{"paper":{"title":"Worst singularities of plane curves of given degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Ivan Cheltsov","submitted_at":"2014-09-22T14:36:19Z","abstract_excerpt":"We prove that $\\frac{2}{d}$, $\\frac{2d-3}{(d-1)^2}$, $\\frac{2d-1}{d(d-1)}$, $\\frac{2d-5}{d^2-3d+1}$ and $\\frac{2d-3}{d(d-2)}$ are the smallest log canonical thresholds of reduced plane curves of degree $d\\geqslant 3$, and we describe reduced plane curves of degree $d$ whose log canonical thresholds are these numbers. As an application, we prove that $\\frac{2}{d}$, $\\frac{2d-3}{(d-1)^2}$, $\\frac{2d-1}{d(d-1)}$, $\\frac{2d-5}{d^2-3d+1}$ and $\\frac{2d-3}{d(d-2)}$ are the smallest values of the $\\alpha$-invariant of Tian of smooth surfaces in $\\mathbb{P}^3$ of degree $d\\geqslant 3$. We also prove t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.6186","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"}