{"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"}