$\newcommand{\Spec}{\operatorname{Spec}}$
Cross-post from [Math.SE](https://math.stackexchange.com/q/3116195/127263), hopefully people more knowledgeable in the field will see the question here on MO.


It is a well-known fact that a smooth projective variety over $\mathbb Q$ has good reduction almost everywhere, i.e. everywhere apart from finitely many points. In the language of schemes, this is almost obvious -- given projective equations of the variety $X/\mathbb Q$, for some $n$ we can construct a projective model $\mathcal X\to\Spec\mathbb Z[1/n]$ of $X$. Since a projective morphism is proper and the singular locus of $\mathcal X$ is closed, the set of points of $\Spec\mathbb Z[1/n]$ with singular fibers is closed, hence finite (since the generic fiber is smooth).


When I've first learned this proof, it seemed somewhat clear to me this works for smooth *proper* varieties over $\mathbb Q$, but my professor has pointed out that while the latter part of the argument goes through, there may be problems in constructing a model over a subscheme of $\Spec\mathbb Z$. He couldn't, off the top of his head, answer whether such varieties could have infinitely many places of bad reduction, but he speculated the answer is yes. This is my question:



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> Can a smooth proper variety over $\mathbb Q$ have infinitely many places of bad reduction?
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I have speculated about a much stronger failure as well:



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> Can a smooth proper variety over $\mathbb Q$ have bad reduction *everywhere* (i.e. at every finite prime)?
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Since every proper curve is projective, the example necessarily has dimension at least $2$.


Note: I have added a reference-request tag because I strongly suspect the answer has been already addressed in the literature, but my searches were unsuccessful, mostly returning results about varieties with everywhere good reduction.



Smooth proper variety over $\mathbb Q$ with everywhere bad reduction
As explained in the comments, there is no such variety.


This is an application of a general set of techniques called "spreading out". You can find a very nice treatment of this in Chapter 3 of the book:


Bjorn Poonen - Rational points on varieties.


I can't really give a clearer treatment than Poonen. But the proof goes roughly as follows.


First you prove that any smooth affine variety over $\mathbb{Q}$ admits a smooth model over some $\mathrm{Spec}(\mathbb{Z}[1/n])$. The proof of this is similar to what you outline, namely choosing equations for the variety and choosing $n$ large enough. Next you obtain the result for any smooth variety by glueing along affine opens. Finally if your original variety is proper, you get properness of the model by spreading out and enlarging $n$ as necessary.


