Factorization of birational maps on steroids
Searching literature you will find the following statement (I'm paraphrasing): "If $X_1,X_2$ are nonsingular schemes proper over a complete DVR $R$ with residue characteristic 0, and $\phi: X_1 \to X_2$ is birational, then $\phi$ can be factored as a sequence of blowups and blowdown between nonsingular schemes proper over $R$, with nonsingular blowup centers." along with a demonstration: "The method of [Włodarczyk] or [AKMW] works word-for-word." In revenge you will find elsewhere (I'm paraphrasing): "Since a proof of weak factorization of birational maps over a complete DVR is not available, we use a different route." Something has to give. I will report on work (in progress, with hopes of imminent release) with Michael Temkin (Jerusalem), in which we prove weak factorization for qe (Not to worry - I'll tell you how to pronounce this.) schemes whenever suitably strong resolution holds (e.g. characteristic 0). This includes the above, and, with some work, similar statement for such geometries as analytic, rigid analytic and formal schemes. Surprisingly, a new version of GAGA seems to be required.