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Suppose I have a (smooth, projective) variety $X/\mathbb{Q}$. Is the notion of a primes of bad/good reduction well-defined from this data?

Motivation and attempt at an answer: The question should be local, so we can base change to get $Y/\mathbb{Q}_p$. Good reduction should mean something like: There exists a regular, proper $\mathcal{Y}/\mathbb{Z}_p$ such that the generic fiber is isomorphic to $Y$ and the special fiber is smooth. This is potentially a bad definition:

This behavior for curves of genus $g\geq 1$ is nice, because in the appropriate category of such models there is a partial order by dominating. One can prove that there is a unique, regular, proper, minimal model of the curve which can be used to determine reduction type.

(Edited paragraph from comments) Note that given such a minimal model, one can blow up points on the special fiber. These blowups are still generically isomorphisms and hence models. They are no longer minimal, though.

Related question 1: For higher dimensional varieties, if you have $2$ minimal, regular, proper models (these may not be unique), if one has a nonsingular special fiber, then must the other as well? This would give a well-defined way to determine reduction type.

Question 2: Is this approach just overly complicated (i.e. has this theory been worked out in some other way)?

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If the special fiber of the minimal model of a curve is singular, after blowing-ups, we can make the special fiber a normal crossing divisor, but never smooth. –  user18119 Oct 31 '12 at 22:33
    
@QiL Thank you so much!! That completely clarifies my confusion on this point. I was reading Brian Conrad's "Minimal Model's for Elliptic Curves" and was confused for hours today on this point. –  Matt Nov 1 '12 at 0:57
    
Ah. I just thought of another approach. Supposing the variety satisfies the hypothesis of Matsusaka-Mumford, then a reformulation of the problem is to ask when two minimal, projective, regular models have a polarized isomorphism on the generic fibers. If the class of varieties being considered always has this happen, then the special fibers will actually be isomorphic by M-M. –  Matt Nov 1 '12 at 18:07
    
if you consider polarized varieties, then the good reduction (together with the polarization) is indeed unique. –  user18119 Nov 1 '12 at 20:15

1 Answer 1

Partial answer.

(1). Smooth projective models (when they exist) for a given smooth projective variety are not unique. But their special fibers are birational at least when one of them is not uniruled. This can be proved by considering the graph of the birational map between two models and applying a theorem of Abhyankar.

(2). This works fine for abelian varieties thanks to Néron models. In general, the problem is that in higher dimension, one should use Mori's minimal model program which involve singular models.

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