# Primes of good reduction for varieties

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