What is an example of a smooth variety over a finite field F_p which does not lift to Z_p?

Question

Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.

Answer 1

Examples are also in my paper "Murphy's Law in Algebraic Geometry", which you can get from my preprints page

Here is a short (not quite complete) description of a construction, with two explanations of why it works. I hope I am remembering this correctly!

In characteristic $>2$, consider the blow up of $\mathbf{P}^2$ at the $\mathbf{F}_p$-valued points of the plane. Take a Galois cover of this surface, with Galois group $(\mathbf{Z}/2)^3$, branched only over the proper transform of the lines, and the transform of another high degree curve with no $\mathbf{F}_p$-points. Then you can check that this surface violates the numerical constraints of the Bogomolov-Miyaoka-Yau inequality, which holds in characteristic zero; hence it doesn't lift. (This is in a paper by Rob Easton.) Alternatively, show that deformations of this surface must always preserve that Galois cover structure, which in turn must preserve the data of the branch locus back in $\mathbf{P}^2$, meaning that any deformation must preserve the data of those $p^2+p+1$ lines meeting $p+1$ to a point, which forces you to live over $\mathbf{Z}/p$.

The two papers mentioned above give more exotic behavior too (of different sorts in the two papers), e.g. you an find a surface that lifts to $\mathbf{Z}/p^{10}$ but still not to $\mathbf{Z}_p$.

Answer 2

There is an article by Illusie in Fundamental Algebraic Geometry: Grothendieck's FGA Explained that contains a construction of a non-liftable surface. The article is online (pdf - see section 6) and the construction is rather complicated. There are additional references to other nonliftability results (section 5F).