Fiber products under toric flops and flips

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.