On Universal derivations for multiarrangements

2025/11/07 by Abe, Takuro, Maehara, Shota, Roehrle, Gerhard +1
#14N20 #32S22 #52C35 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2511.05086

Abstract

The study of universal derivations for arbitrary multiarrangements and multiplicity functions was initiated by Abe, Röhrle, Stump, and Yoshinaga in 2024 which focused on arrangements arising from (well-generated) reflection groups. In this paper we provide a criterion for determining whether a derivation is universal along with a characterization of universal derivations for arbitrary 2-multiarrangements. As an application we give descriptions of universal derivations for several multiarrangements, including the so-called deleted A₃ arrangement. This is the first known example of a non-reflection arrangement that admits a universal derivation distinct from the Euler derivation.

Citations

• Exponents of 2-multiarrangements and Wakefield--Yuzvinsky matrices
• A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups
• Free Multiplicities on the moduli of X₃
• Free and Non-free Multiplicities on the A₃ Arrangement
• Coxeter multiarrangements with quasi-constant multiplicities

Related

• Integral expressions for derivations of multiarrangements
• Explicit description of a basis for derivations of a Coxeter multiarrangement of type B₂
• Freeness of multi-reflection arrangements via primitive vector fields
• Ziegler's Multi-Reflection Arrangements are free
• The primitive derivation and freeness of multi-Coxeter arrangements