{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:73SLQWCNH4JZYVICECLCZH74RK","short_pith_number":"pith:73SLQWCN","schema_version":"1.0","canonical_sha256":"fee4b8584d3f139c550220962c9ffc8ab964da0a8c5df1600b56307fec6ee986","source":{"kind":"arxiv","id":"2105.13951","version":2},"attestation_state":"computed","paper":{"title":"Planes in cubic fourfolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alex Degtyarev, Ilia Itenberg, John Christian Ottem","submitted_at":"2021-05-28T16:19:01Z","abstract_excerpt":"We show that the maximal number of planes in a complex smooth cubic fourfold in ${\\mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the so-called Clebsch--Segre cubic. Altogether, there are but three (up to projective equivalence) cubics with more than $350$ planes."},"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":"2105.13951","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-05-28T16:19:01Z","cross_cats_sorted":[],"title_canon_sha256":"39327d951c52a75080a81b3bfe50b91e70959e5c19510578d240c5b1c7ab8388","abstract_canon_sha256":"16c06f5c0193dcc20e5687dd8df0d3fc66d6f058df275c098750761f3228ae5c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:56:10.282779Z","signature_b64":"IBItOlAO0GgqIQK3MCFqODrx3n9NvdXJAA3u/KCYQdRsEcPUi9YQNBgXeC4fm3wY7SFdcVKZNon+9pbapKvxCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fee4b8584d3f139c550220962c9ffc8ab964da0a8c5df1600b56307fec6ee986","last_reissued_at":"2026-07-05T08:56:10.282393Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:56:10.282393Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Planes in cubic fourfolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alex Degtyarev, Ilia Itenberg, John Christian Ottem","submitted_at":"2021-05-28T16:19:01Z","abstract_excerpt":"We show that the maximal number of planes in a complex smooth cubic fourfold in ${\\mathbb P}^5$ is $405$, realized by the Fermat cubic only; the maximal number of real planes in a real smooth cubic fourfold is $357$, realized by the so-called Clebsch--Segre cubic. Altogether, there are but three (up to projective equivalence) cubics with more than $350$ planes."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.13951","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2105.13951/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2105.13951","created_at":"2026-07-05T08:56:10.282455+00:00"},{"alias_kind":"arxiv_version","alias_value":"2105.13951v2","created_at":"2026-07-05T08:56:10.282455+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.13951","created_at":"2026-07-05T08:56:10.282455+00:00"},{"alias_kind":"pith_short_12","alias_value":"73SLQWCNH4JZ","created_at":"2026-07-05T08:56:10.282455+00:00"},{"alias_kind":"pith_short_16","alias_value":"73SLQWCNH4JZYVIC","created_at":"2026-07-05T08:56:10.282455+00:00"},{"alias_kind":"pith_short_8","alias_value":"73SLQWCN","created_at":"2026-07-05T08:56:10.282455+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2509.06817","citing_title":"Rational cubic fourfolds with a symplectic group of automorphisms","ref_index":5,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/73SLQWCNH4JZYVICECLCZH74RK","json":"https://pith.science/pith/73SLQWCNH4JZYVICECLCZH74RK.json","graph_json":"https://pith.science/api/pith-number/73SLQWCNH4JZYVICECLCZH74RK/graph.json","events_json":"https://pith.science/api/pith-number/73SLQWCNH4JZYVICECLCZH74RK/events.json","paper":"https://pith.science/paper/73SLQWCN"},"agent_actions":{"view_html":"https://pith.science/pith/73SLQWCNH4JZYVICECLCZH74RK","download_json":"https://pith.science/pith/73SLQWCNH4JZYVICECLCZH74RK.json","view_paper":"https://pith.science/paper/73SLQWCN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2105.13951&json=true","fetch_graph":"https://pith.science/api/pith-number/73SLQWCNH4JZYVICECLCZH74RK/graph.json","fetch_events":"https://pith.science/api/pith-number/73SLQWCNH4JZYVICECLCZH74RK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/73SLQWCNH4JZYVICECLCZH74RK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/73SLQWCNH4JZYVICECLCZH74RK/action/storage_attestation","attest_author":"https://pith.science/pith/73SLQWCNH4JZYVICECLCZH74RK/action/author_attestation","sign_citation":"https://pith.science/pith/73SLQWCNH4JZYVICECLCZH74RK/action/citation_signature","submit_replication":"https://pith.science/pith/73SLQWCNH4JZYVICECLCZH74RK/action/replication_record"}},"created_at":"2026-07-05T08:56:10.282455+00:00","updated_at":"2026-07-05T08:56:10.282455+00:00"}