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Let $X$ be the affine line with a double origin over $\mathrm{Spec}\,\mathbb Z$. Let $X_\eta$ be its generic fibre, the affine line with a double origin over $\mathrm{Spec}\,\mathbb Q$.

Let $0$ be one of the origins of $X_\eta$.

Are there infinitely many ways of extending $0$ to a section of $X$?

My feeling is that besides taking the closure of $0$ in $X$, we can also alter our choice of origin at some primes. If that feeling is correct, then we would get infinitely many sections of $X\to \mathrm{Spec}\,\mathbb Z$ inducing the same section generically.

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    $\begingroup$ You should think about this on your own. Do you know a finite atlas of open affines for $X$? For each such open, what is the relative closure of $\{0\}$ in that open? $\endgroup$
    – Jason Starr
    Commented Aug 11, 2015 at 12:01
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    $\begingroup$ ... My last sentence should have been: What are the inverse images of these open affines with respect to your (putative) section? $\endgroup$
    – Jason Starr
    Commented Aug 11, 2015 at 13:51

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