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If $\\kappa(X)=\\kappa(F)+1$, then $s>\\frac{4}m+2$. If $\\kappa(X)\\geq 0$, then $s\\geq\\frac{4}m+2$. In particular, if $m=1$, $s=6$ and $\\kappa(X)=0$, then the family $f$ is Teichm\\\"uller."},"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":"1610.07756","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2016-10-25T07:31:35Z","cross_cats_sorted":[],"title_canon_sha256":"bc55a00ece1a0d3d177089ca8f20dc494916fd01a02bc2627024ea968e4489a7","abstract_canon_sha256":"b0e0954326bea81561ac5969b15fa113ef3e9ab1015c885f037fcb7b190e8c78"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:17.717693Z","signature_b64":"YbBGZkUIrEF65nXAP5M9DBZY86GBDr5fR++D3lq4c337tKG93khuIRtI4wLUcwHyhSCWXPgcvtbP4JbLWX5HBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"562860d2e86378bb9f8104875ffcecdf9bd6628d382afb426bea96bba033431d","last_reissued_at":"2026-05-18T01:01:17.716965Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:17.716965Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Singular Fibers and Kodaira Dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Kang Zuo, Sheng-Li Tan, Xin Lu","submitted_at":"2016-10-25T07:31:35Z","abstract_excerpt":"Let $f:\\,X \\to \\mathbb{P}^1$ be a non-isotrivial semi-stable family of varieties of dimension $m$ over $\\mathbb{P}^1$ with $s$ singular fibers. Assume that the smooth fibers $F$ are minimal, i.e., their canonical line bundles are semiample. Then $\\kappa(X)\\leq \\kappa(F)+1$. If $\\kappa(X)=\\kappa(F)+1$, then $s>\\frac{4}m+2$. If $\\kappa(X)\\geq 0$, then $s\\geq\\frac{4}m+2$. In particular, if $m=1$, $s=6$ and $\\kappa(X)=0$, then the family $f$ is Teichm\\\"uller."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07756","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":"1610.07756","created_at":"2026-05-18T01:01:17.717090+00:00"},{"alias_kind":"arxiv_version","alias_value":"1610.07756v1","created_at":"2026-05-18T01:01:17.717090+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.07756","created_at":"2026-05-18T01:01:17.717090+00:00"},{"alias_kind":"pith_short_12","alias_value":"KYUGBUXIMN4L","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_16","alias_value":"KYUGBUXIMN4LXH4B","created_at":"2026-05-18T12:30:29.479603+00:00"},{"alias_kind":"pith_short_8","alias_value":"KYUGBUXI","created_at":"2026-05-18T12:30:29.479603+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KYUGBUXIMN4LXH4BASDV77HM36","json":"https://pith.science/pith/KYUGBUXIMN4LXH4BASDV77HM36.json","graph_json":"https://pith.science/api/pith-number/KYUGBUXIMN4LXH4BASDV77HM36/graph.json","events_json":"https://pith.science/api/pith-number/KYUGBUXIMN4LXH4BASDV77HM36/events.json","paper":"https://pith.science/paper/KYUGBUXI"},"agent_actions":{"view_html":"https://pith.science/pith/KYUGBUXIMN4LXH4BASDV77HM36","download_json":"https://pith.science/pith/KYUGBUXIMN4LXH4BASDV77HM36.json","view_paper":"https://pith.science/paper/KYUGBUXI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1610.07756&json=true","fetch_graph":"https://pith.science/api/pith-number/KYUGBUXIMN4LXH4BASDV77HM36/graph.json","fetch_events":"https://pith.science/api/pith-number/KYUGBUXIMN4LXH4BASDV77HM36/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KYUGBUXIMN4LXH4BASDV77HM36/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KYUGBUXIMN4LXH4BASDV77HM36/action/storage_attestation","attest_author":"https://pith.science/pith/KYUGBUXIMN4LXH4BASDV77HM36/action/author_attestation","sign_citation":"https://pith.science/pith/KYUGBUXIMN4LXH4BASDV77HM36/action/citation_signature","submit_replication":"https://pith.science/pith/KYUGBUXIMN4LXH4BASDV77HM36/action/replication_record"}},"created_at":"2026-05-18T01:01:17.717090+00:00","updated_at":"2026-05-18T01:01:17.717090+00:00"}