Generalizing Saito's Criterion for Nonfree Arrangements

TL;DR

This paper extends Saito's criterion from free hyperplane arrangements to a broader class called SPOG arrangements, providing algebraic characterizations and a computable framework for non-free cases.

Contribution

It generalizes Saito's criterion to SPOG arrangements, offering a complete algebraic recognition method under certain projective dimension conditions.

Findings

Complete solution for the minor-based recognition problem for SPOG arrangements.

Algebraic characterization of SPOG arrangements in three dimensions.

Provides a computable framework bridging free and non-free arrangement theory.

Abstract

Saito's criterion is a foundational result that algebraically characterizes free hyperplane arrangements via the determinant of a square matrix of logarithmic derivations. It is natural to ask whether this criterion can be generalized to the non-free setting. To address this, we formulate a general problem concerning the maximal minors of a $p \times \ell$ ($p \geq \ell$) derivation matrix and the algebraic relations among their associated coefficients. Focusing on strictly plus-one generated (SPOG) arrangements, we completely solve this minor-based recognition problem under the assumption that $\operatorname{pd} D(\mathcal{A}) \leq 1$. As a direct consequence, we obtain a purely algebraic, necessary and sufficient characterization of SPOG arrangements in dimension three. Ultimately, this framework provides a computable bridge to post-free arrangement theory.