{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2022:XS3YUY5SGFHYLGGEBP4RSM5ONG","short_pith_number":"pith:XS3YUY5S","schema_version":"1.0","canonical_sha256":"bcb78a63b2314f8598c40bf91933ae69879a0f87b49af3a3f9b6b6aca7589753","source":{"kind":"arxiv","id":"2207.09576","version":4},"attestation_state":"computed","paper":{"title":"Relative Nakayama-Zariski decomposition and minimal models of generalized pairs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jihao Liu, Lingyao Xie","submitted_at":"2022-07-19T22:39:41Z","abstract_excerpt":"We prove some basic properties of the relative Nakayama-Zariski decomposition. We apply them to the study of lc generalized pairs. We prove the existence of log minimal models or Mori fiber spaces for (relative) lc generalized pairs polarized by an ample divisor. This extends a result of Hashizume-Hu to generalized pairs. We also show that, for any lc generalized pair $(X,B+A,{\\bf{M}})/Z$ such that $K_X+B+A+{\\bf{M}}_X\\sim_{\\mathbb R,Z}0$ and $B\\geq 0,A\\geq 0$, $(X,B,{\\bf{M}})/Z$ has either a log minimal model or a Mori fiber space. This is an analogue of a result of Birkar/Hacon-Xu and Hashizu"},"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":"2207.09576","kind":"arxiv","version":4},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.AG","submitted_at":"2022-07-19T22:39:41Z","cross_cats_sorted":[],"title_canon_sha256":"3c4e8ed14f23014f983be579186f964b5ee673832c95a2efe744be02ac2f1639","abstract_canon_sha256":"68e57c1408afc9546801397cdcf28b55c57336eddd4c92b6acc697d2508e189e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:11:56.229255Z","signature_b64":"XawkGRFCVT/YKzvWrbx6WKpojRyjg6jGZ97/dBGxMYBLCcSwOvGA6ZI0ZeyTSj96Owal8XturvoYlxmvTOebAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bcb78a63b2314f8598c40bf91933ae69879a0f87b49af3a3f9b6b6aca7589753","last_reissued_at":"2026-07-05T06:11:56.228839Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:11:56.228839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Relative Nakayama-Zariski decomposition and minimal models of generalized pairs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Jihao Liu, Lingyao Xie","submitted_at":"2022-07-19T22:39:41Z","abstract_excerpt":"We prove some basic properties of the relative Nakayama-Zariski decomposition. We apply them to the study of lc generalized pairs. We prove the existence of log minimal models or Mori fiber spaces for (relative) lc generalized pairs polarized by an ample divisor. This extends a result of Hashizume-Hu to generalized pairs. We also show that, for any lc generalized pair $(X,B+A,{\\bf{M}})/Z$ such that $K_X+B+A+{\\bf{M}}_X\\sim_{\\mathbb R,Z}0$ and $B\\geq 0,A\\geq 0$, $(X,B,{\\bf{M}})/Z$ has either a log minimal model or a Mori fiber space. This is an analogue of a result of Birkar/Hacon-Xu and Hashizu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.09576","kind":"arxiv","version":4},"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/2207.09576/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":"2207.09576","created_at":"2026-07-05T06:11:56.228893+00:00"},{"alias_kind":"arxiv_version","alias_value":"2207.09576v4","created_at":"2026-07-05T06:11:56.228893+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.09576","created_at":"2026-07-05T06:11:56.228893+00:00"},{"alias_kind":"pith_short_12","alias_value":"XS3YUY5SGFHY","created_at":"2026-07-05T06:11:56.228893+00:00"},{"alias_kind":"pith_short_16","alias_value":"XS3YUY5SGFHYLGGE","created_at":"2026-07-05T06:11:56.228893+00:00"},{"alias_kind":"pith_short_8","alias_value":"XS3YUY5S","created_at":"2026-07-05T06:11:56.228893+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/XS3YUY5SGFHYLGGEBP4RSM5ONG","json":"https://pith.science/pith/XS3YUY5SGFHYLGGEBP4RSM5ONG.json","graph_json":"https://pith.science/api/pith-number/XS3YUY5SGFHYLGGEBP4RSM5ONG/graph.json","events_json":"https://pith.science/api/pith-number/XS3YUY5SGFHYLGGEBP4RSM5ONG/events.json","paper":"https://pith.science/paper/XS3YUY5S"},"agent_actions":{"view_html":"https://pith.science/pith/XS3YUY5SGFHYLGGEBP4RSM5ONG","download_json":"https://pith.science/pith/XS3YUY5SGFHYLGGEBP4RSM5ONG.json","view_paper":"https://pith.science/paper/XS3YUY5S","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2207.09576&json=true","fetch_graph":"https://pith.science/api/pith-number/XS3YUY5SGFHYLGGEBP4RSM5ONG/graph.json","fetch_events":"https://pith.science/api/pith-number/XS3YUY5SGFHYLGGEBP4RSM5ONG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XS3YUY5SGFHYLGGEBP4RSM5ONG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XS3YUY5SGFHYLGGEBP4RSM5ONG/action/storage_attestation","attest_author":"https://pith.science/pith/XS3YUY5SGFHYLGGEBP4RSM5ONG/action/author_attestation","sign_citation":"https://pith.science/pith/XS3YUY5SGFHYLGGEBP4RSM5ONG/action/citation_signature","submit_replication":"https://pith.science/pith/XS3YUY5SGFHYLGGEBP4RSM5ONG/action/replication_record"}},"created_at":"2026-07-05T06:11:56.228893+00:00","updated_at":"2026-07-05T06:11:56.228893+00:00"}