{"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"}