{"paper":{"title":"Jacobian Ideals of Hyperplane Arrangements and their Graded Betti Numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Juan Migliore, Uwe Nagel","submitted_at":"2025-08-16T17:54:23Z","abstract_excerpt":"A hyperplane arrangement $\\cA$ is said to be free if the corresponding Jacobian ideal $J_\\cA$ is Cohen-Macaulay. If $\\cA$ is free then $J_\\cA$ is unmixed (i.e. equidimensional). Freeness is an important property, yet its presence is not well understood. A conjecture of Terao says that freeness of $\\cA$ depends only on the intersection lattice of $\\cA$. Given an arrangement $\\cA$, we define the ideal $J_\\cA^{top}$ to be the intersection of the codimension 2 primary components of $J_\\cA$. This ideal is unmixed, but not necessarily Cohen-Macaulay; if $\\cA$ is free then $J_\\cA = J_\\cA^{top}$. We d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.12113","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.12113/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}