# Locally Heavy Hyperplanes in Multiarrangements

## TL;DR

This paper proves that arrangements with a locally heavy flag satisfy Terao's conjecture, extending previous results and providing new insights into the freeness of multiarrangements with heavy hyperplanes.

## Contribution

It generalizes the characterization of freeness from heavy hyperplanes to locally heavy hyperplanes and applies this to prove Terao's conjecture in new cases.

## Key findings

- Arrangements with a locally heavy flag satisfy Terao's conjecture.
- Irreducible arrangements with a generic hyperplane are totally non-free.
- Multiarrangements of rank 3 with multiple locally heavy hyperplanes are not free.

## Abstract

Hyperplane Arrangements of rank $3$ admitting an unbalanced Ziegler restriction are known to fulfill Terao's conjecture. This long-standing conjecture asks whether the freeness of an arrangement is determined by its combinatorics. In this note, we prove that arrangements that admit a locally heavy flag satisfy Terao's conjecture which is a generalization of the statement above to arbitrary dimension. To this end, we extend results characterizing the freeness of multiarrangements with a heavy hyperplane to those satisfying the weaker notion of a locally heavy hyperplane. As a corollary, we give a new proof that irreducible arrangements with a generic hyperplane are totally non-free. In another application, we show that an irreducible multiarrangement of rank $3$ with at least two locally heavy hyperplanes is not free.