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!



minimal model of a "surface" over $Spec(\mathbb{Z})$
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. 


