Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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).

share|improve this question

3 Answers 3

up vote 4 down vote accepted

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.

share|improve this answer
David, I think you misread the question (or maybe I am confused). If X is projetive, then it embeds into P^n over Z_p, which is a smooth Z_p scheme. Ravi's example (I believe) is of a variety this doesn't lift to a variety over Z_p. –  David Zureick-Brown Oct 15 '09 at 5:10
David edited his answer so that my comment no longer makes sense, but I can't edit my comment to reflect that so I'll leave this comment instead. –  David Zureick-Brown Oct 16 '09 at 5:52
I had a bounty on this question and had to accept your answer! –  David Zureick-Brown Jul 15 at 18:20
Hmm, can you switch your answer to ulrich's now, and I'll delete mine? (Sorry for not thinking clearly back in '09.) –  David Speyer Jul 15 at 21:23

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).

share|improve this answer

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

share|improve this answer
Also, if you get an informative response, please post it here. –  S. Carnahan Oct 13 '09 at 16:01
Kiran didn't know off the top of his head, but said that he believes should exist. –  David Zureick-Brown Oct 14 '09 at 14:58

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.