What is the current status of de Jong's smooth alteration theorem for a family of schemes?
His 1997 paper 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.