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Fix a function $P: \\mathbb Z_{\\geq 0}\\to \\mathbb Z $, then there exists an integer $N_1>0$ such that if $(X,\\mathcal F)$ is a canonical or nef model of a foliation of general type with Hilbert polynomial $\\chi (X, mK_{\\mathcal F})=P(m)$ for all $m\\in \\mathbb Z_{\\geq 0}$, then $|mK_{\\mathcal F}|$ defines a birational map for all $m\\geq N_1$.\n  We also prove a Grauert-Riemannschneider type vanishing theorem for foliated surfaces with canonical singularities."},"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":"1910.07709","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2019-10-17T04:58:43Z","cross_cats_sorted":[],"title_canon_sha256":"2d8473fdded3c00ff4b3555c1267e7f9a5b141c68d8ea4f42b2d3fbfe85feef4","abstract_canon_sha256":"007cd06772f4199b95bb1a92a72af8762b0aac7ef25f056b1fc5a9a6fae8c11f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:59:10.359912Z","signature_b64":"jd0VbtLvb0SB5SFBa+UFOn0icRfgb7/CUDXVR8MKaYwW5uFkVe1pYs8oz+e2ecn9BA49vPKdjQcL+JIFt9eJCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15d02972e5e09aacad7915db98f640753743130e62c30f1bfad521e230f09e83","last_reissued_at":"2026-07-05T09:59:10.359382Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:59:10.359382Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On birational boundedness of foliated surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Adrian Langer, Christopher D. 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