{"paper":{"title":"Rational curves on elliptic surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Douglas Ulmer","submitted_at":"2014-07-29T19:47:20Z","abstract_excerpt":"We prove that a very general elliptic surface $\\mathcal{E}\\to\\mathbb{P}^1$ over the complex numbers with a section and with geometric genus $p_g\\ge2$ contains no rational curves other than the section and components of singular fibers. Equivalently, if $E/\\mathbb{C}(t)$ is a very general elliptic curve of height $d\\ge3$ and if $L$ is a finite extension of $\\mathbb{C}(t)$ with $L\\cong\\mathbb{C}(u)$, then the Mordell-Weil group $E(L)=0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.7845","kind":"arxiv","version":3},"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"}