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We look into defining equations of $M$ at its singularity $E$, partly because if $M$ admits only canonical singularities, then the Kodaira dimension $\\kappa(M)$ can be calculated. We show the following.\n  (A) $E$ is at worst canonical singularity of $M$ if the restriction of $E_{\\eta}$ to the generic fiber of $X$ has no rank-one subsheaf, and if the number of multiple fibers of $X"},"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":"1908.05027","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-08-14T09:07:12Z","cross_cats_sorted":[],"title_canon_sha256":"fd8a8c7d646fc794bb8b7bd6d48006329afe62e4e7c9a09e9b1f63f4a399ee9b","abstract_canon_sha256":"b32ec080d8f96bcfd186e808ad63c9f10d8964cef8bb20149453f56cb1c1def3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T02:18:03.484213Z","signature_b64":"A83flmxaEbNZJ5D3+zZdkc0NqzMJCx4aF/j7RmbdUbvRkKojSjcmOWwq3+Dg7R4ImeAw4vY0DFyXy2WsX7/nAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21a2a605905bf42b4019dc657f2a6160e914127ef6e2677cfe9777b1ad816213","last_reissued_at":"2026-07-05T02:18:03.483764Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T02:18:03.483764Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Kodaira dimension and singularities of moduli of stable sheaves on some elliptic surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Kimiko Yamada","submitted_at":"2019-08-14T09:07:12Z","abstract_excerpt":"Let $X$ be an elliptic surface over ${\\bf P}^1$ with $\\kappa(X)=1$, and $M=M(c_2)$ be the moduli scheme of rank-two stable sheaves $E$ on $X$ with $(c_1(E),c_2(E))=(0,c_2)$ in $\\operatorname{Pic}(X)\\times\\mathbb{Z}$. We look into defining equations of $M$ at its singularity $E$, partly because if $M$ admits only canonical singularities, then the Kodaira dimension $\\kappa(M)$ can be calculated. We show the following.\n  (A) $E$ is at worst canonical singularity of $M$ if the restriction of $E_{\\eta}$ to the generic fiber of $X$ has no rank-one subsheaf, and if the number of multiple fibers of $X"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.05027","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/1908.05027/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1908.05027","created_at":"2026-07-05T02:18:03.483818+00:00"},{"alias_kind":"arxiv_version","alias_value":"1908.05027v3","created_at":"2026-07-05T02:18:03.483818+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1908.05027","created_at":"2026-07-05T02:18:03.483818+00:00"},{"alias_kind":"pith_short_12","alias_value":"EGRKMBMQLP2C","created_at":"2026-07-05T02:18:03.483818+00:00"},{"alias_kind":"pith_short_16","alias_value":"EGRKMBMQLP2CWQAZ","created_at":"2026-07-05T02:18:03.483818+00:00"},{"alias_kind":"pith_short_8","alias_value":"EGRKMBMQ","created_at":"2026-07-05T02:18:03.483818+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EGRKMBMQLP2CWQAZ3RSX6KTBMD","json":"https://pith.science/pith/EGRKMBMQLP2CWQAZ3RSX6KTBMD.json","graph_json":"https://pith.science/api/pith-number/EGRKMBMQLP2CWQAZ3RSX6KTBMD/graph.json","events_json":"https://pith.science/api/pith-number/EGRKMBMQLP2CWQAZ3RSX6KTBMD/events.json","paper":"https://pith.science/paper/EGRKMBMQ"},"agent_actions":{"view_html":"https://pith.science/pith/EGRKMBMQLP2CWQAZ3RSX6KTBMD","download_json":"https://pith.science/pith/EGRKMBMQLP2CWQAZ3RSX6KTBMD.json","view_paper":"https://pith.science/paper/EGRKMBMQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1908.05027&json=true","fetch_graph":"https://pith.science/api/pith-number/EGRKMBMQLP2CWQAZ3RSX6KTBMD/graph.json","fetch_events":"https://pith.science/api/pith-number/EGRKMBMQLP2CWQAZ3RSX6KTBMD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EGRKMBMQLP2CWQAZ3RSX6KTBMD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EGRKMBMQLP2CWQAZ3RSX6KTBMD/action/storage_attestation","attest_author":"https://pith.science/pith/EGRKMBMQLP2CWQAZ3RSX6KTBMD/action/author_attestation","sign_citation":"https://pith.science/pith/EGRKMBMQLP2CWQAZ3RSX6KTBMD/action/citation_signature","submit_replication":"https://pith.science/pith/EGRKMBMQLP2CWQAZ3RSX6KTBMD/action/replication_record"}},"created_at":"2026-07-05T02:18:03.483818+00:00","updated_at":"2026-07-05T02:18:03.483818+00:00"}