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We prove the subadditivity of Kodaira dimension $\\kappa(X)\\geq\\kappa(Z)+\\kappa(F)$, where $F$ is the generic geometric fiber of $f$, and $\\kappa(F)$ is the Kodaira dimension of the normalization of $F$. Moreover, if $\\dim X=2$ and $\\dim Z=1$, we have a stronger inequality $\\kappa(X)\\geq \\kappa(Z)+\\kappa_1(F)$ where $\\kappa_1(F)=\\kappa(F,\\omega^o_F)$ is the Kodaira dimension of the dualizing sheaf $\\omega_F^o$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1305.6024","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2013-05-26T14:07:31Z","cross_cats_sorted":[],"title_canon_sha256":"fc0cd3a012e00287b18c4c8731278eea8b92c90ce4645815264aee67132a3e4e","abstract_canon_sha256":"38a77bd06491411d5e66c4ef51497e6d624037cc02dd2ffe7d8108e0439134e3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:24:53.133689Z","signature_b64":"P12OGMF6u5TNT/PxBbaitcOtSiN2VmVlvryMiaBI+AqJq7LNrz20mmzmK3H36urmt/h4z7LNOn03Oy6lb6edAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e411cc8d6b48edd57d6018ea34380ffe9d6a575478ba17963ccf52bfe7520254","last_reissued_at":"2026-05-18T03:24:53.133025Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:24:53.133025Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The subadditivity of the Kodaira Dimension for Fibrations of Relative Dimension One in Positive Characteristics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Lei Zhang, Yifei Chen","submitted_at":"2013-05-26T14:07:31Z","abstract_excerpt":"Let $f:X\\rightarrow Z$ be a separable fibration of relative dimension 1 between smooth projective varieties over an algebraically closed field $k$ of positive characteristic. We prove the subadditivity of Kodaira dimension $\\kappa(X)\\geq\\kappa(Z)+\\kappa(F)$, where $F$ is the generic geometric fiber of $f$, and $\\kappa(F)$ is the Kodaira dimension of the normalization of $F$. Moreover, if $\\dim X=2$ and $\\dim Z=1$, we have a stronger inequality $\\kappa(X)\\geq \\kappa(Z)+\\kappa_1(F)$ where $\\kappa_1(F)=\\kappa(F,\\omega^o_F)$ is the Kodaira dimension of the dualizing sheaf $\\omega_F^o$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6024","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1305.6024","created_at":"2026-05-18T03:24:53.133133+00:00"},{"alias_kind":"arxiv_version","alias_value":"1305.6024v1","created_at":"2026-05-18T03:24:53.133133+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1305.6024","created_at":"2026-05-18T03:24:53.133133+00:00"},{"alias_kind":"pith_short_12","alias_value":"4QI4ZDLLJDW5","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"4QI4ZDLLJDW5K7LA","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"4QI4ZDLL","created_at":"2026-05-18T12:27:34.582898+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2009.07158","citing_title":"Pseudo-effectivity of the relative canonical divisor and uniruledness in positive characteristic","ref_index":8,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/4QI4ZDLLJDW5K7LADDVDIOAP72","json":"https://pith.science/pith/4QI4ZDLLJDW5K7LADDVDIOAP72.json","graph_json":"https://pith.science/api/pith-number/4QI4ZDLLJDW5K7LADDVDIOAP72/graph.json","events_json":"https://pith.science/api/pith-number/4QI4ZDLLJDW5K7LADDVDIOAP72/events.json","paper":"https://pith.science/paper/4QI4ZDLL"},"agent_actions":{"view_html":"https://pith.science/pith/4QI4ZDLLJDW5K7LADDVDIOAP72","download_json":"https://pith.science/pith/4QI4ZDLLJDW5K7LADDVDIOAP72.json","view_paper":"https://pith.science/paper/4QI4ZDLL","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1305.6024&json=true","fetch_graph":"https://pith.science/api/pith-number/4QI4ZDLLJDW5K7LADDVDIOAP72/graph.json","fetch_events":"https://pith.science/api/pith-number/4QI4ZDLLJDW5K7LADDVDIOAP72/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4QI4ZDLLJDW5K7LADDVDIOAP72/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4QI4ZDLLJDW5K7LADDVDIOAP72/action/storage_attestation","attest_author":"https://pith.science/pith/4QI4ZDLLJDW5K7LADDVDIOAP72/action/author_attestation","sign_citation":"https://pith.science/pith/4QI4ZDLLJDW5K7LADDVDIOAP72/action/citation_signature","submit_replication":"https://pith.science/pith/4QI4ZDLLJDW5K7LADDVDIOAP72/action/replication_record"}},"created_at":"2026-05-18T03:24:53.133133+00:00","updated_at":"2026-05-18T03:24:53.133133+00:00"}