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Let $X\to\mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism. Is there a closed immersion $X\to Y$ where $Y$ is flat of finite type over $\mathbb{Z}/p^2\mathbb{Z}$?

As mentioned in the comments if $X$ is projective the problem is solved immediately. So a counterexample would have $\mathrm{dim}(X)\geq 3$.

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  • $\begingroup$ For example, if $X$ is projective, you can take $\mathbf P^N_{\mathbf Z/p^2}$. Is that the sort of thing you're looking for? $\endgroup$
    – R. van Dobben de Bruyn
    Aug 1, 2020 at 21:40
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    $\begingroup$ I think it is confusing that the answer to the question implicit in the title, and to the actual question in the body of the text, are opposite (whatever those answers are). This is relevant because people often start an answer with "The answer is Yes", or whatever. $\endgroup$
    – LSpice
    Aug 1, 2020 at 22:05
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    $\begingroup$ As a non-specialist my immediate response is, are there counterexamples to the stronger statements "X smooth implies there exist Y smooth" and "X flat implies there exist Y flat"? $\endgroup$
    – Tom Goodwillie
    Aug 2, 2020 at 11:10
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    $\begingroup$ @TomGoodwillie there were some similar questions (mathoverflow.net/q/410 mathoverflow.net/q/63969) but I don't know a counterexample to either of those two statements. It can also be noted that anything is flat over a field. $\endgroup$
    – user158636
    Aug 2, 2020 at 11:45

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