{"paper":{"title":"A note on smooth $SL_2$-surfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Gene Freudenburg","submitted_at":"2024-11-24T15:37:32Z","abstract_excerpt":"Working over a field $k$ of characteristic zero, we study the ring $\\mathfrak{R}=\\mathfrak{D}^{\\mathbb{Z}_2}$ where $\\mathfrak{D}=k[x_0,x_1,x_2]/(2x_0x_2-x_1^2-1)$ and $\\mathbb{Z}_2$ acts by $x_i\\to -x_i$. $\\mathfrak{D}$ admits an algebraic $SL_2(k)$-action which restricts to $\\mathfrak{R}$. Our results include the following. (1) If $k$ is algebraically closed, the smooth $SL_2$-surface $X={\\rm Spec}(\\mathfrak{R})$ admits an algebraic embedding in $\\mathbb{A}_k^4$, and for any such embedding the $SL_2(k)$-action on $X$ does not extend to $\\mathbb{A}_k^4$. In addition, there is no algebraic emb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.15879","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.15879/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}