{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2002:EPPSOFGA5ANB4QJQMTJS4SJNMI","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":"1badfac0e40e82566b04c7ef6b097a813b2d4abcaff84994dc2bcaa0620e2b84","cross_cats_sorted":[],"license":"","primary_cat":"math.AG","submitted_at":"2002-04-12T01:54:10Z","title_canon_sha256":"fc7bdc4c187e5f79f9630cea9563ea5a234fa9caac58406e766dbe3f3f8e1486"},"schema_version":"1.0","source":{"id":"math/0204160","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0204160","created_at":"2026-05-18T04:11:18Z"},{"alias_kind":"arxiv_version","alias_value":"math/0204160v3","created_at":"2026-05-18T04:11:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0204160","created_at":"2026-05-18T04:11:18Z"},{"alias_kind":"pith_short_12","alias_value":"EPPSOFGA5ANB","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_16","alias_value":"EPPSOFGA5ANB4QJQ","created_at":"2026-05-18T12:25:50Z"},{"alias_kind":"pith_short_8","alias_value":"EPPSOFGA","created_at":"2026-05-18T12:25:50Z"}],"graph_snapshots":[{"event_id":"sha256:db73f6e412fa7b87e2c83a8f299fb1c46ae3e6c977d8b0377e6fb3fb266b324b","target":"graph","created_at":"2026-05-18T04:11:18Z","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"},"paper":{"abstract_excerpt":"We give a survey of the background and recent development on the $K$-equivalence relation among birational manifolds. After a brief historical sketch of birational geometry, we define the $K$-partial ordering and $K$-equivalence in a birational class and discuss geometric situations that lead to these notions. One application to the filling-in problem for threefolds is given.\n  We discuss the motivic aspect of $K$-equivalence relation. We believe that $K$-equivalent manifolds have the same Chow motive though we are unable to prove it at this moment. Instead we discuss various approaches toward","authors_text":"Chin-Lung Wang","cross_cats":[],"headline":"","license":"","primary_cat":"math.AG","submitted_at":"2002-04-12T01:54:10Z","title":"K-equivalence in Birational Geometry"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0204160","kind":"arxiv","version":3},"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":"sha256:567121d0d5949f13013e520cc82abad297061a25b3a757b34ffa2abafe50856f","target":"record","created_at":"2026-05-18T04:11:18Z","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":"1badfac0e40e82566b04c7ef6b097a813b2d4abcaff84994dc2bcaa0620e2b84","cross_cats_sorted":[],"license":"","primary_cat":"math.AG","submitted_at":"2002-04-12T01:54:10Z","title_canon_sha256":"fc7bdc4c187e5f79f9630cea9563ea5a234fa9caac58406e766dbe3f3f8e1486"},"schema_version":"1.0","source":{"id":"math/0204160","kind":"arxiv","version":3}},"canonical_sha256":"23df2714c0e81a1e413064d32e492d6206c5c5b7604ffc5714a93187b6268c6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"23df2714c0e81a1e413064d32e492d6206c5c5b7604ffc5714a93187b6268c6c","first_computed_at":"2026-05-18T04:11:18.375834Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:18.375834Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dJax02bwE5mIV5BT6bM2Lu/EAAhnW+FiWalT+Et9ITIQkqjPLGrBqfNfU4rWfxLJX0AqQMU2BdJXvgz/F9KNAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:18.376435Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0204160","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:567121d0d5949f13013e520cc82abad297061a25b3a757b34ffa2abafe50856f","sha256:db73f6e412fa7b87e2c83a8f299fb1c46ae3e6c977d8b0377e6fb3fb266b324b"],"state_sha256":"851b21290ea06ad568ff040cad6843f3167fc413a1468ddcccc5c33d092c6441"}