Fiber products under toric flops and flips

Abstract

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. 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 $X_{\rm red} = \overline{\Gamma}_f$. A complete numerical criterion to ensure $X = X_{\rm red}$ is given for 3-folds, which is fulfilled when $X_\Sigma$ has at most terminal singularities. In this case, we further conclude that $X$ is normal.