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Let $C_1$ and $C_2$ be rational curves fully tangent to $D$ at the same point $P$ and assume that $C_1$ and $C_2$ are immersed and that $(C_1.C_2)_P=\\min\\{D.C_1, D.C_2\\}$. Then we show that the contribution of $C_1\\cup C_2$ to the virtual count of $\\mathfrak{M}_{[C_1]+[C_2]}$ is $\\min\\{D.C_1, D.C_2\\}$.\n  As an example, we describe genus $0$ relative stable morphisms to $(\\mathb"},"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":"1711.08173","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2017-11-22T08:28:44Z","cross_cats_sorted":[],"title_canon_sha256":"6bc2bda78f049fee74ef6ca02aaf0187d178c168696dbba34400fc6b9b0ba35b","abstract_canon_sha256":"408ee5e47a9c790c7134cdb06cd363bbac6d7561e177feeeae8a8372371844fa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:29:50.485427Z","signature_b64":"8uHwS6lNfnUXyHaC7fgMF987UzeiY51AXJFP4DqobxHXxw7w2p4+HOKJP3ILtQmPghv0VkF7ilkkPTuqsUJUAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1629e4ee74a0ca809dce1128eb9ae0f0606cf8a366f04e0aace047bcb004035a","last_reissued_at":"2026-05-18T00:29:50.484791Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:29:50.484791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the multiplicity of reducible relative stable morphisms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Nobuyoshi Takahashi","submitted_at":"2017-11-22T08:28:44Z","abstract_excerpt":"Let $(Z, D)$ be a pair of a smooth surface and a smooth anti-canonical divisor. Denote by $\\mathfrak{M}_\\beta$ the moduli stack of genus $0$ relative stable morphisms of class $\\beta$ with full tangency to the boundary. Let $C_1$ and $C_2$ be rational curves fully tangent to $D$ at the same point $P$ and assume that $C_1$ and $C_2$ are immersed and that $(C_1.C_2)_P=\\min\\{D.C_1, D.C_2\\}$. Then we show that the contribution of $C_1\\cup C_2$ to the virtual count of $\\mathfrak{M}_{[C_1]+[C_2]}$ is $\\min\\{D.C_1, D.C_2\\}$.\n  As an example, we describe genus $0$ relative stable morphisms to $(\\mathb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.08173","kind":"arxiv","version":1},"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":"1711.08173","created_at":"2026-05-18T00:29:50.484877+00:00"},{"alias_kind":"arxiv_version","alias_value":"1711.08173v1","created_at":"2026-05-18T00:29:50.484877+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1711.08173","created_at":"2026-05-18T00:29:50.484877+00:00"},{"alias_kind":"pith_short_12","alias_value":"CYU6J3TUUDFI","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_16","alias_value":"CYU6J3TUUDFIBHOO","created_at":"2026-05-18T12:31:10.602751+00:00"},{"alias_kind":"pith_short_8","alias_value":"CYU6J3TU","created_at":"2026-05-18T12:31:10.602751+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/CYU6J3TUUDFIBHOOCEUOXGXA6B","json":"https://pith.science/pith/CYU6J3TUUDFIBHOOCEUOXGXA6B.json","graph_json":"https://pith.science/api/pith-number/CYU6J3TUUDFIBHOOCEUOXGXA6B/graph.json","events_json":"https://pith.science/api/pith-number/CYU6J3TUUDFIBHOOCEUOXGXA6B/events.json","paper":"https://pith.science/paper/CYU6J3TU"},"agent_actions":{"view_html":"https://pith.science/pith/CYU6J3TUUDFIBHOOCEUOXGXA6B","download_json":"https://pith.science/pith/CYU6J3TUUDFIBHOOCEUOXGXA6B.json","view_paper":"https://pith.science/paper/CYU6J3TU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1711.08173&json=true","fetch_graph":"https://pith.science/api/pith-number/CYU6J3TUUDFIBHOOCEUOXGXA6B/graph.json","fetch_events":"https://pith.science/api/pith-number/CYU6J3TUUDFIBHOOCEUOXGXA6B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CYU6J3TUUDFIBHOOCEUOXGXA6B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CYU6J3TUUDFIBHOOCEUOXGXA6B/action/storage_attestation","attest_author":"https://pith.science/pith/CYU6J3TUUDFIBHOOCEUOXGXA6B/action/author_attestation","sign_citation":"https://pith.science/pith/CYU6J3TUUDFIBHOOCEUOXGXA6B/action/citation_signature","submit_replication":"https://pith.science/pith/CYU6J3TUUDFIBHOOCEUOXGXA6B/action/replication_record"}},"created_at":"2026-05-18T00:29:50.484877+00:00","updated_at":"2026-05-18T00:29:50.484877+00:00"}