{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:5YZP4I2JXY3XL25SIGNW75XLHS","short_pith_number":"pith:5YZP4I2J","canonical_record":{"source":{"id":"2103.14935","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-03-27T16:08:00Z","cross_cats_sorted":[],"title_canon_sha256":"ba1250034654dd70853fcabd98897b0390306117a8091f41f3881b13897c1fc3","abstract_canon_sha256":"d4b4b6b13cd39d62ba7b432a44c0e57ff4610148df7c3cd9d81a0a437c21c55e"},"schema_version":"1.0"},"canonical_sha256":"ee32fe2349be3775ebb2419b6ff6eb3cb94b3bc8bb0e8d7a55d5be5439747d1f","source":{"kind":"arxiv","id":"2103.14935","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2103.14935","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"arxiv_version","alias_value":"2103.14935v2","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2103.14935","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"pith_short_12","alias_value":"5YZP4I2JXY3X","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"pith_short_16","alias_value":"5YZP4I2JXY3XL25S","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"pith_short_8","alias_value":"5YZP4I2J","created_at":"2026-07-05T02:28:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:5YZP4I2JXY3XL25SIGNW75XLHS","target":"record","payload":{"canonical_record":{"source":{"id":"2103.14935","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-03-27T16:08:00Z","cross_cats_sorted":[],"title_canon_sha256":"ba1250034654dd70853fcabd98897b0390306117a8091f41f3881b13897c1fc3","abstract_canon_sha256":"d4b4b6b13cd39d62ba7b432a44c0e57ff4610148df7c3cd9d81a0a437c21c55e"},"schema_version":"1.0"},"canonical_sha256":"ee32fe2349be3775ebb2419b6ff6eb3cb94b3bc8bb0e8d7a55d5be5439747d1f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:28:56.614625Z","signature_b64":"fzjVv3mD7SrrSrJ/f9Cy/PkuqMcAtrgdZCnaEjMElx702wtFsuddwVssEB+L7xC8KF60U3NyPmp48sEy9A3KCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee32fe2349be3775ebb2419b6ff6eb3cb94b3bc8bb0e8d7a55d5be5439747d1f","last_reissued_at":"2026-07-05T02:28:56.614282Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:28:56.614282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2103.14935","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:28:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DcywNE7jXl8XZYnuB98rmLsWakvdMvqhDEwuFLcFa+4G3LTkVW3cT7RpkAocOcKovIx3xTKWtgucbMFY1MYFDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:47:16.388731Z"},"content_sha256":"476be2bc9da07010ac229e194c210cdd99c88bb627e23995ddb32d76099d3fd0","schema_version":"1.0","event_id":"sha256:476be2bc9da07010ac229e194c210cdd99c88bb627e23995ddb32d76099d3fd0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:5YZP4I2JXY3XL25SIGNW75XLHS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Boundedness and volume of generalised pairs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Caucher Birkar","submitted_at":"2021-03-27T16:08:00Z","abstract_excerpt":"In this paper we investigate boundedness and volumes of generalised pairs, and give applications to usual pairs especially to a class of pairs that we call stable log minimal models.\n  Fixing the dimension and a DCC set controlling coefficients, we will show that the set of volumes of all projective generalised lc pairs $(X,B+M)$ under the given data, satisfies the DCC. Futhermore, we will show that in the klt case, the set of such pairs with ample $K_X+B+M$ and fixed volume, forms a bounded family.\n  We prove a result about descent of nef divisors to bounded families. This is the key to provi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.14935","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/2103.14935/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T02:28:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Sq+f3ZhNYg9laPnvqfvUcGZ7Dhd0zEWUC3lRChSONlqzjxdJ9v1eT/V75MRhfvdT2Opn2QpXPPdIddbHgNP6BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T08:47:16.389252Z"},"content_sha256":"6ac41f75563df23440342c3ce14f1ad2953192ab8e0243db8ea1e15678f5f851","schema_version":"1.0","event_id":"sha256:6ac41f75563df23440342c3ce14f1ad2953192ab8e0243db8ea1e15678f5f851"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5YZP4I2JXY3XL25SIGNW75XLHS/bundle.json","state_url":"https://pith.science/pith/5YZP4I2JXY3XL25SIGNW75XLHS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5YZP4I2JXY3XL25SIGNW75XLHS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-12T08:47:16Z","links":{"resolver":"https://pith.science/pith/5YZP4I2JXY3XL25SIGNW75XLHS","bundle":"https://pith.science/pith/5YZP4I2JXY3XL25SIGNW75XLHS/bundle.json","state":"https://pith.science/pith/5YZP4I2JXY3XL25SIGNW75XLHS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5YZP4I2JXY3XL25SIGNW75XLHS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:5YZP4I2JXY3XL25SIGNW75XLHS","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d4b4b6b13cd39d62ba7b432a44c0e57ff4610148df7c3cd9d81a0a437c21c55e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-03-27T16:08:00Z","title_canon_sha256":"ba1250034654dd70853fcabd98897b0390306117a8091f41f3881b13897c1fc3"},"schema_version":"1.0","source":{"id":"2103.14935","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2103.14935","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"arxiv_version","alias_value":"2103.14935v2","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2103.14935","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"pith_short_12","alias_value":"5YZP4I2JXY3X","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"pith_short_16","alias_value":"5YZP4I2JXY3XL25S","created_at":"2026-07-05T02:28:56Z"},{"alias_kind":"pith_short_8","alias_value":"5YZP4I2J","created_at":"2026-07-05T02:28:56Z"}],"graph_snapshots":[{"event_id":"sha256:6ac41f75563df23440342c3ce14f1ad2953192ab8e0243db8ea1e15678f5f851","target":"graph","created_at":"2026-07-05T02:28:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2103.14935/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper we investigate boundedness and volumes of generalised pairs, and give applications to usual pairs especially to a class of pairs that we call stable log minimal models.\n  Fixing the dimension and a DCC set controlling coefficients, we will show that the set of volumes of all projective generalised lc pairs $(X,B+M)$ under the given data, satisfies the DCC. Futhermore, we will show that in the klt case, the set of such pairs with ample $K_X+B+M$ and fixed volume, forms a bounded family.\n  We prove a result about descent of nef divisors to bounded families. This is the key to provi","authors_text":"Caucher Birkar","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-03-27T16:08:00Z","title":"Boundedness and volume of generalised pairs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2103.14935","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:476be2bc9da07010ac229e194c210cdd99c88bb627e23995ddb32d76099d3fd0","target":"record","created_at":"2026-07-05T02:28:56Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d4b4b6b13cd39d62ba7b432a44c0e57ff4610148df7c3cd9d81a0a437c21c55e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2021-03-27T16:08:00Z","title_canon_sha256":"ba1250034654dd70853fcabd98897b0390306117a8091f41f3881b13897c1fc3"},"schema_version":"1.0","source":{"id":"2103.14935","kind":"arxiv","version":2}},"canonical_sha256":"ee32fe2349be3775ebb2419b6ff6eb3cb94b3bc8bb0e8d7a55d5be5439747d1f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee32fe2349be3775ebb2419b6ff6eb3cb94b3bc8bb0e8d7a55d5be5439747d1f","first_computed_at":"2026-07-05T02:28:56.614282Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T02:28:56.614282Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fzjVv3mD7SrrSrJ/f9Cy/PkuqMcAtrgdZCnaEjMElx702wtFsuddwVssEB+L7xC8KF60U3NyPmp48sEy9A3KCw==","signature_status":"signed_v1","signed_at":"2026-07-05T02:28:56.614625Z","signed_message":"canonical_sha256_bytes"},"source_id":"2103.14935","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:476be2bc9da07010ac229e194c210cdd99c88bb627e23995ddb32d76099d3fd0","sha256:6ac41f75563df23440342c3ce14f1ad2953192ab8e0243db8ea1e15678f5f851"],"state_sha256":"df0c620e0ef604dd7ee73b98e46271b676fd88491791eb4d4b62940e225a98cd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+/lS2ViKbHKeSI6J/1nRQDoHoqc2guLLyUJzUJug5KT+DQRkIzEWmQYKRnZ+OJRj+/bpgAEuOKMIJW6le89oDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T08:47:16.394716Z","bundle_sha256":"4cc86287c09b2d47b8345288efe9ad3f76c2bfe47b556125e1dbd85c2b7e9d0b"}}