Addition theorems for Ziegler pairs of hyperplane arrangements

Technical Summary: Addition Theorems for Ziegler Pairs of Hyperplane Arrangements

Problem Statement
The paper addresses the relationship between the combinatorial and algebraic properties of hyperplane arrangements. Specifically, it investigates Ziegler pairs: pairs of central hyperplane arrangements $(A_1, A_2)$ that share the same intersection lattice (and thus the same underlying matroid, $L(A_1) \cong L(A_2)$) but possess non-isomorphic modules of logarithmic derivations, $D(A_1) \not\cong D(A_2)$.

This investigation is motivated by Terao's Freeness Conjecture, which posits that if two arrangements have the same combinatorics, then one is free if and only if the other is. While the conjecture remains open, Ziegler pairs serve as counter-examples to the stronger claim that the entire derivation module structure is determined by combinatorics. Prior to this work, known Ziegler pairs were limited to low dimensions (specifically line arrangements in $\mathbb{P}^2$) and often relied on specific geometric coincidences, such as hidden collinearities not forced by the matroid. The paper seeks to construct Ziegler pairs in arbitrary dimensions and sizes, specifically focusing on irreducible arrangements.

Methodology
The authors develop a general construction method based on two primary operations:

1. Adding a Generic Hyperplane: Starting with arrangements in $\mathbb{C}^3$, they add a hyperplane $H$ that is "generic" with respect to the arrangement. The paper distinguishes between combinatorially generic hyperplanes (intersecting flats in the expected codimension) and algebraically generic hyperplanes (satisfying specific conditions regarding the evaluation of derivation generators).
2. Coning: The authors utilize the coning operation to lift arrangements from $\mathbb{C}^3$ to higher dimensions $\mathbb{C}^\ell$.

The technical core of the paper relies on analyzing the Euler exact sequence associated with the deletion-restriction triple. A crucial step involves proving that under specific genericity assumptions, the Euler restriction map $\rho: D(A) \to D(A_H)$ is surjective. This allows the authors to determine the minimal free resolution and the degrees of the generators (exponents) of the derivation module for the new arrangement based on the original arrangement.

Key theoretical tools include:

• Theorem 2.1 (Abe et al.): Conditions under which the Euler sequence is right exact.
• Lemma 3.2 & 3.6: Establishing that for irreducible arrangements in $\mathbb{C}^3$, a combinatorially generic hyperplane is also algebraically generic, ensuring the preservation of minimality in the generating sets of derivations.
• Inductive Construction: Using successive coning and the addition of generic hyperplanes to propagate the properties of Ziegler pairs from $\mathbb{C}^3$ to $\mathbb{C}^\ell$.

Key Contributions and Results

1. General Construction in $\mathbb{C}^3$ (Theorem 4.1):
   The authors prove that if $A_1$ and $A_2$ are central arrangements in $\mathbb{C}^3$ with isomorphic lattices but distinct exponent sequences ($\exp(A_1) \neq \exp(A_2)$), and if $H$ is a generic hyperplane such that $L(A_1 \cup \{H\}) \cong L(A_2 \cup \{H\})$, then the resulting pairs $B_1 = A_1 \cup \{H\}$ and $B_2 = A_2 \cup \{H\}$ form a Ziegler pair.

   • Theorem 4.2 provides the explicit formula for the exponents of the new arrangement: if $\exp(A) = (1, a_2, \dots, a_n)$, then $\exp(B) = (1, a_2+1, \dots, a_n+1, |A|-1)$. This shift ensures that distinct initial exponent sequences remain distinct after the addition.

2. Extension to Arbitrary Dimensions (Theorem 5.1):
   The paper presents a general recipe for constructing Ziegler pairs in $\mathbb{C}^\ell$ ($\ell \geq 3$). Starting with an irreducible arrangement $A \subset \mathbb{C}^3$, the authors perform $(\ell-3)$ coning operations to create $A_\ell \subset \mathbb{C}^\ell$, and then add a generic hyperplane $H$.

   • Theorem 5.1 explicitly calculates the degree sequence of the minimal generators for the resulting arrangement $B_\ell = A_\ell \cup \{H\}$. The formula involves a specific distribution of degrees derived from the original exponents and the dimension.

3. Existence of Irreducible Ziegler Pairs (Corollary 5.2):
   By applying the construction in Theorem 5.1 to known Ziegler pairs in $\mathbb{C}^3$ (such as the classical Ziegler example of 9 lines), the authors generate Ziegler pairs in any dimension $\ell$.

   • Example 5.3 explicitly constructs a Ziegler pair in $\mathbb{C}^4$ derived from the classical 9-line arrangement. The resulting arrangements are irreducible (cannot be decomposed as a product of lower-dimensional arrangements), a property not guaranteed in previous examples.

Significance and Claims
The paper claims to provide the first known families of Ziegler pairs in arbitrary dimension and size that are irreducible.

• Prior to this work, known Ziegler pairs were largely confined to line arrangements in $\mathbb{P}^2$ (dimension 2) or reducible arrangements in higher dimensions.
• The authors explicitly state that their examples are the "first irreducible (non-trivial) Ziegler pairs in dimensions greater than three."
• The work demonstrates that the phenomenon of combinatorial equivalence failing to imply algebraic equivalence (non-isomorphic derivation modules) is not an artifact of low-dimensional geometry or specific singularities but can be systematically generated in high-dimensional spaces through generic perturbations and coning.

The paper does not claim to resolve Terao's conjecture but rather expands the landscape of known counter-examples to the idea that the derivation module is purely combinatorial, providing a robust mechanism to generate such examples in higher dimensions.