{"paper":{"title":"The Noether inequality for threefolds and three moduli spaces with minimal volumes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Chen Jiang, Meng Chen, Yong Hu","submitted_at":"2024-07-01T13:31:49Z","abstract_excerpt":"We establish the Noether inequality\n  \\[\\textrm{Vol}(X)\\geq \\frac{4}{3}p_g(X)-\\frac{10}{3}\\] for all projective $3$-folds $X$ of general type with geometric genus $5\\leq p_g(X)\\leq 10$ where $\\textrm{Vol}(X)$ is the canonical volume. This result resolves all remaining cases of the Noether inequality for $3$-folds.\n  We further investigate the moduli spaces of canonical $3$-folds with small genera and minimal volumes. For a $3$-fold of general type with geometric genus $2$ and with minimal canonical volume $\\frac{1}{3}$, we prove that its canonical model is a hypersurface of degree $16$ in $\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2407.01276","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/2407.01276/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"}