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We study the dimensions of higher secant varieties of $X^{(n,m)}_{(1,d)}$ and we prove that there is no defective $s^{th}$ secant variety, except possibly for $n$ values of $s$. Moreover when ${m+d \\choose d}$ is multiple of $(m+n+1)$, the $s^{th}$ secant variety of  $X^{(n,m)}_{(1,d)}$ has the expected dimension for every $s$."},"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":"1004.2614","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-04-15T12:02:04Z","cross_cats_sorted":[],"title_canon_sha256":"7a72e1b1c195bbd25d98188b9f15ac561a38409a11167670bf8d9498828232da","abstract_canon_sha256":"d018c65ccf5b21a39e2134c0bcc779553827e3890bcd71b2b60bfbf78741c1e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:54.344982Z","signature_b64":"8Nxv896u2Xe0pYsukUy5bElV8Vb8LbY9xL/ZPkuGYcqAT7PUb/xVSiBxIbLMk1cnqmMNWOV9sNntobVnbR4TBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"22a067f10a05d6da255e558c064fe77bda54c39aeed91162a1296d39946f1472","last_reissued_at":"2026-05-18T04:07:54.344468Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:54.344468Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Higher secant varieties of $\\mathbb{P}^n \\times \\mathbb{P}^m$ embedded in bi-degree $(1,d)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alessandra Bernardi, Enrico Carlini, Maria Virginia Catalisano","submitted_at":"2010-04-15T12:02:04Z","abstract_excerpt":"Let $X^{(n,m)}_{(1,d)}$ denote the Segre-Veronese embedding of $\\mathbb{P}^n \\times \\mathbb{P}^m$ via the sections of the sheaf $\\mathcal{O}(1,d)$. 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