{"paper":{"title":"On an open problem of characterizing the birationality of 4K","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Meng Chen, Yong Hu","submitted_at":"2017-11-13T11:44:37Z","abstract_excerpt":"We answer an open problem raised by Chen and Zhang in 2008 and prove that, for any minimal projective 3-fold $X$ of general type with the geometric genus $\\geq 5$, $X$ is birationally fibred by a pencil of $(1,2)$-surfaces (i.e. $c_1^2=1$, $p_g=2$) if and only if the $4$-canonical map $\\varphi_{4,X}$ is non-birational. The statement does not hold for those with the geometric genus $\\leq 4$ according to our examples."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.04535","kind":"arxiv","version":3},"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"}