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In this paper, we prove an analogous statement for the hyperplane sections of unions general curves. More specifically, if H is a general hyperplane, we show that H^0(O_H(m)) \\to H^0(O_{(C_1 \\cup C_2 \\cup \\cdots \\cup C_n) \\cap H}(m)) is of maximal rank, except for some counterexamples when m = 2.\n  As explained in"},"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":"1208.2730","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-08-14T00:07:50Z","cross_cats_sorted":[],"title_canon_sha256":"704cc4f4e3647f2ed8acb7f095fb724545292764083872ec8de1112046b66ab8","abstract_canon_sha256":"a3bc42d7cc71e52c5e04394a48610d3f5f5830166bc73b8004b554f0279bcab8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:24.398172Z","signature_b64":"lcXMqN7KO0mbT30EH+eKInGGpJBA9/xSmZuDjRxN6EfqYaW/Eu8sddHFks0hI+KcGVPOgwV1ujdZqOCjcKuTCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee99d9feb0bc770dcd9bda7ff84bb15c1185dadb3eab28f7bd6956f50620c6f5","last_reissued_at":"2026-05-18T00:05:24.397674Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:24.397674Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Maximal Rank Conjecture for Sections of Curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Eric Larson","submitted_at":"2012-08-14T00:07:50Z","abstract_excerpt":"Let be a general curve of genus g embedded via a general linear series of degree d in P^r. The well-known Maximal Rank Conjecture asserts that the restriction maps H^0(O_{P^r}(m)) \\to H^0(O_C(m) are of maximal rank; if known, this conjecture would determine the Hilbert function of C. In this paper, we prove an analogous statement for the hyperplane sections of unions general curves. More specifically, if H is a general hyperplane, we show that H^0(O_H(m)) \\to H^0(O_{(C_1 \\cup C_2 \\cup \\cdots \\cup C_n) \\cap H}(m)) is of maximal rank, except for some counterexamples when m = 2.\n  As explained in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.2730","kind":"arxiv","version":5},"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":"1208.2730","created_at":"2026-05-18T00:05:24.397759+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.2730v5","created_at":"2026-05-18T00:05:24.397759+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.2730","created_at":"2026-05-18T00:05:24.397759+00:00"},{"alias_kind":"pith_short_12","alias_value":"52M5T7VQXR3Q","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"52M5T7VQXR3Q3TM3","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"52M5T7VQ","created_at":"2026-05-18T12:26:53.410803+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/52M5T7VQXR3Q3TM33J77QS5RLQ","json":"https://pith.science/pith/52M5T7VQXR3Q3TM33J77QS5RLQ.json","graph_json":"https://pith.science/api/pith-number/52M5T7VQXR3Q3TM33J77QS5RLQ/graph.json","events_json":"https://pith.science/api/pith-number/52M5T7VQXR3Q3TM33J77QS5RLQ/events.json","paper":"https://pith.science/paper/52M5T7VQ"},"agent_actions":{"view_html":"https://pith.science/pith/52M5T7VQXR3Q3TM33J77QS5RLQ","download_json":"https://pith.science/pith/52M5T7VQXR3Q3TM33J77QS5RLQ.json","view_paper":"https://pith.science/paper/52M5T7VQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.2730&json=true","fetch_graph":"https://pith.science/api/pith-number/52M5T7VQXR3Q3TM33J77QS5RLQ/graph.json","fetch_events":"https://pith.science/api/pith-number/52M5T7VQXR3Q3TM33J77QS5RLQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/52M5T7VQXR3Q3TM33J77QS5RLQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/52M5T7VQXR3Q3TM33J77QS5RLQ/action/storage_attestation","attest_author":"https://pith.science/pith/52M5T7VQXR3Q3TM33J77QS5RLQ/action/author_attestation","sign_citation":"https://pith.science/pith/52M5T7VQXR3Q3TM33J77QS5RLQ/action/citation_signature","submit_replication":"https://pith.science/pith/52M5T7VQXR3Q3TM33J77QS5RLQ/action/replication_record"}},"created_at":"2026-05-18T00:05:24.397759+00:00","updated_at":"2026-05-18T00:05:24.397759+00:00"}