{"paper":{"title":"On the universal family of Hilbert schemes of points on a surface","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Lei Song","submitted_at":"2014-07-21T13:39:00Z","abstract_excerpt":"For a smooth quasi-projective surface $X$ and an integer $n\\ge 3$, we show that the universal family $Z^n$ over the Hilbert scheme $\\text{Hilb}^{n}(X)$ of $n$ points has non $\\mathbb{Q}$-Gorenstein, rational singularities, and that the Samuel multiplicity $\\mu$ at a closed point on $Z^n$ can be computed in terms of the dimension of the socle. We also show that $\\mu\\le n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.5490","kind":"arxiv","version":2},"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"}