Such an example of course could not be projective and would not itself lift to Z\_p. The context is that one can compute p-adic cohomology of a variety X over a finite field F\_p via the cohomology of an embedding of X into a smooth Z\_p scheme. 


This is similar in spirit to my questions [here](https://mathoverflow.net/questions/201/) and [here](https://mathoverflow.net/questions/423/what-is-an-example-of-a-smooth-variety-over-a-finite-field-fp-which-does-not-lif) (but a different question than the second link).



What is an example of a smooth variety over a finite field F_p which does not embed into a smooth scheme over Z_p?
EDIT 7/15/14 I was just looking back at this old answer, and I don't think I ever answered the stated question. I can't delete an accepted answer, but I'll point at that, as far as I can tell, the Vakil reference I give also only address the question of deforming $X$ over $\mathbb{Z}\_p$, not of embedding it in some larger flat family over $\mathbb{Z}\_p$.


EDIT Oops! David Brown points out below that I misread the question. I was answering the question of finding a smooth scheme which does not deform in a smooth family over Z\_p.


Well, to make up for that, I'll point to some references which definitely contain answers. Look at section 2.3 of Ravi Vakil's paper *[Murphy's Law in algebraic geometry: Badly-behaved deformation spaces](https://arxiv.org/abs/math/0411469)* for some history, and several good references. Moreover, Ravi describes how to build an explicit cover of P^2 in characteristic p which does not deform to characteristic 0. Basically, the idea is to take a collection of lines in P^2 which doesn't deform to characteristic 0 and take a branched cover over those lines. For example, you could take that p^2+p+1 lines that have coefficients in F\_p.


