Fiber products under toric flops and flips

Let $\mathrm{\Sigma }$ and ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ be two refinements of a fan ${\mathrm{\Sigma }}_0$ and $f\text{\hspace{0.17em}⁣}:X_{\mathrm{\Sigma }}⇢X_{{\mathrm{\Sigma }}^{\mathrm{\prime }}}$ be the birational map induced by $X_{\mathrm{\Sigma }}\to X_{{\mathrm{\Sigma }}_0}\leftarrow X_{{\mathrm{\Sigma }}^{\mathrm{\prime }}}$. We show that the graph closure ${\bar{\mathrm{\Gamma }}}_f$ 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 $X:=X_{\mathrm{\Sigma }}\underset{X_{{\mathrm{\Sigma }}_0}}{}{X}_{{\mathrm{\Sigma }}^{\mathrm{\prime }}}$ is in general not toric, though it is still irreducible and $X_{\mathrm{r}\mathrm{e}\mathrm{d}}={\bar{\mathrm{\Gamma }}}_f$. A complete numerical criterion to ensure $X=X_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is given for 3-folds, which is fulfilled when $X_{\mathrm{\Sigma }}$ has at most terminal singularities. In this case, we further conclude that $X$ is normal.