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We show that the graph closure $\\overline{\\Gamma}_f$ is a not necessarily normal toric variety and we give a combinatorial criterion for its normality.\n  In contrast to it, for $f$ being a toric flop/flip, we show that the scheme-theoretic fiber product $X:=X_{\\Sigma}\\mathop{\\times}\\limits_{X_{\\Sigma_0}}X_{\\Sigma'}$ is in general not toric, though it is still irreducible and $X_{\\rm"},"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":"2411.12446","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2024-11-19T12:08:23Z","cross_cats_sorted":[],"title_canon_sha256":"d3912aa3f730d2ca9277ab08ec8c101706e9def667f2666d7fa6bf1623b89f28","abstract_canon_sha256":"c0c1c30c39d5024dad23473f77edfced981c158cf9f663e53d949073f568e781"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:04.211694Z","signature_b64":"j7KPQDtZbeDBn2Ig01Vx9Ie4N5DmpwKcLnzR6jiQNHDpZjfbSu2YEbGtOHDr6E9uB2I55qUvwTu98kwjEZwEDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"230de2139bb9c2b8d393bc90b4a1e645443ec093d04ce4bb6a127d444fa9a7c5","last_reissued_at":"2026-06-09T01:05:04.211184Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:04.211184Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Fiber products under toric flops and flips","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Hui-Wen Lin, Sz-Sheng Wang, Tsung-Chen Chen","submitted_at":"2024-11-19T12:08:23Z","abstract_excerpt":"Let $\\Sigma$ and $\\Sigma'$ be two refinements of a fan $\\Sigma_0$ and $f \\colon X_{\\Sigma} \\dashrightarrow X_{\\Sigma'}$ be the birational map induced by $X_{\\Sigma} \\rightarrow X_{\\Sigma_0} \\leftarrow X_{\\Sigma'}$. We show that the graph closure $\\overline{\\Gamma}_f$ is a not necessarily normal toric variety and we give a combinatorial criterion for its normality.\n  In contrast to it, for $f$ being a toric flop/flip, we show that the scheme-theoretic fiber product $X:=X_{\\Sigma}\\mathop{\\times}\\limits_{X_{\\Sigma_0}}X_{\\Sigma'}$ is in general not toric, though it is still irreducible and $X_{\\rm"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.12446","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/2411.12446/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":"2411.12446","created_at":"2026-06-09T01:05:04.211247+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.12446v3","created_at":"2026-06-09T01:05:04.211247+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.12446","created_at":"2026-06-09T01:05:04.211247+00:00"},{"alias_kind":"pith_short_12","alias_value":"EMG6EE43XHBL","created_at":"2026-06-09T01:05:04.211247+00:00"},{"alias_kind":"pith_short_16","alias_value":"EMG6EE43XHBLRU4T","created_at":"2026-06-09T01:05:04.211247+00:00"},{"alias_kind":"pith_short_8","alias_value":"EMG6EE43","created_at":"2026-06-09T01:05:04.211247+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/EMG6EE43XHBLRU4TXSILJIPGIV","json":"https://pith.science/pith/EMG6EE43XHBLRU4TXSILJIPGIV.json","graph_json":"https://pith.science/api/pith-number/EMG6EE43XHBLRU4TXSILJIPGIV/graph.json","events_json":"https://pith.science/api/pith-number/EMG6EE43XHBLRU4TXSILJIPGIV/events.json","paper":"https://pith.science/paper/EMG6EE43"},"agent_actions":{"view_html":"https://pith.science/pith/EMG6EE43XHBLRU4TXSILJIPGIV","download_json":"https://pith.science/pith/EMG6EE43XHBLRU4TXSILJIPGIV.json","view_paper":"https://pith.science/paper/EMG6EE43","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.12446&json=true","fetch_graph":"https://pith.science/api/pith-number/EMG6EE43XHBLRU4TXSILJIPGIV/graph.json","fetch_events":"https://pith.science/api/pith-number/EMG6EE43XHBLRU4TXSILJIPGIV/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EMG6EE43XHBLRU4TXSILJIPGIV/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EMG6EE43XHBLRU4TXSILJIPGIV/action/storage_attestation","attest_author":"https://pith.science/pith/EMG6EE43XHBLRU4TXSILJIPGIV/action/author_attestation","sign_citation":"https://pith.science/pith/EMG6EE43XHBLRU4TXSILJIPGIV/action/citation_signature","submit_replication":"https://pith.science/pith/EMG6EE43XHBLRU4TXSILJIPGIV/action/replication_record"}},"created_at":"2026-06-09T01:05:04.211247+00:00","updated_at":"2026-06-09T01:05:04.211247+00:00"}