# Combinatorially formal arrangements are not determined by their points and lines

## TL;DR

This paper investigates the relationship between combinatorial properties of hyperplane arrangements and their geometric realizations, demonstrating that certain combinatorial conditions are not necessary for formality.

## Contribution

It provides a counterexample showing that the absence of a proper lift with the same points and lines does not imply non-formality, challenging previous assumptions.

## Key findings

- Counterexample matroid with a lift but no non-formal realization
- Formality is not solely determined by combinatorial data
- Sufficient conditions for formality are not necessary

## Abstract

An arrangement of hyperplanes is called formal, if the relations between the hyperplanes are generated by relations in codimension 2. Formality is not a combinatorial property, raising the question for a characterization for combinatorial formality. A sufficient condition for this is if the underlying matroid has no proper lift with the same points and lines. We present an example of a matroid with such a lift but no non-formal realization, thus showing that above condition is not necessary for combinatorial formality.