 Jun 2026).