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We prove that $X$ is linearly normal if $\\rho (X)''\\ge \\lceil (r+2)/2\\rceil$ and that $\\rho (X)'' < 2\\lceil (r+1)/(n+1)\\rceil$, unless either $n=r$ or $X$ is a rational normal curve."},"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":"1905.08149","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-05-20T14:54:27Z","cross_cats_sorted":[],"title_canon_sha256":"bf52e100d077fdec9c00fa87f04483a1834e24cbb48f74c9459ffb576b5790fa","abstract_canon_sha256":"57ddf16e946e02841ae9164e1a1fba8eb70a5feba3554866e4fbc068f3b4d91d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:45:47.455952Z","signature_b64":"MzzpsatRZ4mL/bw8puDT6H7atV/FqIP88B3MBKeWF51O3gBYbdbQ+aYFgxZVpM8nn3FyVgm2zKZrVDg0RxjWDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"12a41282b3af42696c4f664585a781a6b7e5f07f087553f9d8960e6e4da85479","last_reissued_at":"2026-05-17T23:45:47.455405Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:45:47.455405Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dependent subsets of embedded projective varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Edoardo Ballico","submitted_at":"2019-05-20T14:54:27Z","abstract_excerpt":"Let $X\\subset \\mathbb {P}^r$ be an integral and non-degenerate variety. Set $n:= \\dim (X)$. Let $\\rho (X)''$ be the maximal integer such that every zero-dimensional scheme $Z\\subset X$ smoothable in $X$ is linearly independent. We prove that $X$ is linearly normal if $\\rho (X)''\\ge \\lceil (r+2)/2\\rceil$ and that $\\rho (X)'' < 2\\lceil (r+1)/(n+1)\\rceil$, unless either $n=r$ or $X$ is a rational normal curve."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.08149","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1905.08149","created_at":"2026-05-17T23:45:47.455476+00:00"},{"alias_kind":"arxiv_version","alias_value":"1905.08149v1","created_at":"2026-05-17T23:45:47.455476+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1905.08149","created_at":"2026-05-17T23:45:47.455476+00:00"},{"alias_kind":"pith_short_12","alias_value":"CKSBFAVTV5BG","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_16","alias_value":"CKSBFAVTV5BGS3CP","created_at":"2026-05-18T12:33:15.570797+00:00"},{"alias_kind":"pith_short_8","alias_value":"CKSBFAVT","created_at":"2026-05-18T12:33:15.570797+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CKSBFAVTV5BGS3CPMZCYLJ4BU2","json":"https://pith.science/pith/CKSBFAVTV5BGS3CPMZCYLJ4BU2.json","graph_json":"https://pith.science/api/pith-number/CKSBFAVTV5BGS3CPMZCYLJ4BU2/graph.json","events_json":"https://pith.science/api/pith-number/CKSBFAVTV5BGS3CPMZCYLJ4BU2/events.json","paper":"https://pith.science/paper/CKSBFAVT"},"agent_actions":{"view_html":"https://pith.science/pith/CKSBFAVTV5BGS3CPMZCYLJ4BU2","download_json":"https://pith.science/pith/CKSBFAVTV5BGS3CPMZCYLJ4BU2.json","view_paper":"https://pith.science/paper/CKSBFAVT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1905.08149&json=true","fetch_graph":"https://pith.science/api/pith-number/CKSBFAVTV5BGS3CPMZCYLJ4BU2/graph.json","fetch_events":"https://pith.science/api/pith-number/CKSBFAVTV5BGS3CPMZCYLJ4BU2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CKSBFAVTV5BGS3CPMZCYLJ4BU2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CKSBFAVTV5BGS3CPMZCYLJ4BU2/action/storage_attestation","attest_author":"https://pith.science/pith/CKSBFAVTV5BGS3CPMZCYLJ4BU2/action/author_attestation","sign_citation":"https://pith.science/pith/CKSBFAVTV5BGS3CPMZCYLJ4BU2/action/citation_signature","submit_replication":"https://pith.science/pith/CKSBFAVTV5BGS3CPMZCYLJ4BU2/action/replication_record"}},"created_at":"2026-05-17T23:45:47.455476+00:00","updated_at":"2026-05-17T23:45:47.455476+00:00"}