{"paper":{"title":"Birationality of \\'etale morphisms via surgery","license":"","headline":"","cross_cats":["math.GT"],"primary_cat":"math.AG","authors_text":"Frederico Xavier, Laurence R. Taylor, Scott Nollet","submitted_at":"2007-05-03T19:49:20Z","abstract_excerpt":"We use a counting argument and surgery theory to show that if $D$ is a sufficiently general algebraic hypersurface in $\\Bbb C^n$, then any local diffeomorphism $F:X \\to \\Bbb C^n$ of simply connected manifolds which is a $d$-sheeted cover away from $D$ has degree $d=1$ or $d=\\infty$ (however all degrees $d > 1$ are possible if $F$ fails to be a local diffeomorphism at even a single point). In particular, any \\'etale morphism $F:X \\to \\Bbb C^n$ of algebraic varieties which covers away from such a hypersurface $D$ must be birational."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0705.0524","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":""},"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"}