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If I have a smooth compact algebraic scheme of dimention $2$ over $Spec(\mathbb{Z})$ whose generic fiber is a surface in minimal model (say of general type). Then:

(a) Is it true that the special fibers are in minimal model as well? (I would guess the answer is no in general but at least in an open subscheme of the base this should be true).

(b) If the asnwer to (a) is negative, does there exist some scheme whose fibers are minimal? (it is clear that in each special fiber I can do this, but I want the resulting scheme to be a smooth scheme over $Spec(\mathbb{Z})$).

I had no luck while searching for references for this sort of questions (since this is not a general threefold) so references on the subject are more than welcome!

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  • $\begingroup$ By "compact", do you mean "proper over $\mathbf{Z}$"? And by "of dimension 2", do you mean "of relative dimension 2"? Sorry to ask these questions. $\endgroup$
    – Ariyan Javanpeykar
    Commented Jul 22, 2011 at 5:21
  • $\begingroup$ Ariyan: the OP says "whose generic fibre is a surface..." which suggests the relative dimension is indeed 2. A. Pacetti: do you know the answer to your question in the case where Spec(Z) is replaced by a smooth curve defined over a field? $\endgroup$
    – user5117
    Commented Jul 22, 2011 at 8:54
  • $\begingroup$ @Ariyan, yes to both question. @Artie, I don't know the answer even in the case you mentioned. In the particular case I have in mind, I more or less can prove that in an open set the special fibers are minimal, but it is something particular to the case. My question is if there is something general and if this (natural?) question was studied before... $\endgroup$
    – A. Pacetti
    Commented Jul 22, 2011 at 9:40

1 Answer 1

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The question is local on the base, so you can replace $\mathbb Z$ with a discrete valuation ring $R$. Moreover, the Kodaira dimension and the minimality of surfaces are stable by field extension, so one can suppose the residue field of $R$ is algebraically closed.

Then the positive answer is given in Katsura and Ueno: "On elliptic surfaces in characteristic $p$". Math. Ann. 272 (1985), Lemma 9.4 (see also Lemma 9.6). This holds under the hypothesis that the generic fiber has non-negative Kodaira dimension.

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