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Aug 10 at 13:31 answer added HYL timeline score: 1
Aug 1 at 10:35 comment added George That's the first time I've heard about alteration . Thank you.
Aug 1 at 3:25 comment added Karl Schwede I assume 1. does not hold but I'd have to think hard to find an example. 2. is open outside of when we know resolution of singularities holds (ie, char 0 or dim $\leq 3$). If you can weaken $\pi$ birational to $\pi$ and alteration (I assume you want $\pi$ to be proper/projective) then you can make $Y$ nonsingular.
Jul 31 at 15:10 history asked George CC BY-SA 4.0