
> 
> Does there exist a smooth proper morphism $E \to \operatorname{Spec} \mathbb Z$ whose fibers are Enriques surfaces?
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By a theorem of, independently, [Fontaine](http://www.ams.org/mathscinet-getitem?mr=1274493) and [Abrashkin](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=1152&option_lang=eng), combined with the Enriques-Kodaira classification of complex surfaces, the only possible surface of Kodaira dimension $0$ or $1$ that can appear this way is an Enriques surface.


In particular, no K3 surfaces are smooth over $\mathbb Z$. Hence the K3 double cover of $E$ is not smooth. The cover is etale away from $\mathbb F\_2$, hence must be singular over $\mathbb F\_2$. This means $E$ most be a classical Enriques surface (a $\mu\_2$-torsor rather than a $\mathbb Z/2$-torsor or $\alpha\_2$-torsor).


It is possible to get some other information about $E$:


The Picard group of this surface has rank $10$. The Galois action on the Picard group must be unramified at each prime, hence trivial, so the full lattice of cycles is defined over $\mathbb Q$. Thus by the Lefschetz trace formula, $E$ has exactly $25$ $\mathbb F\_2$-points.


Some other questions that might be helpful to solve this one are:



> 
> How many K3 surfaces are there with good reduction away from $2$ (and Picard rank at least $10$, and a fixed-point-free involution, etc.)?
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> Given an Enriques surface over $\mathbb Q\_2$, what are obstructions to good reduction over $\mathbb Z\_2$, other than ramification of the cohomology?
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> Which classical Enriques surfaces over $\mathbb F\_2$ are liftable to $\mathbb Z\_2$? Can something be said about the singularities and Galois representations of their $K3$ double covers?
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> Can we compute the discriminants of [explicit families](http://en.wikipedia.org/wiki/Enriques_surface#Examples) of Enriques surfaces and try to solve the Diophantine equation $\Delta=1$?
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One example of an Enriques surface over $\mathbb Z[1/2]$ whose cohomology is unramified at $2$ can be constructed as the quotient of a Kummer surface with good reduction away from $2$. Let $E\_1$ and $E\_2$ be two elliptic curves that are either $y^2=x^3-x$ or $y^2=x^3-4x$. Let $e\_1$ and $e\_2$ be $2$-torsion points on $E\_1$ and $E\_2$ respectively. Then we [can construct](http://projecteuclid.org/download/pdf_1/euclid.nmj/1118781595) (Example 3.1) a fixed-point free involution on the Kummer surface of $E\_1 \times E\_2$, giving an Enriques surface.


Because $E\_1$ and $E\_2$ have good reduction away from $2$, this surface has good reduction also. $H^2$ of the Kummer surface comes from $H^2(E\_1 \times E\_2)$ plus the exceptional classes of the 16 blown-up $2$-torsion points. Because these points are defined over $\mathbb Q$, the cohomology classes are unramified. $H^2(E\_1)$ and $H^2(E\_2)$ are unramified as well, so the only ramified part of the cohomology of the Kummer surface is $H^1(E\_1) \times H^1(E\_2)$. Because the involution acts as reflection on $E\_1$ and translation on $E\_2$, it acts as $-1$ on $H^1(E\_1) \times H^1(E\_2)$, so that does not descend to the Enriques surface, hence its cohomology is unramified. This surface also has $\mathbb Q$-points, thus $\mathbb Q\_2$ points.


Thus I cannot see any obstruction to good reduction at $2$. However, the construction certainly does not produce a smooth model of the surface over $2$.



> 
> Does this surface have good reduction at $2$?
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Enriques surfaces over $\mathbb Z$
A preprint by Stefan Schröer came out today with the answer to this question: [arXiv:2004.07025](https://arxiv.org/abs/2004.07025).


No such Enriques surface exists. In fact, there is no classical Enriques surface over $\mathbb F\_2$ with 25 $\mathbb F\_2$-points (and with the extension of $\mathbb Z^{10}$ by $\mathbb Z/2$ split in the Picard group, which you can also deduce).


