Fiber products under toric flops and flips
Tsung-Chen Chen, Hui-Wen Lin, Sz-Sheng Wang
TL;DR
The paper analyzes how graph closures and scheme-theoretic fiber products behave under toric flops and flips. It shows that the graph closure $\overline{\Gamma}_f$ is always a toric variety, with its normalization given by a coarsest common refinement $X_{\widetilde{\Sigma}}$; a precise combinatorial criterion determines when $\overline{\Gamma}_f$ is normal. For the fiber product $X = X_{\Sigma} \times_{X_{\Sigma_0}} X_{\Sigma'}$, irreducibility holds in toric flop/flip contexts, and $X_{\mathrm{red}} = \overline{\Gamma}_f$, with $X_{\mathrm{red}}$ normal exactly when $\overline{\Gamma}_f$ is normal. In dimension three, a complete numerical criterion ensures $X = X_{\mathrm{red}}$ and hence normality when $X_{\Sigma}$ has terminal singularities, while higher-dimensional criteria require smoothness and divisibility conditions on wall data. Collectively, these results connect graph closures, fiber products, and minimal model program moves in toric geometry, offering concrete criteria for normality and reducedness in both threefolds and higher dimensions.
Abstract
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.