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Assume that either m>\\frac{n}{2} and X is a complete intersection or that m\\geq\\frac{N}{2}, we show deg(X)|deg(Y) and codim_{span(Y)}Y\\geq codim_{\\mathbb{P}^{N}}X, where span(Y) is the linear span of Y. These bounds are sharp. As an application, we classify smooth projective n-dimensional quadratic varieties swept out by m\\geq[\\frac{n}{2}]+1 dimensional quadrics passing through one point."},"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":"1207.5621","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-07-24T08:51:12Z","cross_cats_sorted":[],"title_canon_sha256":"9c366b2c19bbf6155d39ce5b724ada0353a433d2cb9b77e2b324438e181ab1a8","abstract_canon_sha256":"a7a548e08ae72c5a63c36d1e7bd7c1c198be234b8fcc71b5db6001cb9eaab956"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:20:50.159972Z","signature_b64":"AZ8tQg87gzev5XP96H4xIk2BxP4LoiT1Mnqlbd102xCOd8rWcXJMFnm8sBmr+RertbDzVYyivMgf5O5GcQ9gCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d99264bc69fb80ab6af7b1d7f6910277696126fb3530a60fedc7b6944a6c215","last_reissued_at":"2026-05-18T02:20:50.159193Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:20:50.159193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Subvarieties of small codimension in smooth projective varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Qifeng Li","submitted_at":"2012-07-24T08:51:12Z","abstract_excerpt":"Let X\\subsetneq\\mathbb{P}_{\\mathbb{C}}^{N} be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m>\\frac{n}{2} and X is a complete intersection or that m\\geq\\frac{N}{2}, we show deg(X)|deg(Y) and codim_{span(Y)}Y\\geq codim_{\\mathbb{P}^{N}}X, where span(Y) is the linear span of Y. These bounds are sharp. 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