{"paper":{"title":"Algebraic rank on hyperelliptic graphs and graphs of genus $3$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.AG","authors_text":"Kazuhiko Yamaki, Shu Kawaguchi","submitted_at":"2014-01-16T08:47:30Z","abstract_excerpt":"Let $\\bar{G} = (G, \\omega)$ be a vertex-weighted graph, and $\\delta$ a divisor class on $G$. Let $r_{\\bar{G}}(\\delta)$ denote the combinatorial rank of $\\delta$. Caporaso has introduced the algebraic rank $r_{\\bar{G}}^{\\operatorname{alg}}(\\delta)$ of $\\delta$, by using nodal curves with dual graph $\\bar{G}$. In this paper, when $\\bar{G}$ is hyperelliptic or of genus $3$, we show that $r_{\\bar{G}}^{\\operatorname{alg}}(\\delta) \\geq r_{\\bar{G}}(\\delta)$ holds, generalizing our previous result. We also show that, with respect to the specialization map from a non-hyperelliptic curve of genus $3$ to"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.3935","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"}