{"paper":{"title":"A short proof of the G\\\"ottsche conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["hep-th","math.SG"],"primary_cat":"math.AG","authors_text":"M. Kool, R. P. Thomas, V. Shende","submitted_at":"2010-10-15T16:36:33Z","abstract_excerpt":"We prove that for a sufficiently ample line bundle $L$ on a surface $S$, the number of $\\delta$-nodal curves in a general $\\delta$-dimensional linear system is given by a universal polynomial of degree $\\delta$ in the four numbers $L^2,\\,L.K_S,\\,K_S^2$ and $c_2(S)$.\n  The technique is a study of Hilbert schemes of points on curves on a surface, using the BPS calculus of [PT3] and the computation of tautological integrals on Hilbert schemes by Ellingsrud, G\\\"ottsche and Lehn.\n  We are also able to weaken the ampleness required, from G\\\"ottsche's $(5\\delta-1)$-very ample to $\\delta$-very ample."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.3211","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"}