Derivation degree sequences of non-free arrangements

TL;DR

This paper investigates the derivation degree sequences of non-free arrangements, establishing a generalized addition theorem that links free and non-free arrangements, with applications to graphic and hypersolvable arrangements.

Contribution

It introduces a generalized addition theorem for all arrangements, providing new relationships and bounds for derivation degrees in non-free arrangements.

Findings

Established a generalized addition theorem for arrangements.

Derived lower bounds for maximal degree generators in graphic arrangements.

Applied results to hypersolvable arrangements to find degree bounds.

Abstract

In this note we study the logarithmic derivation module of a non-free arrangement. We prove a generalized addition theorem for all arrangements. This addition theorem allows us to find various relationships between non-free arrangements, free arrangements and restriction counts. For graphic arrangements we can use these results to find a lower bound for the maximal degree generator in terms of triangles in the associated graph. We also apply these results to the case of hypersolvable arrangements where we define hyperexponents and use them to find a lower bound for their maximal degree generator.