{"paper":{"title":"Classification of Rank 2 Cluster Varieties","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Travis Mandel","submitted_at":"2014-07-23T14:37:21Z","abstract_excerpt":"We classify rank $2$ cluster varieties (those for which the span of the rows of the exchange matrix is $2$-dimensional) according to the deformation type of a generic fiber $U$ of their ${\\mathcal X}$-spaces, as defined by Fock and Goncharov [Ann. Sci. \\'Ec. Norm. Sup\\'er. (4) 42 (2009), 865-930]. Our approach is based on the work of Gross, Hacking, and Keel for cluster varieties and log Calabi-Yau surfaces. Call $U$ positive if $\\dim[\\Gamma(U,{\\mathcal O}_U)] = \\dim(U)$ (which equals 2 in these rank 2 cases). This is the condition for the Gross-Hacking-Keel construction [Publ. Math. Inst. Hau"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.6241","kind":"arxiv","version":5},"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"}