On homological properties of some Cynk-Szemberg octic hyperplane arrangements

TL;DR

This paper investigates the homological properties of Cynk-Szemberg octic hyperplane arrangements, introducing a classification of their types and providing a combinatorial criterion for non-freeness in four-dimensional complex space.

Contribution

It introduces a new notion of arrangement types and offers a combinatorial criterion for non-freeness of hyperplane arrangements in our-dimensional complex space.

Findings

Defined the notion of arrangement types for Cynk-Szemberg octic arrangements.

Provided a combinatorial criterion for non-freeness in our-dimensional space.

Analyzed homological properties of derivation modules of these arrangements.

Abstract

In this paper we study Cynk-Szemberg octic hyperplane arrangements from the perspective of homological properties of their derivation modules. In particular, we define the notion of the type of hyperplane arrangements that will be used in our characterization of rigid Cynk-Szemberg octic hyperplane arrangements. Moreover, we deliver a combinatorial non-freeness criterion for essential hyperplane arrangements in $\mathbb{C}^{4}$.