{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:NU2S7BPVMQVDLMST7YHMURNCQ3","short_pith_number":"pith:NU2S7BPV","canonical_record":{"source":{"id":"1207.7153","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-07-31T03:17:30Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"0ce3531b102d83a0757265d2bbc41ce677937dbed3dc1ec1be443c784596b50f","abstract_canon_sha256":"4435bcaa8edc74dc8e92c06cd163e4ad5ce262533abd0515513cf722e94cbb09"},"schema_version":"1.0"},"canonical_sha256":"6d352f85f5642a35b253fe0eca45a286c038c3781fb7918da3212d9fb90b3444","source":{"kind":"arxiv","id":"1207.7153","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.7153","created_at":"2026-05-18T03:25:43Z"},{"alias_kind":"arxiv_version","alias_value":"1207.7153v2","created_at":"2026-05-18T03:25:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.7153","created_at":"2026-05-18T03:25:43Z"},{"alias_kind":"pith_short_12","alias_value":"NU2S7BPVMQVD","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NU2S7BPVMQVDLMST","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NU2S7BPV","created_at":"2026-05-18T12:27:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:NU2S7BPVMQVDLMST7YHMURNCQ3","target":"record","payload":{"canonical_record":{"source":{"id":"1207.7153","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-07-31T03:17:30Z","cross_cats_sorted":["math.AC"],"title_canon_sha256":"0ce3531b102d83a0757265d2bbc41ce677937dbed3dc1ec1be443c784596b50f","abstract_canon_sha256":"4435bcaa8edc74dc8e92c06cd163e4ad5ce262533abd0515513cf722e94cbb09"},"schema_version":"1.0"},"canonical_sha256":"6d352f85f5642a35b253fe0eca45a286c038c3781fb7918da3212d9fb90b3444","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:25:43.809289Z","signature_b64":"85kwEv91zyBljzMv4XtJRdJynOr/amUGkED+KwngVkcWomcURpvzfade2tU3IuEz7qeWV0kzKXOunweiQQNrCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d352f85f5642a35b253fe0eca45a286c038c3781fb7918da3212d9fb90b3444","last_reissued_at":"2026-05-18T03:25:43.808497Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:25:43.808497Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1207.7153","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":"2026-05-18T03:25:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"64uycJ5eRQRtZmsHEu+vDxrSlaD9Hl98HTtPwiUKWQBBtgqqFR7bhj9TOn/tcuqSEd9pianX6Hulf+q/LCpWDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T19:13:19.939719Z"},"content_sha256":"57dc23c58dd0d5e09ac1c25b9e2a29922658e7eefac28f22f0880af2237d83ad","schema_version":"1.0","event_id":"sha256:57dc23c58dd0d5e09ac1c25b9e2a29922658e7eefac28f22f0880af2237d83ad"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:NU2S7BPVMQVDLMST7YHMURNCQ3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Containment problem for points on a reducible conic in $\\mathbb{P}^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC"],"primary_cat":"math.AG","authors_text":"Annika Denkert, Mike Janssen","submitted_at":"2012-07-31T03:17:30Z","abstract_excerpt":"Given an ideal $I$ in a Noetherian ring, one can ask the containment question: for which $m$ and $r$ is the symbolic power $I^{(m)}$ contained in the ordinary power $I^r$? C. Bocci and B. Harbourne study the containment question in a geometric setting, where the ideal $I$ is in a polynomial ring over a field. Like them, we will consider special geometric constructs. In particular, we obtain a complete solution in two extreme cases of ideals of points on a pair of lines in $\\mathbb{P}^2$; in one case, the number of points on each line is the same, while in the other all the points but one are o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.7153","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":""},"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":"2026-05-18T03:25:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FqlwnAfscOmpyY7BqY6/zgqViRx+tZGwC4NO8F94Uho3Oz6pF0hUvtMkQ0aNg3C/iJlYx1wmeGpGF36cJSkJAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T19:13:19.940495Z"},"content_sha256":"c015d60e1ba943c572878dcd28c75376c284433ed5226adc65f1db6e0bae2006","schema_version":"1.0","event_id":"sha256:c015d60e1ba943c572878dcd28c75376c284433ed5226adc65f1db6e0bae2006"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NU2S7BPVMQVDLMST7YHMURNCQ3/bundle.json","state_url":"https://pith.science/pith/NU2S7BPVMQVDLMST7YHMURNCQ3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NU2S7BPVMQVDLMST7YHMURNCQ3/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":"2026-05-28T19:13:19Z","links":{"resolver":"https://pith.science/pith/NU2S7BPVMQVDLMST7YHMURNCQ3","bundle":"https://pith.science/pith/NU2S7BPVMQVDLMST7YHMURNCQ3/bundle.json","state":"https://pith.science/pith/NU2S7BPVMQVDLMST7YHMURNCQ3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NU2S7BPVMQVDLMST7YHMURNCQ3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:NU2S7BPVMQVDLMST7YHMURNCQ3","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":"4435bcaa8edc74dc8e92c06cd163e4ad5ce262533abd0515513cf722e94cbb09","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-07-31T03:17:30Z","title_canon_sha256":"0ce3531b102d83a0757265d2bbc41ce677937dbed3dc1ec1be443c784596b50f"},"schema_version":"1.0","source":{"id":"1207.7153","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.7153","created_at":"2026-05-18T03:25:43Z"},{"alias_kind":"arxiv_version","alias_value":"1207.7153v2","created_at":"2026-05-18T03:25:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.7153","created_at":"2026-05-18T03:25:43Z"},{"alias_kind":"pith_short_12","alias_value":"NU2S7BPVMQVD","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_16","alias_value":"NU2S7BPVMQVDLMST","created_at":"2026-05-18T12:27:16Z"},{"alias_kind":"pith_short_8","alias_value":"NU2S7BPV","created_at":"2026-05-18T12:27:16Z"}],"graph_snapshots":[{"event_id":"sha256:c015d60e1ba943c572878dcd28c75376c284433ed5226adc65f1db6e0bae2006","target":"graph","created_at":"2026-05-18T03:25:43Z","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":"Given an ideal $I$ in a Noetherian ring, one can ask the containment question: for which $m$ and $r$ is the symbolic power $I^{(m)}$ contained in the ordinary power $I^r$? C. Bocci and B. Harbourne study the containment question in a geometric setting, where the ideal $I$ is in a polynomial ring over a field. Like them, we will consider special geometric constructs. In particular, we obtain a complete solution in two extreme cases of ideals of points on a pair of lines in $\\mathbb{P}^2$; in one case, the number of points on each line is the same, while in the other all the points but one are o","authors_text":"Annika Denkert, Mike Janssen","cross_cats":["math.AC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-07-31T03:17:30Z","title":"Containment problem for points on a reducible conic in $\\mathbb{P}^2$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.7153","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":"sha256:57dc23c58dd0d5e09ac1c25b9e2a29922658e7eefac28f22f0880af2237d83ad","target":"record","created_at":"2026-05-18T03:25:43Z","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":"4435bcaa8edc74dc8e92c06cd163e4ad5ce262533abd0515513cf722e94cbb09","cross_cats_sorted":["math.AC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-07-31T03:17:30Z","title_canon_sha256":"0ce3531b102d83a0757265d2bbc41ce677937dbed3dc1ec1be443c784596b50f"},"schema_version":"1.0","source":{"id":"1207.7153","kind":"arxiv","version":2}},"canonical_sha256":"6d352f85f5642a35b253fe0eca45a286c038c3781fb7918da3212d9fb90b3444","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6d352f85f5642a35b253fe0eca45a286c038c3781fb7918da3212d9fb90b3444","first_computed_at":"2026-05-18T03:25:43.808497Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:25:43.808497Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"85kwEv91zyBljzMv4XtJRdJynOr/amUGkED+KwngVkcWomcURpvzfade2tU3IuEz7qeWV0kzKXOunweiQQNrCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:25:43.809289Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.7153","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:57dc23c58dd0d5e09ac1c25b9e2a29922658e7eefac28f22f0880af2237d83ad","sha256:c015d60e1ba943c572878dcd28c75376c284433ed5226adc65f1db6e0bae2006"],"state_sha256":"43e25449bc1ddb1a7a106f296d214e565a88986b40f08b23a727267e7e0c963d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7W2Umdnu/kRyAUPhDckOsLTKD4t8US2ZN+ZzWkPZ6CnxOfL7f+rzug5+5TnNIqSMESRWCdHyqKGzgIS+zrfhAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T19:13:19.943667Z","bundle_sha256":"6ffb9aba5b42ccfb8d2af376b72a3b29613813032997dc7255aa6caaa501b85b"}}