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Let $R$ be a normal noetherian domain and $K$ the quotient field of $R$. Let $X$ be a smooth algebraic $K$-scheme and $x\in X(K)$ a $K$-rational point of $X$. Does there exist a tripel $(U, s, i)$ consisting of a smooth algebraic $R$-scheme $U$, a section $s:Spec(R)\to U$ and an open immersion $i: U\times_R Spec(K)\to X$ such that $s$ corresponds to $x$, i.e. such that $i\circ (s\times_R Spec(K)): Spec(K)\to X$ agrees with $x$?

My main interest is the case where $R$ is local and $dim(X)=1$.

(Maybe it is easy. I read something in a paper which more or less comes down to that, but I do not get it at the moment.)

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    $\begingroup$ When R is a Dedekind domain, I believe this follows from the "smoothening process" which is the first step in constructing Néron models. Chapters 1 & 3 of Bosch, Lütkebohmert & Raynaud, "Néron models", contain all you could want to know about this case. $\endgroup$
    – Martin Bright
    Commented Nov 25, 2011 at 12:18
  • $\begingroup$ Assume that $X$ is projective over $K$. Then there is a closed immersion $X\to \mathbf{P}^n_K$ for some $n\geq 0$. Let $\mathcal{X}$ be the Zariski closure of $X$ in $\mathbf{P}^n_{R}$. Note that $\mathcal{X}\times_R K =X$. By the valuative criterion of properness, there exists a unique $s \in \mathcal{X}(R)$ corresponding to $x$ on the generic fibre. Let $U\subset \mathcal{X}$ be any non-empty open subscheme such that $U\to \mathrm{Spec} \ R$ is smooth. (The set $\mathcal{X}_{sm}$ of points where $\pi:\mathcal{X}\to \mathrm{Spec} \ R$ is smooth is open in $\mathcal{X}$. See, for example.... $\endgroup$
    – Ariyan Javanpeykar
    Commented Nov 25, 2011 at 13:10
  • $\begingroup$ ...Corollary 6.2.12 in Q. Liu's book. Now, you can take $U$ to be the inverse image of $\mathrm{Spec} \ R \backslash \pi(\mathcal{X} \backslash \mathcal{X}_{sm} )$ under $\pi$. This is non-empty because $X$ is smooth over $K$. Note that $U$ is the "biggest" choice possible.) It is clear that $x$ extends to a section of $U$ over $\mathrm{Spec} \ R \backslash \pi(\mathcal{X} \backslash \mathcal{X}_{sm} )$. I'm not sure if one can extend $x$ to a section of $U$ over $\mathrm{Spec} \ R$. $\endgroup$
    – Ariyan Javanpeykar
    Commented Nov 25, 2011 at 13:10
  • $\begingroup$ @Sebastian: Maybe the following notes of M. Romagny on the construction of the Neron model math.jussieu.fr/~romagny/exposes/Neron_models.pdf might come in handy. I would actually like to see how this works myself.... $\endgroup$
    – Ariyan Javanpeykar
    Commented Nov 25, 2011 at 19:32
  • $\begingroup$ Dear Ariyan, thanks for the helpful references! There is also the paper of Pop "Henselian implies large" which contains useful information in the first section after the intro. My impression is that the answer is yes. Maybe I write a brief note on this for internal use (i.e. of course not to be published). I this sees the day, I send you a copy. Keep me informed if you have any news. Best, Sebastian $\endgroup$
    – Sebastian Petersen
    Commented Dec 2, 2011 at 14:19

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