On the numerical Terao's conjecture and Ziegler pairs for line arrangements

TL;DR

This work investigates when freeness and related homological properties of line arrangements are determined by combinatorial data. By exhibiting a 13-line counterexample to the Numerical Terao's Conjecture and conducting extensive matroid-realization analyses, it shows that weak-combinatorial data can fail to determine freeness. It then introduces and demonstrates weak Ziegler and Ziegler pairs, including new examples arising from singular realization spaces of matroids, revealing nuanced non-combinatorial behavior in Milnor algebra resolutions. Overall, the paper highlights both the existence of counterexamples to combinatorial freeness principles and the rich geometry of realization spaces in guiding such phenomena.

Abstract

In this paper we present a smallest possible counterexample to the Numerical Terao's Conjecture in the class of line arrangements in the complex projective plane. Our example consists of a pair of two arrangements with $13$ lines. Moreover, we use the newly discovered singular matroid realization spaces to construct new examples of pairs of line arrangements having the same underlying matroid but different free resolutions of the Milnor algebras. Such rare arrangements are called Ziegler pairs in the literature.

Figures (3)

• Figure 1: The black lines are projectivized picture of $\mathcal{B}$ with the circle the line at infinity. Adding the red line $y+\varphi z=0$ yields a free arrangement.
• Figure 2: Projectivized picture of the arrangement $\mathcal{L}_{1}$.
• Figure 3: Projectivized picture of the arrangement $\mathcal{L}_{2}$.

Theorems & Definitions (39)

• Conjecture 1.1: Terao
• Definition 1.2
• Conjecture 1.3: Numerical Terao's Conjecture
• Theorem 1.4: Marchesi-Vallés
• Theorem 1.5
• Theorem 1.6: see Theorem \ref{['WZ']}
• Theorem 1.7: see Theorem \ref{['ZZ']}
• Definition 2.1
• Definition 2.2
• Definition 2.3