{"paper":{"title":"The minimal projective bundle dimension and toric $2$-Fano manifolds","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Ana-Maria Castravet, Carolina Araujo, Enrica Mazzon, Kelly Jabbusch, Nivedita Viswanathan, Roya Beheshti, Svetlana Makarova, Will Reynolds","submitted_at":"2023-01-02T21:48:14Z","abstract_excerpt":"Motivated by the problem of classifying toric $2$-Fano manifolds, we introduce a new invariant for smooth projective toric varieties, the minimal projective bundle dimension. This invariant $m(X)\\in\\{1, \\dots,\\dim(X)\\}$ captures the minimal degree of a dominating family of rational curves on $X$ or, equivalently, the minimal length of a centrally symmetric primitive relation for the fan of $X$. We classify smooth projective toric varieties with $m(X)\\geq \\dim(X)-2$, and show that projective spaces are the only $2$-Fano manifolds among smooth projective toric varieties with $m(X)\\in\\{1, \\dim(X)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2301.00883","kind":"arxiv","version":2},"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/2301.00883/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"}