Let $R$ be a DVR and $K$ its fraction field. Let $X$ be a proper integral separated scheme of finite type over $Spec(K)$. Does it always exists a proper integral separated scheme $Y$ of finite type over $Spec(R)$, such that its generic fiber is isomorphic to $X$?


Of course, if the variety is projective, the answer is yes. But I would like to know if there are proper non-projective varieties that do not admit a proper integral model as before. 



Existence of proper integral models.
From Nagata's compactification theorem, we can find a proper map $\overline{X} \to \mathop{\rm Spec}R$, containing $X$ as an open dense subscheme; then it is immediate to check that the generic fiber must be $X$.


