# Extending a model to a given compactification of its generic fiber

Let $R$ be a discrete valuation ring and $K$ its field of fraction. Let $X$ be a proper $K$-variety, $U$ a dense open and consider an $R$-model $\mathcal{U}$ of $U$. Can we embed $\mathcal{U}$ in a proper $R$-model of $X$?

Nagata's embedding theorem ensures the existence of a compactification for $\mathcal{U}$, but I don't see any way to force its generic fiber to be $X$.

• It is possible to apply Nagata's theorem to the $R$-scheme obtained by glueing $\mathcal{U}$ and $X$ along $U$. – Olivier Benoist Jan 25 '15 at 12:53
• @OlivierBenoist: right, but you just have to check that this $R$-scheme is separated, e.g. by the valuative criterion. – Laurent Moret-Bailly Jan 25 '15 at 19:50
• Oh, that's great! Thanks you two for the tips! – Piet Jan 26 '15 at 19:18