What is the current status of de Jong's smooth alteration theorem for a family of schemes?


[His 1997 paper](http://aif.cedram.org/item?id=AIF_1997__47_2_599_0) shows that given any family of curves $X/S$ with $S$ of finite type (and, say, local) over a field, there exists a pair of alterations $S'\to S$ and $X'\to X\times\_S S'$. However the general case (with $X/S$ arbitrary of finite type) doesn't seem to follow. Could someone explain why not? After all, can't any such family be written inductively as a sequence of families of curves?


I'm interested in a slightly stronger result. Namely, I'd like the map $S'\to S$ to be 'etale, and the map $X'\to X\times\_S S'$ to be *fiberwise* an alteration. Does anyone know if there's any hope of this being true?


The motivation for this comes from trying to control the behavior of 'etale cohomology over families --- ultimately, I'm interested in a similar generalization of his semistability theorem to families over a Dedekind domain.


Thanks!



de jong's alteration theorem for families
Theorem 5.9 in de Jong's paper is pretty general ($S$ need not be local, only excellent integral of finite dimension) with $X$ proper over $S$. If $X$ is only of finite type over $S$ but separated, using a compactification $\overline{X}$ of $X$, you should get alterations $X' \to X$, $S' \to S$ with $S'\to S$ generically étale. 


**EDIT**: change ''birational'' to ''generically finite'' 


If you want $S'\to S$ be étale, it is possible just by shrinking $S'$ to the étale locus. If you want the condition on the fibers, it should still be possible by shrinking $S'$ (working with proper $X$, the locus where the fiber is not generically finite is closed and projects to a closed subset in $S$, the same holds for fibers $X'\to S'$ which are not regular, at least if the residue fields of $S$ are perfect). But if you want $S'\to S$ be surjective, it is impossible in general: let $S$ be the spectrum of a DVR with perfect residue field, what you want would imply that the generic fiber has potentially good reduction. This is false in general if $X$ is proper (with integral fibers to make sense for "fiberwise alteration").


