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.
Categories: math.AG