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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?

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 and here (but a different question than the second link).

MathOverflow Asked by David Zureick-Brown on August 9, 2020

3 Answers

3 Answers

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 on Murphy's law for deformation spaces http://front.math.ucdavis.edu/0411.5469 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.

Correct answer by David E Speyer on August 9, 2020

A theorem of Wlodarczyk in "Embedding varieties in toric varieties" says that any smooth variety such that any two points are contained in an affine open set can be embedded in a smooth toric variety. Toric varieties can be lifted to Z_p so any variety over F_p with the above property can be embedded in a smooth scheme over Z_p.

Unfortunately, not all smooth varieties have this property; the example in Hartshorne of a smooth proper 3-fold which is not projective appears to be one where this fails (though for suitable choices these could lift to Z_p).

Answered by ulrich on August 9, 2020

I know that Kiran Kedlaya knows the answer to this question. You might e-mail him.

Answered by David E Speyer on August 9, 2020

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