# Combinatorially equivalent hyperplane arrangements

## TL;DR

This paper investigates the combinatorial properties of hyperplane arrangements over various fields, establishing conditions for lattice isomorphisms via Gr"obner bases and linking Terao's conjecture over finite fields to the rational case.

## Contribution

It introduces a criterion for when arrangements and their reductions have isomorphic lattices and connects Terao's conjecture over finite fields with the conjecture over the rationals.

## Key findings

- Determines when arrangements and their reductions modulo primes have isomorphic lattices.
- Proves that Terao's conjecture over finite fields implies the conjecture over the rationals.

## Abstract

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $\sigma$-Gr\"obner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals.