{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:TMGYV5DJOL7Q7ABZFNQGJTLMGK","short_pith_number":"pith:TMGYV5DJ","canonical_record":{"source":{"id":"1506.07407","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-06-24T15:03:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d52289627c42d4e7dd4b28060a389a1aae2effb31a2b830e3ff776b091397e90","abstract_canon_sha256":"445bae88d09119be87935cd4ba0eefc0333c04debbfdfd998626988031378f52"},"schema_version":"1.0"},"canonical_sha256":"9b0d8af46972ff0f80392b6064cd6c32b1cbe205f35665769b7672d397ff8d6c","source":{"kind":"arxiv","id":"1506.07407","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.07407","created_at":"2026-05-18T01:39:54Z"},{"alias_kind":"arxiv_version","alias_value":"1506.07407v1","created_at":"2026-05-18T01:39:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.07407","created_at":"2026-05-18T01:39:54Z"},{"alias_kind":"pith_short_12","alias_value":"TMGYV5DJOL7Q","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TMGYV5DJOL7Q7ABZ","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TMGYV5DJ","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:TMGYV5DJOL7Q7ABZFNQGJTLMGK","target":"record","payload":{"canonical_record":{"source":{"id":"1506.07407","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-06-24T15:03:17Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"d52289627c42d4e7dd4b28060a389a1aae2effb31a2b830e3ff776b091397e90","abstract_canon_sha256":"445bae88d09119be87935cd4ba0eefc0333c04debbfdfd998626988031378f52"},"schema_version":"1.0"},"canonical_sha256":"9b0d8af46972ff0f80392b6064cd6c32b1cbe205f35665769b7672d397ff8d6c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:39:54.138602Z","signature_b64":"GqrYvLib0PHQT2jJTx+aX1E390c9MPkKJEHuFHAOqrpoxqNyU9JHSJN61VCGGjcahRZ0eEzR53SIfZ5cMIBQAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b0d8af46972ff0f80392b6064cd6c32b1cbe205f35665769b7672d397ff8d6c","last_reissued_at":"2026-05-18T01:39:54.137923Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:39:54.137923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1506.07407","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":"2026-05-18T01:39:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"04VzFqf+drSe3Yuw4vI/NXvpso8EAMcufp44qQj8Ile+YPza3XG+sKTi502kO327VYt9CeedupDG9O038J1WDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T05:46:54.643014Z"},"content_sha256":"cada0adac10798349e2dc14a526f8b39f04ac33ad388bdaf2e39c74256fdee5b","schema_version":"1.0","event_id":"sha256:cada0adac10798349e2dc14a526f8b39f04ac33ad388bdaf2e39c74256fdee5b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:TMGYV5DJOL7Q7ABZFNQGJTLMGK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Tropical surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Kristin Shaw","submitted_at":"2015-06-24T15:03:17Z","abstract_excerpt":"We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and tropical Chern cycles and numbers. We provide a new method for constructing tropical surfaces, called the tropical sum, similar to the fiber sum of usual manifolds. We prove a tropical adjunction formula for curves in compact tropical surfaces satisfying a local condition, a partial Castelnuovo-Enriques criterion for contracting (-1)-curves, and also invariance of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.07407","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":""},"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":"2026-05-18T01:39:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Bx34oVtgwULFQREOJqzv+eelTy0E+GrfM/e4d4+tqsWDV1TTdNK+92+sCmx3KlbLc0ly2IqoNe8yi8BYD6CGCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T05:46:54.643378Z"},"content_sha256":"6cb72810fe6c996f103227969a6083471995aeb497401afb7549f79e2e88f986","schema_version":"1.0","event_id":"sha256:6cb72810fe6c996f103227969a6083471995aeb497401afb7549f79e2e88f986"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TMGYV5DJOL7Q7ABZFNQGJTLMGK/bundle.json","state_url":"https://pith.science/pith/TMGYV5DJOL7Q7ABZFNQGJTLMGK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TMGYV5DJOL7Q7ABZFNQGJTLMGK/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":"2026-06-05T05:46:54Z","links":{"resolver":"https://pith.science/pith/TMGYV5DJOL7Q7ABZFNQGJTLMGK","bundle":"https://pith.science/pith/TMGYV5DJOL7Q7ABZFNQGJTLMGK/bundle.json","state":"https://pith.science/pith/TMGYV5DJOL7Q7ABZFNQGJTLMGK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TMGYV5DJOL7Q7ABZFNQGJTLMGK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:TMGYV5DJOL7Q7ABZFNQGJTLMGK","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":"445bae88d09119be87935cd4ba0eefc0333c04debbfdfd998626988031378f52","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-06-24T15:03:17Z","title_canon_sha256":"d52289627c42d4e7dd4b28060a389a1aae2effb31a2b830e3ff776b091397e90"},"schema_version":"1.0","source":{"id":"1506.07407","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.07407","created_at":"2026-05-18T01:39:54Z"},{"alias_kind":"arxiv_version","alias_value":"1506.07407v1","created_at":"2026-05-18T01:39:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.07407","created_at":"2026-05-18T01:39:54Z"},{"alias_kind":"pith_short_12","alias_value":"TMGYV5DJOL7Q","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"TMGYV5DJOL7Q7ABZ","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"TMGYV5DJ","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:6cb72810fe6c996f103227969a6083471995aeb497401afb7549f79e2e88f986","target":"graph","created_at":"2026-05-18T01:39:54Z","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 present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and tropical Chern cycles and numbers. We provide a new method for constructing tropical surfaces, called the tropical sum, similar to the fiber sum of usual manifolds. We prove a tropical adjunction formula for curves in compact tropical surfaces satisfying a local condition, a partial Castelnuovo-Enriques criterion for contracting (-1)-curves, and also invariance of ","authors_text":"Kristin Shaw","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-06-24T15:03:17Z","title":"Tropical surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.07407","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:cada0adac10798349e2dc14a526f8b39f04ac33ad388bdaf2e39c74256fdee5b","target":"record","created_at":"2026-05-18T01:39:54Z","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":"445bae88d09119be87935cd4ba0eefc0333c04debbfdfd998626988031378f52","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-06-24T15:03:17Z","title_canon_sha256":"d52289627c42d4e7dd4b28060a389a1aae2effb31a2b830e3ff776b091397e90"},"schema_version":"1.0","source":{"id":"1506.07407","kind":"arxiv","version":1}},"canonical_sha256":"9b0d8af46972ff0f80392b6064cd6c32b1cbe205f35665769b7672d397ff8d6c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b0d8af46972ff0f80392b6064cd6c32b1cbe205f35665769b7672d397ff8d6c","first_computed_at":"2026-05-18T01:39:54.137923Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:39:54.137923Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"GqrYvLib0PHQT2jJTx+aX1E390c9MPkKJEHuFHAOqrpoxqNyU9JHSJN61VCGGjcahRZ0eEzR53SIfZ5cMIBQAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:39:54.138602Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.07407","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cada0adac10798349e2dc14a526f8b39f04ac33ad388bdaf2e39c74256fdee5b","sha256:6cb72810fe6c996f103227969a6083471995aeb497401afb7549f79e2e88f986"],"state_sha256":"4264fa167067df189c230f1fd5a269eb1b68039090e6ccd2359a55004bd5dcd6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LgPG031oqlahx7yXQx2sWNY5YKpz3LNDtXaXS2FxghXK1uevz6QLpDe3/gmATwpjaKn4rHWo7aAZc0IsZMdNBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T05:46:54.645629Z","bundle_sha256":"ff90b59a1c787eb381d72730df6f79c5bb0d6f0951b67e2cbc387477a03aaf29"}}