Restrictions of free arrangements and the division theorem

TL;DR

This paper surveys and investigates the division theorem in free arrangements, exploring its implications for the modified Orlik conjecture and providing partial answers to related problems.

Contribution

It introduces a generalized division theorem for free arrangements and examines its role in the modified Orlik conjecture with new insights and reformulations.

Findings

The division theorem generalizes classical addition-deletion theorems.

Partial results on the modified Orlik conjecture are presented.

The paper reformulates key problems related to free arrangements.

Abstract

This is a survey and research note on the modified Orlik conjecture derived from the division theorem introduced in [2]. The division theorem is a generalization of classical addition-deletion theorems for free arrangements. The division theorem can be regarded as a modified converse of the Orlik's conjecture with a combinatorial condition, i.e., an arrangement is free if the restriction is free and the characteristic polynomial of the restriction divides that of an arrangement. In this article we recall, summarize, pose and re-formulate some of results and problems related to the division theorem based on [2], and study the modified Orlik's conjecture with partial answers.