{"paper":{"title":"Divisionally free arrangements of hyperplanes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AC","authors_text":"Takuro Abe","submitted_at":"2015-02-26T12:13:01Z","abstract_excerpt":"We consider the triple $(\\mathcal{A},\\mathcal{A}',\\mathcal{A}^H)$ of hyperplane arrangements and the division of their characteristic polynomials. We show that the freeness of $\\mathcal{A}^H$ and the division of $\\chi(\\mathcal{A};t)$ by $\\chi(\\mathcal{A}^H;t)$ confirm the freeness of $\\mathcal{A}$. The key ingredient of this \"division theorem\" on freeness is the fact that, if $\\chi(\\mathcal{A}^H;t)$ divides $\\chi(\\mathcal{A};t)$, then the same holds for the localization at the codimension three flat in $H$. This implies the local-freeness of $\\mathcal{A}$ in codimension three along $H$. Based "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1502.07520","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}