{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:24AYIJTNRZ3RK6PGM32GCT4W67","short_pith_number":"pith:24AYIJTN","schema_version":"1.0","canonical_sha256":"d70184266d8e771579e666f4614f96f7fa9515e0c63e0d0c72c3a8e2f6e9014e","source":{"kind":"arxiv","id":"1012.2354","version":3},"attestation_state":"computed","paper":{"title":"Hilbert-Kunz theory for nodal cubics, via sheaves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Paul Monsky","submitted_at":"2010-12-10T19:17:22Z","abstract_excerpt":"Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J.\n  Let q=p^n. When J=(x,y,z), Pardue (see R. Buchweitz, Q. Chen. Hilbert-Kunz functions of cubic curves and surfaces. J. Algebra 197 (1997). 246-267) showed that e_n=(7q^2)/3-q/3-R where R=5/3 if q is congruent to 2 (3), and is 1 otherwise. We generalize this, showing that e_n= (mu q^2) + (alpha q) - R where R only depends on q mod 3. We describe alpha and R 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":"1012.2354","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2010-12-10T19:17:22Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"f48ff94bd1be19c4a36e469dcdcc24e7d70a5e043d915a63ea26278710722780","abstract_canon_sha256":"482f7423f3a22caa7891107d3404021bda30b40b2ac27ce93f00248ca85db410"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:31:50.987288Z","signature_b64":"IESDxLFKFAhYxEst/cuOuP7B6S8KWbRmsKXZGhE5POXXU9Tjzfp2ba3bKg3Cot9HC6WMUfXLw1YWpYRwLcX8AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d70184266d8e771579e666f4614f96f7fa9515e0c63e0d0c72c3a8e2f6e9014e","last_reissued_at":"2026-05-18T04:31:50.986676Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:31:50.986676Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hilbert-Kunz theory for nodal cubics, via sheaves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AC","authors_text":"Paul Monsky","submitted_at":"2010-12-10T19:17:22Z","abstract_excerpt":"Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J.\n  Let q=p^n. When J=(x,y,z), Pardue (see R. Buchweitz, Q. Chen. Hilbert-Kunz functions of cubic curves and surfaces. J. Algebra 197 (1997). 246-267) showed that e_n=(7q^2)/3-q/3-R where R=5/3 if q is congruent to 2 (3), and is 1 otherwise. We generalize this, showing that e_n= (mu q^2) + (alpha q) - R where R only depends on q mod 3. We describe alpha and R in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.2354","kind":"arxiv","version":3},"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":"1012.2354","created_at":"2026-05-18T04:31:50.986742+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.2354v3","created_at":"2026-05-18T04:31:50.986742+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.2354","created_at":"2026-05-18T04:31:50.986742+00:00"},{"alias_kind":"pith_short_12","alias_value":"24AYIJTNRZ3R","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_16","alias_value":"24AYIJTNRZ3RK6PG","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_8","alias_value":"24AYIJTN","created_at":"2026-05-18T12:26:03.138858+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/24AYIJTNRZ3RK6PGM32GCT4W67","json":"https://pith.science/pith/24AYIJTNRZ3RK6PGM32GCT4W67.json","graph_json":"https://pith.science/api/pith-number/24AYIJTNRZ3RK6PGM32GCT4W67/graph.json","events_json":"https://pith.science/api/pith-number/24AYIJTNRZ3RK6PGM32GCT4W67/events.json","paper":"https://pith.science/paper/24AYIJTN"},"agent_actions":{"view_html":"https://pith.science/pith/24AYIJTNRZ3RK6PGM32GCT4W67","download_json":"https://pith.science/pith/24AYIJTNRZ3RK6PGM32GCT4W67.json","view_paper":"https://pith.science/paper/24AYIJTN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.2354&json=true","fetch_graph":"https://pith.science/api/pith-number/24AYIJTNRZ3RK6PGM32GCT4W67/graph.json","fetch_events":"https://pith.science/api/pith-number/24AYIJTNRZ3RK6PGM32GCT4W67/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/24AYIJTNRZ3RK6PGM32GCT4W67/action/timestamp_anchor","attest_storage":"https://pith.science/pith/24AYIJTNRZ3RK6PGM32GCT4W67/action/storage_attestation","attest_author":"https://pith.science/pith/24AYIJTNRZ3RK6PGM32GCT4W67/action/author_attestation","sign_citation":"https://pith.science/pith/24AYIJTNRZ3RK6PGM32GCT4W67/action/citation_signature","submit_replication":"https://pith.science/pith/24AYIJTNRZ3RK6PGM32GCT4W67/action/replication_record"}},"created_at":"2026-05-18T04:31:50.986742+00:00","updated_at":"2026-05-18T04:31:50.986742+00:00"}