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Let $R$ be a regular local ring, $\hat{R}$ be its completion, $X$ be a proper scheme over $\text{Spec}(\hat{R})$. In what case there exist a proper scheme $Y$ over $\text{Spec}(R)$, such that $X$ is the pullback of $Y$?

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    $\begingroup$ Even if $R$ is Henselian, this can fail to be the case. Let $R$ be the Henselization of $\mathbb{C}[t]_{\langle t \rangle}$, let $\widehat{R}$ be $\mathbb{C}[[t]]$, and let $X$ be the blowing up of $\mathbb{P}^2_{\widehat{R}}$ at $[1,0,0]$, $[0,1,0]$, $[0,0,1]$, $[1,1,1]$ and $[1,0,e^t]$. $\endgroup$ – Jason Starr May 29 '15 at 14:27
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    $\begingroup$ There are uncountably many isomorphism classes of genus-$1$ smooth proper (geometrically connected) curves over $\mathbf{Q}_p$ but only countably many over $\mathbf{Q}$. $\endgroup$ – grghxy May 29 '15 at 14:40

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