{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:RQEXFCI5IQIFESZ54FS3O476HT","short_pith_number":"pith:RQEXFCI5","canonical_record":{"source":{"id":"1309.4698","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-18T16:36:53Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"abafbf4d84b58f022f42d399ac22c3e229a4b208f3c18397b167ad8332772bc4","abstract_canon_sha256":"ee1c7571355c9fa9802bf42e72f69163edd6ab8a47491403f63a71f3e1487ee6"},"schema_version":"1.0"},"canonical_sha256":"8c0972891d4410524b3de165b773fe3cee443fc2256404ee4ccfd6a09fdc0268","source":{"kind":"arxiv","id":"1309.4698","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4698","created_at":"2026-05-18T02:49:31Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4698v4","created_at":"2026-05-18T02:49:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4698","created_at":"2026-05-18T02:49:31Z"},{"alias_kind":"pith_short_12","alias_value":"RQEXFCI5IQIF","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"RQEXFCI5IQIFESZ5","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"RQEXFCI5","created_at":"2026-05-18T12:27:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:RQEXFCI5IQIFESZ54FS3O476HT","target":"record","payload":{"canonical_record":{"source":{"id":"1309.4698","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-18T16:36:53Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"abafbf4d84b58f022f42d399ac22c3e229a4b208f3c18397b167ad8332772bc4","abstract_canon_sha256":"ee1c7571355c9fa9802bf42e72f69163edd6ab8a47491403f63a71f3e1487ee6"},"schema_version":"1.0"},"canonical_sha256":"8c0972891d4410524b3de165b773fe3cee443fc2256404ee4ccfd6a09fdc0268","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:31.734811Z","signature_b64":"rvIi4+NKLmaPj5fvro1xOhqMl+n8cAKQ4ny3jXsN96BnIw1QPUFx3yUEJ8E7rcMIjbpJIVpM51OAZ44mUho0CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c0972891d4410524b3de165b773fe3cee443fc2256404ee4ccfd6a09fdc0268","last_reissued_at":"2026-05-18T02:49:31.734388Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:31.734388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.4698","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:49:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ViJADxa0exqn06VkgAnWfng0z0w6ssgVq4nqtCSJXP9J61EaLPEPq7Gh88IMPCDn1mShnCHOmcTa2t+a6uX3DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T05:52:46.530717Z"},"content_sha256":"e281cbe6fdc9398e957955d68139a403dea661d6a2ba259cb9b2e8d5f5e0eb96","schema_version":"1.0","event_id":"sha256:e281cbe6fdc9398e957955d68139a403dea661d6a2ba259cb9b2e8d5f5e0eb96"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:RQEXFCI5IQIFESZ54FS3O476HT","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Koszul determinantal rings and $2\\times e$ matrices of linear forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AC","authors_text":"Hop D. Nguyen, Phong Dinh Thieu, Thanh Vu","submitted_at":"2013-09-18T16:36:53Z","abstract_excerpt":"Let $k$ be an algebraically closed field of characteristic $0$. Let $X$ be a $2\\times e$ matrix of linear forms over a polynomial ring $k[\\mathsf{x}_1, \\ldots,\\mathsf{x}_n]$ (where $e,n\\ge 1$). We prove that the determinantal ring $R = k[\\mathsf{x}_1,\\ldots,\\mathsf{x}_n]/I_2(X)$ is Koszul if and only if in the Kronecker-Weierstrass normal form of $X$, the largest length of a nilpotent block is at most twice the smallest length of a scroll block. As an application, we classify rational normal scrolls whose all section rings by natural coordinates are Koszul. This result settles a conjecture due"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4698","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":""},"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":"2026-05-18T02:49:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g84WvL/5QT/qrRWHo4+MWFXaHLPmrjuOkvPpPdFLDbM6uCQ/wHjFP5IAAjhcqFtLAkB2fnS1zgW+AC9wLg4vBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T05:52:46.531417Z"},"content_sha256":"8dfa10ec997fbd2cdc32d7c5b161216a732a5936e5e9ff8e37e7063b38bf9afd","schema_version":"1.0","event_id":"sha256:8dfa10ec997fbd2cdc32d7c5b161216a732a5936e5e9ff8e37e7063b38bf9afd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RQEXFCI5IQIFESZ54FS3O476HT/bundle.json","state_url":"https://pith.science/pith/RQEXFCI5IQIFESZ54FS3O476HT/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RQEXFCI5IQIFESZ54FS3O476HT/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":"2026-05-30T05:52:46Z","links":{"resolver":"https://pith.science/pith/RQEXFCI5IQIFESZ54FS3O476HT","bundle":"https://pith.science/pith/RQEXFCI5IQIFESZ54FS3O476HT/bundle.json","state":"https://pith.science/pith/RQEXFCI5IQIFESZ54FS3O476HT/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RQEXFCI5IQIFESZ54FS3O476HT/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:RQEXFCI5IQIFESZ54FS3O476HT","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":"ee1c7571355c9fa9802bf42e72f69163edd6ab8a47491403f63a71f3e1487ee6","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-18T16:36:53Z","title_canon_sha256":"abafbf4d84b58f022f42d399ac22c3e229a4b208f3c18397b167ad8332772bc4"},"schema_version":"1.0","source":{"id":"1309.4698","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.4698","created_at":"2026-05-18T02:49:31Z"},{"alias_kind":"arxiv_version","alias_value":"1309.4698v4","created_at":"2026-05-18T02:49:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.4698","created_at":"2026-05-18T02:49:31Z"},{"alias_kind":"pith_short_12","alias_value":"RQEXFCI5IQIF","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_16","alias_value":"RQEXFCI5IQIFESZ5","created_at":"2026-05-18T12:27:59Z"},{"alias_kind":"pith_short_8","alias_value":"RQEXFCI5","created_at":"2026-05-18T12:27:59Z"}],"graph_snapshots":[{"event_id":"sha256:8dfa10ec997fbd2cdc32d7c5b161216a732a5936e5e9ff8e37e7063b38bf9afd","target":"graph","created_at":"2026-05-18T02:49:31Z","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":"Let $k$ be an algebraically closed field of characteristic $0$. Let $X$ be a $2\\times e$ matrix of linear forms over a polynomial ring $k[\\mathsf{x}_1, \\ldots,\\mathsf{x}_n]$ (where $e,n\\ge 1$). We prove that the determinantal ring $R = k[\\mathsf{x}_1,\\ldots,\\mathsf{x}_n]/I_2(X)$ is Koszul if and only if in the Kronecker-Weierstrass normal form of $X$, the largest length of a nilpotent block is at most twice the smallest length of a scroll block. As an application, we classify rational normal scrolls whose all section rings by natural coordinates are Koszul. This result settles a conjecture due","authors_text":"Hop D. Nguyen, Phong Dinh Thieu, Thanh Vu","cross_cats":["math.RA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-18T16:36:53Z","title":"Koszul determinantal rings and $2\\times e$ matrices of linear forms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.4698","kind":"arxiv","version":4},"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:e281cbe6fdc9398e957955d68139a403dea661d6a2ba259cb9b2e8d5f5e0eb96","target":"record","created_at":"2026-05-18T02:49:31Z","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":"ee1c7571355c9fa9802bf42e72f69163edd6ab8a47491403f63a71f3e1487ee6","cross_cats_sorted":["math.RA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-09-18T16:36:53Z","title_canon_sha256":"abafbf4d84b58f022f42d399ac22c3e229a4b208f3c18397b167ad8332772bc4"},"schema_version":"1.0","source":{"id":"1309.4698","kind":"arxiv","version":4}},"canonical_sha256":"8c0972891d4410524b3de165b773fe3cee443fc2256404ee4ccfd6a09fdc0268","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c0972891d4410524b3de165b773fe3cee443fc2256404ee4ccfd6a09fdc0268","first_computed_at":"2026-05-18T02:49:31.734388Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:49:31.734388Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rvIi4+NKLmaPj5fvro1xOhqMl+n8cAKQ4ny3jXsN96BnIw1QPUFx3yUEJ8E7rcMIjbpJIVpM51OAZ44mUho0CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:49:31.734811Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.4698","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e281cbe6fdc9398e957955d68139a403dea661d6a2ba259cb9b2e8d5f5e0eb96","sha256:8dfa10ec997fbd2cdc32d7c5b161216a732a5936e5e9ff8e37e7063b38bf9afd"],"state_sha256":"61a0fa4eda2e96e00344534067c25084f439502bfd985a264bc1f04db6720da5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"d3nwQjGKda36BqvLsW64bjbUa3x0YX0RjnLCks0OB4tW8VqFGIon0vE74XvkOPl0/ryGD8EiWlfj3KulVJOqBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T05:52:46.535102Z","bundle_sha256":"cd211e79e16b1d67b64672f78dbc437a994a131a56e1b58c0db715b72c30af8c"}}