{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:FMHJB4P5MGJMYD3FU3DAU2WRHY","short_pith_number":"pith:FMHJB4P5","schema_version":"1.0","canonical_sha256":"2b0e90f1fd6192cc0f65a6c60a6ad13e218e2ad3984d88c62743c1bb798af5e1","source":{"kind":"arxiv","id":"2209.08797","version":1},"attestation_state":"computed","paper":{"title":"Boundedness of Fano type fibrations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Caucher Birkar","submitted_at":"2022-09-19T07:01:24Z","abstract_excerpt":"In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano over $Z$, that is, $X$ has \"good\" singularities and $-K_X$ is ample over $Z$. A Fano type fibration is similarly defined where $X$ is assumed to be close to being Fano over $Z$. This class includes many central ingredients of birational geometry such as Fano varieties, Mori fibre spaces, flipping and divisorial contractions, crepant models, germs of singulari"},"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":"2209.08797","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2022-09-19T07:01:24Z","cross_cats_sorted":[],"title_canon_sha256":"47627a7a33f6a87b7c74fd1aa85037e6447443fedc5d65b82aad354d16f7dc46","abstract_canon_sha256":"75c6dc15bfc4561d1742c7b042b07c6795837ccc7b5bb4557f99af684a640f67"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T04:58:46.159103Z","signature_b64":"vf1so1g0AjJMO3mJ3smBW/BkUolBd9y6rsUp8IN58yCYfIV0Wehou1jkEe5bOFb5RMTtQsXj1h7lrwbbh2AtBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2b0e90f1fd6192cc0f65a6c60a6ad13e218e2ad3984d88c62743c1bb798af5e1","last_reissued_at":"2026-07-05T04:58:46.158676Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T04:58:46.158676Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Boundedness of Fano type fibrations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Caucher Birkar","submitted_at":"2022-09-19T07:01:24Z","abstract_excerpt":"In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano over $Z$, that is, $X$ has \"good\" singularities and $-K_X$ is ample over $Z$. A Fano type fibration is similarly defined where $X$ is assumed to be close to being Fano over $Z$. This class includes many central ingredients of birational geometry such as Fano varieties, Mori fibre spaces, flipping and divisorial contractions, crepant models, germs of singulari"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.08797","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2209.08797/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":"2209.08797","created_at":"2026-07-05T04:58:46.158730+00:00"},{"alias_kind":"arxiv_version","alias_value":"2209.08797v1","created_at":"2026-07-05T04:58:46.158730+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2209.08797","created_at":"2026-07-05T04:58:46.158730+00:00"},{"alias_kind":"pith_short_12","alias_value":"FMHJB4P5MGJM","created_at":"2026-07-05T04:58:46.158730+00:00"},{"alias_kind":"pith_short_16","alias_value":"FMHJB4P5MGJMYD3F","created_at":"2026-07-05T04:58:46.158730+00:00"},{"alias_kind":"pith_short_8","alias_value":"FMHJB4P5","created_at":"2026-07-05T04:58:46.158730+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2605.09898","citing_title":"Quasi-Projective Moduli for Polarized klt Good Minimal Models","ref_index":55,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FMHJB4P5MGJMYD3FU3DAU2WRHY","json":"https://pith.science/pith/FMHJB4P5MGJMYD3FU3DAU2WRHY.json","graph_json":"https://pith.science/api/pith-number/FMHJB4P5MGJMYD3FU3DAU2WRHY/graph.json","events_json":"https://pith.science/api/pith-number/FMHJB4P5MGJMYD3FU3DAU2WRHY/events.json","paper":"https://pith.science/paper/FMHJB4P5"},"agent_actions":{"view_html":"https://pith.science/pith/FMHJB4P5MGJMYD3FU3DAU2WRHY","download_json":"https://pith.science/pith/FMHJB4P5MGJMYD3FU3DAU2WRHY.json","view_paper":"https://pith.science/paper/FMHJB4P5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2209.08797&json=true","fetch_graph":"https://pith.science/api/pith-number/FMHJB4P5MGJMYD3FU3DAU2WRHY/graph.json","fetch_events":"https://pith.science/api/pith-number/FMHJB4P5MGJMYD3FU3DAU2WRHY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FMHJB4P5MGJMYD3FU3DAU2WRHY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FMHJB4P5MGJMYD3FU3DAU2WRHY/action/storage_attestation","attest_author":"https://pith.science/pith/FMHJB4P5MGJMYD3FU3DAU2WRHY/action/author_attestation","sign_citation":"https://pith.science/pith/FMHJB4P5MGJMYD3FU3DAU2WRHY/action/citation_signature","submit_replication":"https://pith.science/pith/FMHJB4P5MGJMYD3FU3DAU2WRHY/action/replication_record"}},"created_at":"2026-07-05T04:58:46.158730+00:00","updated_at":"2026-07-05T04:58:46.158730+00:00"}