{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:BPY5YFYYQXIPDW4FKLSLRRNN4I","short_pith_number":"pith:BPY5YFYY","canonical_record":{"source":{"id":"math/0610957","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AG","submitted_at":"2006-10-31T05:37:40Z","cross_cats_sorted":[],"title_canon_sha256":"e31346ea283ea15e0815192308a5d8c8a00e502ea497c485d11ede903e7446df","abstract_canon_sha256":"b28bfc805a15455952ff4a709fb8c5678a93868494eb52215856e6387459d637"},"schema_version":"1.0"},"canonical_sha256":"0bf1dc171885d0f1db8552e4b8c5ade2059e6f0832b3dc11bc5ce64a2ad8b85a","source":{"kind":"arxiv","id":"math/0610957","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610957","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610957v1","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610957","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"pith_short_12","alias_value":"BPY5YFYYQXIP","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"pith_short_16","alias_value":"BPY5YFYYQXIPDW4F","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"pith_short_8","alias_value":"BPY5YFYY","created_at":"2026-07-04T14:57:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:BPY5YFYYQXIPDW4FKLSLRRNN4I","target":"record","payload":{"canonical_record":{"source":{"id":"math/0610957","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AG","submitted_at":"2006-10-31T05:37:40Z","cross_cats_sorted":[],"title_canon_sha256":"e31346ea283ea15e0815192308a5d8c8a00e502ea497c485d11ede903e7446df","abstract_canon_sha256":"b28bfc805a15455952ff4a709fb8c5678a93868494eb52215856e6387459d637"},"schema_version":"1.0"},"canonical_sha256":"0bf1dc171885d0f1db8552e4b8c5ade2059e6f0832b3dc11bc5ce64a2ad8b85a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T14:57:11.837693Z","signature_b64":"CAvLfLDHkMwubyaes+NYTEw3S5lnef057MMauajlpHdhYq+7/aaPXKB8MjlBvG/iI5hFVLqiw3ZHyh49X8omBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0bf1dc171885d0f1db8552e4b8c5ade2059e6f0832b3dc11bc5ce64a2ad8b85a","last_reissued_at":"2026-07-04T14:57:11.837220Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T14:57:11.837220Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0610957","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-04T14:57:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IvLWAphb79mV/7+IGJVbwx655sp03CKKs2LGI/56HxJtmPmveEltSa60MSKZi9pGjHyhK0dQl8ru88qYBPMkCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T10:12:43.948295Z"},"content_sha256":"b51fdaf3716b38a64ec406ea12adc1f10a248344d718953f9f01b6050fcb625c","schema_version":"1.0","event_id":"sha256:b51fdaf3716b38a64ec406ea12adc1f10a248344d718953f9f01b6050fcb625c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:BPY5YFYYQXIPDW4FKLSLRRNN4I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Homological projective duality for Grassmannians of lines","license":"","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Alexander Kuznetsov","submitted_at":"2006-10-31T05:37:40Z","abstract_excerpt":"We show that homologically projectively dual varieties for Grassmannians Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived categories of linear sections of these Grassmannians and Pfaffians. In particular, we show that\n  (1) the derived category of a Pfaffian cubic 4-fold admits a semiorthogonal decompositions consisting of 3 exceptional line bundles, and of the derived category of a K3-surface;\n  (2) mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and of the correspo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610957","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/math/0610957/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":"2026-07-04T14:57:11Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wrFc7uxWXcmOwImIiYTWuuDiAsLpG5Jv50x9URpi9o30U7NylxDMTj4YFBXdrWcCysmdH9kZPSsfZ5MGFAgwBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-16T10:12:43.948891Z"},"content_sha256":"ba549f2f3b33d7e4c9c5ce8dc2d9004a78c1af36a4e4445046156f27c9eec9b1","schema_version":"1.0","event_id":"sha256:ba549f2f3b33d7e4c9c5ce8dc2d9004a78c1af36a4e4445046156f27c9eec9b1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BPY5YFYYQXIPDW4FKLSLRRNN4I/bundle.json","state_url":"https://pith.science/pith/BPY5YFYYQXIPDW4FKLSLRRNN4I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BPY5YFYYQXIPDW4FKLSLRRNN4I/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":"2026-08-16T10:12:43Z","links":{"resolver":"https://pith.science/pith/BPY5YFYYQXIPDW4FKLSLRRNN4I","bundle":"https://pith.science/pith/BPY5YFYYQXIPDW4FKLSLRRNN4I/bundle.json","state":"https://pith.science/pith/BPY5YFYYQXIPDW4FKLSLRRNN4I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BPY5YFYYQXIPDW4FKLSLRRNN4I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:BPY5YFYYQXIPDW4FKLSLRRNN4I","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":"b28bfc805a15455952ff4a709fb8c5678a93868494eb52215856e6387459d637","cross_cats_sorted":[],"license":"","primary_cat":"math.AG","submitted_at":"2006-10-31T05:37:40Z","title_canon_sha256":"e31346ea283ea15e0815192308a5d8c8a00e502ea497c485d11ede903e7446df"},"schema_version":"1.0","source":{"id":"math/0610957","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0610957","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"arxiv_version","alias_value":"math/0610957v1","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0610957","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"pith_short_12","alias_value":"BPY5YFYYQXIP","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"pith_short_16","alias_value":"BPY5YFYYQXIPDW4F","created_at":"2026-07-04T14:57:11Z"},{"alias_kind":"pith_short_8","alias_value":"BPY5YFYY","created_at":"2026-07-04T14:57:11Z"}],"graph_snapshots":[{"event_id":"sha256:ba549f2f3b33d7e4c9c5ce8dc2d9004a78c1af36a4e4445046156f27c9eec9b1","target":"graph","created_at":"2026-07-04T14:57:11Z","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/math/0610957/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that homologically projectively dual varieties for Grassmannians Gr(2,6) and Gr(2,7) are given by certain noncommutative resolutions of singularities of the corresponding Pfaffian varieties. As an application we describe the derived categories of linear sections of these Grassmannians and Pfaffians. In particular, we show that\n  (1) the derived category of a Pfaffian cubic 4-fold admits a semiorthogonal decompositions consisting of 3 exceptional line bundles, and of the derived category of a K3-surface;\n  (2) mutually orthogonal Calabi-Yau linear sections of Gr(2,7) and of the correspo","authors_text":"Alexander Kuznetsov","cross_cats":[],"headline":"","license":"","primary_cat":"math.AG","submitted_at":"2006-10-31T05:37:40Z","title":"Homological projective duality for Grassmannians of lines"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0610957","kind":"arxiv","version":1},"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:b51fdaf3716b38a64ec406ea12adc1f10a248344d718953f9f01b6050fcb625c","target":"record","created_at":"2026-07-04T14:57:11Z","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":"b28bfc805a15455952ff4a709fb8c5678a93868494eb52215856e6387459d637","cross_cats_sorted":[],"license":"","primary_cat":"math.AG","submitted_at":"2006-10-31T05:37:40Z","title_canon_sha256":"e31346ea283ea15e0815192308a5d8c8a00e502ea497c485d11ede903e7446df"},"schema_version":"1.0","source":{"id":"math/0610957","kind":"arxiv","version":1}},"canonical_sha256":"0bf1dc171885d0f1db8552e4b8c5ade2059e6f0832b3dc11bc5ce64a2ad8b85a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0bf1dc171885d0f1db8552e4b8c5ade2059e6f0832b3dc11bc5ce64a2ad8b85a","first_computed_at":"2026-07-04T14:57:11.837220Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T14:57:11.837220Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CAvLfLDHkMwubyaes+NYTEw3S5lnef057MMauajlpHdhYq+7/aaPXKB8MjlBvG/iI5hFVLqiw3ZHyh49X8omBA==","signature_status":"signed_v1","signed_at":"2026-07-04T14:57:11.837693Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0610957","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b51fdaf3716b38a64ec406ea12adc1f10a248344d718953f9f01b6050fcb625c","sha256:ba549f2f3b33d7e4c9c5ce8dc2d9004a78c1af36a4e4445046156f27c9eec9b1"],"state_sha256":"46b1b5886c659bded98218ed3944d1188cc0e1c2dcdbeb202b0a9e6928cb0b2c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Ohyc6fGnwiv3wKO98peRecLlm67rXvU+le65s8AHb54ZQx9lrAQ6V2gedOuEAMmBIao8UTlFT/pxSiC/dodNDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-16T10:12:43.953612Z","bundle_sha256":"9a168ce60765ea6d6fcfbbbf1fc7646775e7c9a5d415293c43bb3d8c35a80593"}}