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Generalizing Saito's Criterion for Nonfree Arrangements

Junyan Chu

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.

Generalizing Saito's Criterion for Nonfree Arrangements

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 () 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 . 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.
Paper Structure (4 sections, 10 theorems, 48 equations)

This paper contains 4 sections, 10 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of Main Results
  4. Future Work

Key Result

Theorem 1.1

Assume $g_{\ell+1} \in S_1 \setminus \{0\}$, and that $g_1, \ldots, g_\ell \in S_{>0}$ have no non-trivial common divisor modulo $g_{\ell+1}$. If $\operatorname{pd} D(\mathcal{A}) \le 1$, then $\mathcal{A}$ is SPOG. That is, $\theta_1, \dots, \theta_{\ell+1}$ form a minimal generating set for $D(\ma

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 2
  • Definition 3
  • Theorem 2.1: Saito's criterion Saito1980
  • Definition 4: Definition 1.1 in spog
  • Remark 1
  • Theorem 2.2: Theorem 0.2 in graded_betti2010
  • Proposition 1
  • ...and 11 more