Combinatorial data of a free arrangement and the Terao conjecture

TL;DR

This paper introduces a combinatorial framework for free arrangements that enables verification of Terao's conjecture using only combinatorial data, providing new insights into the relationship between combinatorics and algebraic properties.

Contribution

It offers a novel combinatorial structure of generators for the module of derivations, leading to a proof of Terao's conjecture and a sufficient condition for combinatorial properties.

Findings

Established a basis of the module using combinatorial data in free arrangements

Provided a proof of Terao's conjecture based on combinatorial structures

Verified the Ziegler example and derived a sufficient condition for combinatorial generators

Abstract

We present a combinatorial structure of generators of $D(\mathcal{A}).$ This structure permits us to detect the relationship between the combinatorial determined property and the singularity of vector field. Consequently, by using only combinatorial data, we have a basis of the module in free case and that yields a proof for the Terao's conjecture. We also verify the example of Ziegler and give a sufficient condition on combinatorial determined property of generators.