Fiber products under toric flops and flips

Abstract

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 $X_{\hat{I}}\pounds \to X_{\hat{I}{\pounds }_0}\leftarrow X_{\hat{I}{\pounds }^{\prime }}$. We show that the graph closure $\overline{\hat{I}}{“}_f$ 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 $X:=X_{\hat{I}}\pounds \underset{X_{\hat{I}{\pounds }_0}}{\times }{X}_{\hat{I}{\pounds }^{\prime }}$ is in general not toric, though it is still irreducible and $X_{\mathrm{red}}=\overline{\hat{I}}{“}_f$. A complete numerical criterion to ensure $X=X_{\mathrm{red}}$ is given for 3-folds, which is fulfilled when $X_{\hat{I}}\pounds$ has at most terminal singularities. In this case, we further conclude that $X$ is normal.

29 pages. Final version accepted for publication in MathZ. Minor revisions based on referee comments