Fiber products under toric flops and flips

Tsung-Chen Chen, Hui-Wen Lin, Sz-Sheng Wang

Let $\hat{I}\pounds$ and $\hat{I}{\pounds }^{\prime }$ be two refinements of a fan $\hat{I}{\pounds }_0$ and $f:X_{\hat{I}}\pounds ⇢X_{\hat{I}{\pounds }^{\prime }}$ be the birational map induced by $X_{\hat{I}}\pounds \to X_{\hat{I}{\pounds }_0}\leftarrow X_{\hat{I}{\pounds }^{\prime }}$. W…