{"paper":{"title":"A Noether-Lefschetz theorem for varieties of r-planes in complete intersections","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Zhi Jiang","submitted_at":"2010-10-25T17:55:25Z","abstract_excerpt":"Let X be a very general complete intersection in complex projective space and we denote by $F_r(X)$ the variety of r-planes in X, for $r\\geq 1$. We show that the Picard number of $F_r(X)$ is 1, as soon as $\\dim F_r(X)\\geq 2$, except when X is a quadric of dimension 2r or 2r+2, or X is a complete intersection of two quadrics of dimension 2r+2. We also apply this result to determine the cohomology class of the variety of planes of a cubic fivefold contained (by the Abel-Jacobi map) in the intermediate Jacobian."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.5210","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":""},"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"}