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Assume that a general member $F\\in |-K_X|$ meets $C$ only at one point $P$ and furthermore $(F,P)$ is Du Val of type A if index$(X,P)=4$. We classify all such germs in terms of a general member $H\\in |O_X|$ containing $C$."},"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":"1004.4188","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2010-04-23T17:30:26Z","cross_cats_sorted":[],"title_canon_sha256":"07265d8e18b79bf881ef11bb45d63b194bbabf9c37b8af49a3fd623b55fb1ffa","abstract_canon_sha256":"0f73b24a96420f49e3420db3846be6ab7c4ada56996e403a1e6c770042020e50"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:18:46.274461Z","signature_b64":"TCUJn91hd7GVcJ8RjX4gTZ6CewpxP+9A33eeRj7D86/BFlXOIag7yyk6l78M4aKY2o44rRJdAJgeVp/7+fC9CA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c22bfa852e7970ba173fbe54558bf73a465cd6d7d1217c44a6d6765c4fd7f169","last_reissued_at":"2026-05-18T04:18:46.274022Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:18:46.274022Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Threefold extremal contractions of type IA","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Shigefumi Mori, Yuri Prokhorov","submitted_at":"2010-04-23T17:30:26Z","abstract_excerpt":"Let $(X,C)$ be a germ of a threefold $X$ with terminal singularities along an irreducible reduced complete curve $C$ with a contraction $f: (X,C)\\to (Z,o)$ such that $C=f^{-1}(o)_{\\red}$ and $-K_X$ is ample. Assume that a general member $F\\in |-K_X|$ meets $C$ only at one point $P$ and furthermore $(F,P)$ is Du Val of type A if index$(X,P)=4$. 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