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!