{"paper":{"title":"Categorical aspects of the Koll\\'ar--Shepherd-Barron correspondence","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Giancarlo Urz\\'ua, Jenia Tevelev","submitted_at":"2022-04-27T23:02:38Z","abstract_excerpt":"It is well known that a $2$-dimensional cyclic quotient singularity $\\overline{W}$ has the same singularity category as a finite dimensional associative algebra $\\overline{R}$ introduced by Kalck and Karmazyn. We study the deformations of the algebra $\\overline{R}$ induced by the deformations of the surface $\\overline{W}$ to a smooth surface. We show that they are Morita--equivalent to path algebras $\\hat{R}$ of acyclic quivers for general smoothings within each irreducible component of the versal deformation space of $\\overline{W}$ (as described by Koll\\'ar and Shepherd-Barron). Furthermore, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2204.13225","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2204.13225/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"}