2018/06/13 by Michael DiPasquale, DiPasquale, Michael
Mathematics · #13D02 #13N15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #Primary 14Q10 #Secondary 13P20
paper · pdf · doi:10.48550/arxiv.1806.05295
openalex publication_date 2018/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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≥ 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.