Generalizing Saito's Criterion for Nonfree Arrangements
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910067518865408 |
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| author | Chu, Junyan |
| author_facet | Chu, Junyan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21101 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Generalizing Saito's Criterion for Nonfree Arrangements Chu, Junyan Commutative Algebra Algebraic Geometry Combinatorics 32S22, 13N15, 13D02 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. |
| title | Generalizing Saito's Criterion for Nonfree Arrangements |
| topic | Commutative Algebra Algebraic Geometry Combinatorics 32S22, 13N15, 13D02 |
| url | https://arxiv.org/abs/2603.21101 |