What is known about regular proper models for smooth projective varieties over Q? Results for other global fields would also be interesting, as well as general comments and suggested references for integral models.

This is a followup to this question on smooth models, and here is part of what I wrote as an answer to the previous question: Nekovar's survey article on the Beilinson conjectures from the early 90s mentions some results for varieties over Q. He says in section 5.3 that given a smooth projective variety over Q, there always exists a proper flat model over Z, but that a regular such model is rarely known to exist. However, in the published version of the same survey, there is an added note at the very end of the article saying that "Spivakovsky recently announced a general result on resolution of singularities, which implies that a regular proper flat model of X mentioned in 5.3 always exist". However, I have never seen this result of Spivakovsky mentioned anywhere else, so I doubt that it is true. Does anyone else know more about this?

The survey is available here. For the published version, google "Serre Jannsen Motives", click at the Google Books link, and then search for "Spivakovsky" within the book.

  • $\begingroup$ Some people might be interested in reading §3.5.15 in Rational points on varieties by B. Poonen. $\endgroup$
    – Watson
    Commented Oct 31, 2018 at 20:27

1 Answer 1


Regular proper models over Z are only known to exist for curves (by a theorem of Abhyankar). Spivakovsky's claims were never substantiated; as far as I know, no preprint was ever circulated.

For many purposes, de Jong's theorem on alterations suffices: For a proper variety X over a number field K there exists a variety Y with a proper generically finite map to X such that Y has a regular proper model over the ring of integers of K. (The actual theorem is more general.)


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