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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1010.5210","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-10-25T17:55:25Z","cross_cats_sorted":[],"title_canon_sha256":"7dab01d35b1964587ce1af734d19a84678289f8130815bff24a0b867e83876d6","abstract_canon_sha256":"000346e19c60927519900a38722db60fedf6db49b128023cc79c6b4e42be0bbd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:38:49.147735Z","signature_b64":"DoXZ5GECLd4JTB9av0hypPKtoHEMEMMp7qn+eu5hUBaa0MHy2Js/wms6m2BsHwv0Y8h6fkUQjS8pamtff/5oDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"19e7786c6993437dc8574c41260ed505912393e8c836249393e71449f9209973","last_reissued_at":"2026-05-18T04:38:49.147306Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:38:49.147306Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $r\\geq 1$. 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