[Paper Review] Free, not recursively free and non rigid arrangements

Michael Cuntz|arXiv (Cornell University)|Jun 24, 2014

Advanced Combinatorial Mathematics5 references3 citations

TL;DR

This paper constructs the first known counterexamples to Yoshinaga's conjecture that every free arrangement in characteristic zero is either inductively free or rigid. Using a 13-hyperplane arrangement with a one-dimensional moduli space, the authors demonstrate a free arrangement that is neither recursively free nor rigid, providing a minimal example that challenges long-standing assumptions in hyperplane arrangement theory.

ABSTRACT

We construct counterexamples to Yoshinaga's conjecture that every free arrangement is either inductively free or rigid in characteristic zero. The smallest example has $13$ hyperplanes, its intersection lattice has a one dimensional moduli space, and it is free but not recursively free.

Motivation & Objective

• To disprove Yoshinaga's conjecture that every free arrangement in characteristic zero is either inductively free or rigid.
• To construct explicit examples of free arrangements that are not recursively free and not rigid, challenging the prevailing intuition about the structure of free arrangements.
• To investigate the freeness properties within one-dimensional moduli spaces of intersection lattices, particularly in dimension three.
• To clarify the relationship between freeness, recursive freeness, rigidity, and combinatorial invariance in hyperplane arrangements.

Proposed method

• Constructing a family of arrangements parameterized by a complex parameter $ t $, with fixed intersection lattice $ L $, using a specific set of linear forms in $ \mathbb{C}^3 $.
• Computing the intersection lattice $ L(\mathcal{A}_t) $ and showing that the moduli space $ \mathcal{V}_{\mathbb{C}}(L) $ is one-dimensional by proving $ \mathcal{V}_{\mathbb{C}}(L) = \{ \mathcal{A}_\omega \mid \omega \in \mathbb{C} \setminus Z \} $ for a finite set $ Z $ of exceptional values.
• Using the Singular computer algebra system to verify that all arrangements in the moduli space are free, with consistent exponents $ \{1,5,7\} $, and that the arrangement $ \mathcal{A}_t $ is not recursively free.
• Analyzing the automorphism group of the lattice $ L(\mathcal{A}_t) $, showing it is isomorphic to $ \mathbb{Z}/3\mathbb{Z} \rtimes S_3 $, indicating non-rigidity.
• Comparing arrangements at special values $ \omega \in Z $, showing that while some have the same number of hyperplanes, their intersection lattices differ, confirming the moduli space is non-trivial.
• Extending the construction to a 15-hyperplane example with moduli space isomorphic to $ \mathbb{C} \setminus Z' $, where $ Z' $ includes values like $ \frac{3}{2} \pm \sqrt{2} $, and verifying similar freeness and non-recursive freeness properties.

Experimental results

Research questions

• RQ1 Does there exist a free arrangement in characteristic zero that is neither recursively free nor rigid?
• RQ2 Can a one-dimensional moduli space contain both free and non-free arrangements, thereby providing a counterexample to Terao’s conjecture?
• RQ3 Is it possible to construct a free arrangement that is not inductively free and has a non-trivial moduli space?
• RQ4 What is the minimal number of hyperplanes required for a free but not recursively free arrangement in dimension three?
• RQ5 How do the symmetry groups and lattice structures of such arrangements relate to their freeness and recursive freeness properties?

Key findings

• The paper presents a 13-hyperplane arrangement that is free but not recursively free, with a one-dimensional moduli space, serving as the smallest known counterexample to Yoshinaga’s conjecture.
• All arrangements in the moduli space $ \mathcal{V}_{\mathbb{C}}(L) $ for the 13-hyperplane example are free, with exponents $ \{1,5,7\} $, confirming that freeness is preserved across the family.
• The arrangement $ \mathcal{A}_t $ is not recursively free, as verified by computational algebraic methods using Singular, despite being free.
• The moduli space $ \mathcal{V}_{\mathbb{C}}(L) $ is one-dimensional, with only finitely many exceptional parameters $ \omega \in Z $ where the number or lattice structure differs.
• For the 15-hyperplane example, the moduli space is also one-dimensional, with $ \mathcal{V}_{\mathbb{C}}(L) = \{ \mathcal{A}_\omega \mid \omega \in \mathbb{C} \setminus Z' \} $, and the arrangement $ \mathcal{A}_t $ is free with exponents $ \{1,7,7\} $, but not recursively free.
• The automorphism group of the 13-hyperplane lattice is $ \mathbb{Z}/3\mathbb{Z} \rtimes S_3 $, and for the 15-hyperplane lattice it is $ \mathbb{Z}/2\mathbb{Z} \wr S_3 $, both non-trivial, confirming the arrangements are not rigid.