{"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"}