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We prove that every $G$-torsor $Y$ over the generic fibre $X_{\\eta}$ of $X$ can be extended to a torsor over ${X'}$ under the action of an affine and flat $K$-group scheme of finite type $G'$ where $X'$ is obtained by $X$ after a finite number of N\\'eron blowing ups. Moreover if $G$ is finite and \\'etale (resp. admits a finite and flat model) we fin"},"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":"1210.1522","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2012-10-04T17:21:45Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"e0608e639bbb65b2f0ef2ae59ec5d58e08524cdea43aa49d1cf706d7aba517d4","abstract_canon_sha256":"00a0290f154e1bcc48fe6a579e60fdf0817560b45e1a61c6b1879d9dfa169c24"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:11:52.068436Z","signature_b64":"Me4zcs+QbpfJ183UVMADv0BJ/IJKnbp+smC6lIi3cgiMzHPAeh0NVwSGck53u6v8wFdV42w+TOZSbP22PT8yAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8221ca61fa2f52931e35caf3ed84fe5eaf98bd959a3bee467a88f7d15a6b56f9","last_reissued_at":"2026-05-18T01:11:52.068048Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:11:52.068048Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Models of torsors over curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Marco Antei","submitted_at":"2012-10-04T17:21:45Z","abstract_excerpt":"Let $R$ be a complete discrete valuation ring with fraction field $K$ and with algebraically closed residue field. Let $X$ be a faithfully flat $R$-scheme of finite type of relative dimension 1 and $G$ be any affine $K$-group scheme of finite type. We prove that every $G$-torsor $Y$ over the generic fibre $X_{\\eta}$ of $X$ can be extended to a torsor over ${X'}$ under the action of an affine and flat $K$-group scheme of finite type $G'$ where $X'$ is obtained by $X$ after a finite number of N\\'eron blowing ups. Moreover if $G$ is finite and \\'etale (resp. admits a finite and flat model) we fin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.1522","kind":"arxiv","version":4},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1210.1522","created_at":"2026-05-18T01:11:52.068103+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.1522v4","created_at":"2026-05-18T01:11:52.068103+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.1522","created_at":"2026-05-18T01:11:52.068103+00:00"},{"alias_kind":"pith_short_12","alias_value":"QIQ4UYP2F5JJ","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"QIQ4UYP2F5JJGHRV","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"QIQ4UYP2","created_at":"2026-05-18T12:27:18.751474+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/QIQ4UYP2F5JJGHRVZLZ63BH6L2","json":"https://pith.science/pith/QIQ4UYP2F5JJGHRVZLZ63BH6L2.json","graph_json":"https://pith.science/api/pith-number/QIQ4UYP2F5JJGHRVZLZ63BH6L2/graph.json","events_json":"https://pith.science/api/pith-number/QIQ4UYP2F5JJGHRVZLZ63BH6L2/events.json","paper":"https://pith.science/paper/QIQ4UYP2"},"agent_actions":{"view_html":"https://pith.science/pith/QIQ4UYP2F5JJGHRVZLZ63BH6L2","download_json":"https://pith.science/pith/QIQ4UYP2F5JJGHRVZLZ63BH6L2.json","view_paper":"https://pith.science/paper/QIQ4UYP2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.1522&json=true","fetch_graph":"https://pith.science/api/pith-number/QIQ4UYP2F5JJGHRVZLZ63BH6L2/graph.json","fetch_events":"https://pith.science/api/pith-number/QIQ4UYP2F5JJGHRVZLZ63BH6L2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QIQ4UYP2F5JJGHRVZLZ63BH6L2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QIQ4UYP2F5JJGHRVZLZ63BH6L2/action/storage_attestation","attest_author":"https://pith.science/pith/QIQ4UYP2F5JJGHRVZLZ63BH6L2/action/author_attestation","sign_citation":"https://pith.science/pith/QIQ4UYP2F5JJGHRVZLZ63BH6L2/action/citation_signature","submit_replication":"https://pith.science/pith/QIQ4UYP2F5JJGHRVZLZ63BH6L2/action/replication_record"}},"created_at":"2026-05-18T01:11:52.068103+00:00","updated_at":"2026-05-18T01:11:52.068103+00:00"}