{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:5DL4PKQMHGVVODFCDFVGU6MYCZ","short_pith_number":"pith:5DL4PKQM","canonical_record":{"source":{"id":"1706.03633","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-12T13:44:58Z","cross_cats_sorted":[],"title_canon_sha256":"5c46a69dbeef38fecacb510a8bb4790ff2cd217a5173bd95127c91323dde8d68","abstract_canon_sha256":"a6c114a50cd9a8180746eee503bf4662abf20717937a3ac8a6bec6047d0fb1ea"},"schema_version":"1.0"},"canonical_sha256":"e8d7c7aa0c39ab570ca2196a6a7998167ee2c393783ac005f1d429c42ff0c4c0","source":{"kind":"arxiv","id":"1706.03633","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.03633","created_at":"2026-05-18T00:39:12Z"},{"alias_kind":"arxiv_version","alias_value":"1706.03633v2","created_at":"2026-05-18T00:39:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.03633","created_at":"2026-05-18T00:39:12Z"},{"alias_kind":"pith_short_12","alias_value":"5DL4PKQMHGVV","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"5DL4PKQMHGVVODFC","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"5DL4PKQM","created_at":"2026-05-18T12:31:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:5DL4PKQMHGVVODFCDFVGU6MYCZ","target":"record","payload":{"canonical_record":{"source":{"id":"1706.03633","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-12T13:44:58Z","cross_cats_sorted":[],"title_canon_sha256":"5c46a69dbeef38fecacb510a8bb4790ff2cd217a5173bd95127c91323dde8d68","abstract_canon_sha256":"a6c114a50cd9a8180746eee503bf4662abf20717937a3ac8a6bec6047d0fb1ea"},"schema_version":"1.0"},"canonical_sha256":"e8d7c7aa0c39ab570ca2196a6a7998167ee2c393783ac005f1d429c42ff0c4c0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:12.048509Z","signature_b64":"SGqlTGpEaXmSsYQMBeWGVtSHQNbw6hK5nvIpbRkCrcc+NZWQpFM/QS1K2/+On0XvI4NXTWZi+qzGm3TcuxwtAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e8d7c7aa0c39ab570ca2196a6a7998167ee2c393783ac005f1d429c42ff0c4c0","last_reissued_at":"2026-05-18T00:39:12.047898Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:12.047898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.03633","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":"2026-05-18T00:39:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mU1ov+aJumJlrhj7ABuquHw1SjIFsEL31j/5x0wu9fYcw1rdPj1pT56W0TOL6am0PJVAzR8LSfUDGUe1oP1kAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:07:07.233584Z"},"content_sha256":"4c0eff6c818f3ccf52eb4f57135ae74b4476bc1cf18a7966b89af99735919ced","schema_version":"1.0","event_id":"sha256:4c0eff6c818f3ccf52eb4f57135ae74b4476bc1cf18a7966b89af99735919ced"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:5DL4PKQMHGVVODFCDFVGU6MYCZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The $b$-secant variety of a smooth curve has a codimension $1$ locally closed subset whose points have rank at least $b+1$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"E. Ballico","submitted_at":"2017-06-12T13:44:58Z","abstract_excerpt":"Take a smooth, connected and non-degenerate projective curve $X\\subset \\mathbb {P}^r$, $r\\ge 2b+2\\ge 6$, defined over an algebraically closed field with characteristic $0$ and let $\\sigma _b(X)$ be the $b$-secant variety of $X$. We prove that the $X$-rank of $q$ is at least $b+1$ for a non-empty codimension $1$ locally closed subset of $\\sigma _b(X)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03633","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":"2026-05-18T00:39:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"41/ATXtNqqJoGzy5LgA30L/TRpncNVsF6adabdDcitxxvTNNIVFvYSPufgGpDqp+By+ygE0jBbukw27iwK1ABQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T12:07:07.233926Z"},"content_sha256":"2a968e0719055dd93cbdba1549f02b4d8c6c07058e2e8d271e76f901ed442cbd","schema_version":"1.0","event_id":"sha256:2a968e0719055dd93cbdba1549f02b4d8c6c07058e2e8d271e76f901ed442cbd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5DL4PKQMHGVVODFCDFVGU6MYCZ/bundle.json","state_url":"https://pith.science/pith/5DL4PKQMHGVVODFCDFVGU6MYCZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5DL4PKQMHGVVODFCDFVGU6MYCZ/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":"2026-06-25T12:07:07Z","links":{"resolver":"https://pith.science/pith/5DL4PKQMHGVVODFCDFVGU6MYCZ","bundle":"https://pith.science/pith/5DL4PKQMHGVVODFCDFVGU6MYCZ/bundle.json","state":"https://pith.science/pith/5DL4PKQMHGVVODFCDFVGU6MYCZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5DL4PKQMHGVVODFCDFVGU6MYCZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:5DL4PKQMHGVVODFCDFVGU6MYCZ","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":"a6c114a50cd9a8180746eee503bf4662abf20717937a3ac8a6bec6047d0fb1ea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-12T13:44:58Z","title_canon_sha256":"5c46a69dbeef38fecacb510a8bb4790ff2cd217a5173bd95127c91323dde8d68"},"schema_version":"1.0","source":{"id":"1706.03633","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.03633","created_at":"2026-05-18T00:39:12Z"},{"alias_kind":"arxiv_version","alias_value":"1706.03633v2","created_at":"2026-05-18T00:39:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.03633","created_at":"2026-05-18T00:39:12Z"},{"alias_kind":"pith_short_12","alias_value":"5DL4PKQMHGVV","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"5DL4PKQMHGVVODFC","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"5DL4PKQM","created_at":"2026-05-18T12:31:00Z"}],"graph_snapshots":[{"event_id":"sha256:2a968e0719055dd93cbdba1549f02b4d8c6c07058e2e8d271e76f901ed442cbd","target":"graph","created_at":"2026-05-18T00:39:12Z","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":"Take a smooth, connected and non-degenerate projective curve $X\\subset \\mathbb {P}^r$, $r\\ge 2b+2\\ge 6$, defined over an algebraically closed field with characteristic $0$ and let $\\sigma _b(X)$ be the $b$-secant variety of $X$. We prove that the $X$-rank of $q$ is at least $b+1$ for a non-empty codimension $1$ locally closed subset of $\\sigma _b(X)$.","authors_text":"E. Ballico","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-12T13:44:58Z","title":"The $b$-secant variety of a smooth curve has a codimension $1$ locally closed subset whose points have rank at least $b+1$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.03633","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":"sha256:4c0eff6c818f3ccf52eb4f57135ae74b4476bc1cf18a7966b89af99735919ced","target":"record","created_at":"2026-05-18T00:39:12Z","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":"a6c114a50cd9a8180746eee503bf4662abf20717937a3ac8a6bec6047d0fb1ea","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-06-12T13:44:58Z","title_canon_sha256":"5c46a69dbeef38fecacb510a8bb4790ff2cd217a5173bd95127c91323dde8d68"},"schema_version":"1.0","source":{"id":"1706.03633","kind":"arxiv","version":2}},"canonical_sha256":"e8d7c7aa0c39ab570ca2196a6a7998167ee2c393783ac005f1d429c42ff0c4c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e8d7c7aa0c39ab570ca2196a6a7998167ee2c393783ac005f1d429c42ff0c4c0","first_computed_at":"2026-05-18T00:39:12.047898Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:12.047898Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SGqlTGpEaXmSsYQMBeWGVtSHQNbw6hK5nvIpbRkCrcc+NZWQpFM/QS1K2/+On0XvI4NXTWZi+qzGm3TcuxwtAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:12.048509Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.03633","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4c0eff6c818f3ccf52eb4f57135ae74b4476bc1cf18a7966b89af99735919ced","sha256:2a968e0719055dd93cbdba1549f02b4d8c6c07058e2e8d271e76f901ed442cbd"],"state_sha256":"b42fb5750fee8ed498a954b8f07c815363c38f1ac41b99637f7aba5a10537264"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6fYtEH0XirSWHWTjWFWmL1i7B+1UryZsch700m98xu4DAfqZHNFJgWLDtkJN+KIFDNXzZwbqeow7WY4CwEdCAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T12:07:07.235822Z","bundle_sha256":"5a05cb6ac7c0827b6f5138d63394e3081d9608b2718f8eb2c21adc63c5f0cb97"}}