Let $K$ be a number field or a field of rational functions of a curve. Let $S$ be its subring, regular and of dimension $1$. Let $X_S$ be a projective model over $S$ of $X$, where $X$ is a variety over $K$. We need to establish the bijection between $K$ points of $X$ and $S$ points of $X_S$.
In the case $X$ is function field of a curve and $S=k[C]$ then it's not hard to see that this follows from the fact that any rational map from a curve to a projective variety extends to a morphism. (T)
How to extend this to the remaining cases? I think in the function field case we just need to choose an adequate open subset of the curve that is projective and birational to the starting curve. Do we have similar statement for (T) in the arithmetic case? Is it true that a rational map from a curve to a projective scheme extends to a morphism? (I guess the definition of a rational map in the scheme case is just morphism from an open subscheme).
The fact follows from the valuative criteria of properness.