{"paper":{"title":"Tschirnhausen bundles of covers of the projective line","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Ravi Vakil, Sameera Vemulapalli","submitted_at":"2024-10-29T20:53:36Z","abstract_excerpt":"A degree $d$ genus $g$ cover of the complex projective line by a smooth curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. Which bundles are possible? Equivalently, which $\\mathbb{P}^{d-2}$-bundles over $\\mathbb{P}^1$ contain such covers? (In the language of many previous papers: what are the scrollar invariants of the cover?)\n  We give a complete answer in degree $4$, which exhibits the expected pathologies. We describe a polytope (one per degree) which we propose gives the complete answer for primitive covers, i.e. covers that don't factor through "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.22531","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2410.22531/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"}