Restrictions of free arrangements and the division theorem

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.