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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.


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up vote 7 down vote accepted

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").

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Thanks! I didn't see his Theorem 5.9. Is it kosher on MO to accept an answer and rewrite an edited version of a question? Qing, you're right that I'd like $S'\to S$ to be surjective (and not necessarily smooth), and it's enough for $X'/S'$ to be semistable rather than smooth. This is more like Theorem 6.5 in dJ's original paper, and for this the general case is equivalent to the $S$ local one. Is there a counterexample with some $S$ some higher-dimensional local ring? – Dmitry Vaintrob Jul 30 '11 at 14:36
For higher dimension $S$, a semi-stable alteration $X'/S'$ exists if $X/S$ have relative dimension $1$. Otherwise what de Jong proves in this Ann. Fourier paper is weaker then semi-stability. I don't known whether a counterexample to semi-stable alteration exists. – Qing Liu Jul 30 '11 at 21:07
See however Abramovich and Karu: Weak semistable reduction in characteristic 0. Invent. Math. 139 (2000). – Qing Liu Jul 30 '11 at 21:17

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