# Alterations of regular varieties

Let $X$ be a regular quasi-projective variety over a perfect field $k$. The existence of a "good compactification" of $X$, i.e. a regular projective variety $\bar{X}$ with an embedding $X\hookrightarrow \bar{X}$ such that $\bar{X}\setminus X$ is a strict normal crossings divisor, is not known in positive characteristic.

Instead, even if $X$ is not regular, there is de Jong's theorem on alterations which says that there is a dominant proper generically finite morphism (this is called an alteration) $\phi:X'\rightarrow X$, with $X'$ regular such that $X'$ has a good compactification.

In a review of de Jong's paper, Oort writes (bottom of p.5, emphasis mine):

"Also, when starting with a singular X, it might be that the morphism $\phi : X' \rightarrow X$ thus constructed need not be ﬁnite above non-singular points of X."

Even though he doesn't explicitly claim it, this sentence seems to suggest that for non-singular $X$ the situation is different. Hence my question (even though I suspect the answer is 'no'):

If $X$ is non-singular, does there always exist a finite alteration $X'\rightarrow X$, such that $X'$ is regular and has a good compactification?

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deJong's proof gives no control over the dense open locus in the base over which the alteration is finite (equivalently, has finite fibers), in contrast with what Hironaka proved. This is what Oort is surely referring to (as a warning to the reader) with his "might be" comment; it seems doubtful to me that he was intending to suggest any finer control is possible for regular $X$, since the whole structure of the proof involves paying no attention at all to where $X$ is regular. I know this doesn't answer your question; it is just a comment on the apparent motivation for the question. – BCnrd Jun 11 '10 at 15:38
This helps me a lot, thanks! When I looked over the (expositions) of the proof, I saw no reason for my question to have a positive answer, but then again, I've only just begun learning about alterations. – Lars Jun 11 '10 at 16:10
Hi again, in light of BCnrd's answer, can at least something be said about the "size" of the locus where the alteration is not finite, e.g. about it's codimension? – Lars Jun 15 '10 at 13:24