{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:TDI4JDPLXVETQH2A3VKTK6DO3D","short_pith_number":"pith:TDI4JDPL","schema_version":"1.0","canonical_sha256":"98d1c48debbd49381f40dd5535786ed8df07d1c2cc773775cfe455fc1eb0d9d5","source":{"kind":"arxiv","id":"1201.5464","version":1},"attestation_state":"computed","paper":{"title":"Numerical invariants of Fano 4-folds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"C. Casagrande","submitted_at":"2012-01-26T09:18:57Z","abstract_excerpt":"Let X be a (smooth, complex) Fano 4-fold. For any prime divisor D in X, consider the image of N_1(D) in N_1(X) under the push-forward of 1-cycles, and let c_D be its codimension in N_1(X). We define an integral invariant c_X of X as the maximal c_D, where D varies among all prime divisors in X. One easily sees that c_X is at most rho_X-1 (where rho is the Picard number), and that c_X is greater or equal than rho_X-rho_D, for any prime divisor D in X. We know from previous works that if c_X > 2, then either X is a product of Del Pezzo surfaces and rho_X is at most 18, or c_X=3 and rho_X is at m"},"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":"1201.5464","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-01-26T09:18:57Z","cross_cats_sorted":[],"title_canon_sha256":"abda8eaf235f5056cdc8f2b526f1453278bb556744a2ff2ffe33302534c26805","abstract_canon_sha256":"96cc7360a54519a74247b3099abeba466f3e40c8a2261c459a80fd53c16d807f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:53.942023Z","signature_b64":"2kg6CEe4dj45DNmFCOuOT1H/beCFLEuu3oNNZ7YUuzHae1E+82pxBwpkv0JofW8HQ0LjWv7RweasjuOSc2uKDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"98d1c48debbd49381f40dd5535786ed8df07d1c2cc773775cfe455fc1eb0d9d5","last_reissued_at":"2026-05-18T04:03:53.941474Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:53.941474Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Numerical invariants of Fano 4-folds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"C. Casagrande","submitted_at":"2012-01-26T09:18:57Z","abstract_excerpt":"Let X be a (smooth, complex) Fano 4-fold. For any prime divisor D in X, consider the image of N_1(D) in N_1(X) under the push-forward of 1-cycles, and let c_D be its codimension in N_1(X). We define an integral invariant c_X of X as the maximal c_D, where D varies among all prime divisors in X. One easily sees that c_X is at most rho_X-1 (where rho is the Picard number), and that c_X is greater or equal than rho_X-rho_D, for any prime divisor D in X. We know from previous works that if c_X > 2, then either X is a product of Del Pezzo surfaces and rho_X is at most 18, or c_X=3 and rho_X is at m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5464","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":"1201.5464","created_at":"2026-05-18T04:03:53.941551+00:00"},{"alias_kind":"arxiv_version","alias_value":"1201.5464v1","created_at":"2026-05-18T04:03:53.941551+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.5464","created_at":"2026-05-18T04:03:53.941551+00:00"},{"alias_kind":"pith_short_12","alias_value":"TDI4JDPLXVET","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_16","alias_value":"TDI4JDPLXVETQH2A","created_at":"2026-05-18T12:27:23.164592+00:00"},{"alias_kind":"pith_short_8","alias_value":"TDI4JDPL","created_at":"2026-05-18T12:27:23.164592+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/TDI4JDPLXVETQH2A3VKTK6DO3D","json":"https://pith.science/pith/TDI4JDPLXVETQH2A3VKTK6DO3D.json","graph_json":"https://pith.science/api/pith-number/TDI4JDPLXVETQH2A3VKTK6DO3D/graph.json","events_json":"https://pith.science/api/pith-number/TDI4JDPLXVETQH2A3VKTK6DO3D/events.json","paper":"https://pith.science/paper/TDI4JDPL"},"agent_actions":{"view_html":"https://pith.science/pith/TDI4JDPLXVETQH2A3VKTK6DO3D","download_json":"https://pith.science/pith/TDI4JDPLXVETQH2A3VKTK6DO3D.json","view_paper":"https://pith.science/paper/TDI4JDPL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1201.5464&json=true","fetch_graph":"https://pith.science/api/pith-number/TDI4JDPLXVETQH2A3VKTK6DO3D/graph.json","fetch_events":"https://pith.science/api/pith-number/TDI4JDPLXVETQH2A3VKTK6DO3D/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TDI4JDPLXVETQH2A3VKTK6DO3D/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TDI4JDPLXVETQH2A3VKTK6DO3D/action/storage_attestation","attest_author":"https://pith.science/pith/TDI4JDPLXVETQH2A3VKTK6DO3D/action/author_attestation","sign_citation":"https://pith.science/pith/TDI4JDPLXVETQH2A3VKTK6DO3D/action/citation_signature","submit_replication":"https://pith.science/pith/TDI4JDPLXVETQH2A3VKTK6DO3D/action/replication_record"}},"created_at":"2026-05-18T04:03:53.941551+00:00","updated_at":"2026-05-18T04:03:53.941551+00:00"}