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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2411.12446 |
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| _version_ | 1866916609582432256 |
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| author | Chen, Tsung-Chen Lin, Hui-Wen Wang, Sz-Sheng |
| author_facet | Chen, Tsung-Chen Lin, Hui-Wen Wang, Sz-Sheng |
| contents | Let $Σ$ and $Σ'$ be two refinements of a fan $Σ_0$ and $f \colon X_Σ \dashrightarrow X_{Σ'}$ be the birational map induced by $X_Σ \rightarrow X_{Σ_0} \leftarrow X_{Σ'}$. We show that the graph closure $\overlineΓ_f$ is a not necessarily normal toric variety and we give a combinatorial criterion for its normality.
In contrast to it, for $f$ being a toric flop/flip, we show that the scheme-theoretic fiber product $X:=X_Σ\mathop{\times}\limits_{X_{Σ_0}}X_{Σ'}$ is in general not toric, though it is still irreducible and $X_{\rm red} = \overlineΓ_f$.
A complete numerical criterion to ensure $X = X_{\rm red}$ is given for 3-folds, which is fulfilled when $X_Σ$ has at most terminal singularities. In this case, we further conclude that $X$ is normal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12446 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fiber products under toric flops and flips Chen, Tsung-Chen Lin, Hui-Wen Wang, Sz-Sheng Algebraic Geometry Let $Σ$ and $Σ'$ be two refinements of a fan $Σ_0$ and $f \colon X_Σ \dashrightarrow X_{Σ'}$ be the birational map induced by $X_Σ \rightarrow X_{Σ_0} \leftarrow X_{Σ'}$. We show that the graph closure $\overlineΓ_f$ is a not necessarily normal toric variety and we give a combinatorial criterion for its normality. In contrast to it, for $f$ being a toric flop/flip, we show that the scheme-theoretic fiber product $X:=X_Σ\mathop{\times}\limits_{X_{Σ_0}}X_{Σ'}$ is in general not toric, though it is still irreducible and $X_{\rm red} = \overlineΓ_f$. A complete numerical criterion to ensure $X = X_{\rm red}$ is given for 3-folds, which is fulfilled when $X_Σ$ has at most terminal singularities. In this case, we further conclude that $X$ is normal. |
| title | Fiber products under toric flops and flips |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2411.12446 |