{"paper":{"title":"The geometry of sporadic $\\mathbb{C}^*$-embeddings into $\\mathbb{C}^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Karol Palka, Mariusz Koras, Peter Russell","submitted_at":"2014-05-27T11:30:36Z","abstract_excerpt":"A closed algebraic embedding of $\\mathbb{C}^*=\\mathbb{C}^1\\setminus\\{0\\}$ into $\\mathbb{C}^2$ is 'sporadic' if for every curve $A\\subseteq \\mathbb{C}^2$ isomorphic to an affine line the intersection with $\\mathbb{C}^*$ is at least $2$. Non-sporadic embeddings have been classified. There are very few known sporadic embeddings. We establish geometric and algebraic tools to classify them based on the analysis of the minimal log resolution $(X,D)\\to (\\mathbb{P}^2,U)$, where $U$ is the closure of $\\mathbb{C}^*$ on $\\mathbb{P}^2$. We show in particular that one can choose coordinates on $\\mathbb{C}^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.6872","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"}