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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1409.6186","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-09-22T14:36:19Z","cross_cats_sorted":[],"title_canon_sha256":"af5cfa1b53b335bcccd49fbd9b73bec40e4dc6723822723c378b31f4f8e9a272","abstract_canon_sha256":"d4f71fcf941c003379e82c57902abd7eed93eea8d676bbfe35eba9f535e7d7e8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:03:37.062571Z","signature_b64":"mtDa/3jU5cyz95+zSMxRPzAVUayGnsLfSqtEGH0nTcycsqowWjHleecAZbA9clxkICqIs2226lkDY8G16zUgDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fb767ed8d61a75b35d922865abe0c89db917c56a33fe1a14c567eebed313e2f","last_reissued_at":"2026-05-18T01:03:37.061843Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:03:37.061843Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $d\\geqslant 3$. 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