{"paper":{"title":"Algebraic Stability for Skew Products","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.DS","authors_text":"Richard A. P. Birkett","submitted_at":"2024-08-05T17:47:12Z","abstract_excerpt":"In this article we study algebraic stability for rational skew products in two dimensions $\\phi : X \\dashrightarrow X$, i.e. maps of the form $\\phi(x, y) = (\\phi_1(x), \\phi_2(x, y))$. We prove that when $X$ is a birationally ruled surface and $\\phi_1$ has no superattracting cycles, then we can always find a smooth surface $\\hat X$ and an algebraic stabilisation $\\pi : (\\hat \\phi, \\hat X) \\to (\\phi, X)$ which is a birational morphism. We provide an example of a skew product $\\phi$ where $\\phi_1$ has a superattracting fixed point and $\\phi$ is not algebraically stable on any model.\n  Our techniq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.02658","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/2408.02658/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"}