A homological characterization for freeness of multi-arrangements

TL;DR

This paper introduces a homological approach using a co-chain complex to determine the freeness of multi-arrangements, linking algebraic properties to geometric and combinatorial structures.

Contribution

It develops a new homological method based on cohomology to characterize the freeness of multi-arrangements, extending previous work on $k$-formality.

Findings

Freeness implies the underlying arrangement is $k$-formal for all $k extgreater 1

Complete characterization of freeness for certain multi-arrangement families

Counter-examples to Orlik's conjecture related to geometric properties

Abstract

Building on work of Brandt and Terao in their study of $k$-formality, we introduce a co-chain complex associated to a multi-arrangement and prove that its cohomologies determine freeness of the associated module of multi-derivations. This provides a new homological method for determining freeness of arrangements and multi-arrangements. We work out many applications of this homological method. For instance, we prove that if a multi-arrangement is free then the underlying arrangement is $k$-formal for all $k\ge 2$. We also use this method to completely characterize freeness of certain families of multi-arrangements in moduli, showcasing how the geometry of multi-arrangements with the same intersection lattice may have considerable impact on freeness. New counter-examples to Orlik's conjecture also arise in connection to this latter analysis.