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We also prove an analogous result for $k$-planes: if $n \\geq 2 \\binom{d+k-1}{k} + k$, then the space of $k$-planes on $X$ will be irreducible of the expected dimension. As applications, we prove that an arbitrary smooth hypersurface satisfying $n \\geq 2^{d!}$ is unirational, and we prove that the space of degree $e$ curves on $X$ will be irreducible of the expected dimension provided that $d \\leq \\fr"},"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":"1903.02481","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-03-06T16:26:53Z","cross_cats_sorted":[],"title_canon_sha256":"931c33641e46cb156ebc48e0f2ab837af54f96ddcc23f3630f74bb26546926d4","abstract_canon_sha256":"c8c6a541325fe3f37c445beeb9570b55933f55ac6c414f3c2eaa5d140d2a42f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:42:51.520060Z","signature_b64":"RBKnGOmD5dAjMHPhhDITTHTBCTQvXBhkbfgX9hWzwQzo+Uwm733gvaprGl16nhEFyLpDAcp8tex62dYHTq5gBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1126f20859c730e80712a37355e4bf1efb158e06aa86fb7f89100739f5fd10b9","last_reissued_at":"2026-07-05T01:42:51.519575Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:42:51.519575Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Linear subspaces of hypersurfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Eric Riedl, Roya Beheshti","submitted_at":"2019-03-06T16:26:53Z","abstract_excerpt":"Let $X$ be an arbitrary smooth hypersurface in $\\mathbb{C} \\mathbb{P}^n$ of degree $d$. 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