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

TL;DR

This paper provides the smallest known counterexample to the Numerical Terao's Conjecture for line arrangements and introduces new Ziegler pairs with identical matroids but different algebraic properties.

Contribution

It presents the first minimal counterexample to the conjecture and constructs novel Ziegler pairs using singular matroid realization spaces.

Findings

Counterexample with 13 lines disproves the conjecture

New Ziegler pairs with identical matroids but different free resolutions

Application of singular matroid realization spaces in line arrangements

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.