Given a smooth projective variety $X / \mathbb F\_p$, can one find a smooth projective $\mathcal X / \mathbb Z\_p$ such that $\mathcal X \times\_{\mathbb Z\_p} \mathbb F\_p = X$? (or similarly with $\mathbb Z$ instead of $\mathbb Z\_p$).


Thank you!or



Lift smooth projective varieties over $\mathbb F_p$ to $\mathbb Z_p$?
Of course not. The only smooth projective curve over $\mathbf{Z}$ is $\mathbf{P}\_1$, and for every prime $p$ there is a smooth projective curve over $\mathbf{F}\_p$ other than $\mathbf{P}\_1$. 


**Addendum** A bit of googling brings up the overview of Serre's work written on the occasion of his getting the Abel prize. It is mentioned there that recently (2005) Serre has improved his 1961 result mentioned in the comment below 



> 
> by showing that if the variety can be lifted (as a flat scheme) to a local ring $A$, then $p.A=0$. The basic idea consists in transposing the problem to the context of finite groups.
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**Reference** I've come across [these notes](http://staff.ustc.edu.cn/~yiouyang/Illusie.pdf) by [Yi Ouyang](http://staff.ustc.edu.cn/~yiouyang/) of a course by [Luc Illusie](http://www.math.u-psud.fr/~illusie/) on *Topics in Algebraic Geometry* ; the final section deals with Serre's example.


