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

2024/07/09 by Lukas Kühne, Kühne, Lukas, Dante Luber +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2407.07070

openalex publication_date 2024/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

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.

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• Addition theorems for Ziegler pairs of hyperplane arrangements

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• Weak Ziegler pairs of conic-line arrangements with ordinary singularities
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• Examples of strong Ziegler pairs of conic-line arrangements of degree 7 and 8