Let all schemes below be excellent. Let $X_0$ be a regular (not necessarily smooth, projective) non-empty scheme of finite type over the generic point $\eta$ of a regular connected scheme $S$. As the answers to my question For a morphism f from a regular scheme, should there exist an open subscheme U of the target such that fibre of f at each point of U is regular show, there does not have to exist a dense open $U\subset S$ such that $X_0$ possesses a fibrewise regular model over $U$. Yet, should there always exist a pseudo-finite dominant morphism $j:U\to S$ and some model $X$ of $X_0$ over $S$ such that $U$ is regular and the reduced scheme associated to $X_U$ is smooth over $U$? Can we assume that the morphism $U\to S$ is radiciel?

Upd.: it is probably sufficient for my purposes to find such a $U$ for a variety $X_0$ that is smooth over the perfect closure of $\eta$. It seems that (the first) BCnrd comment solves the problem; thanks!!

4more comments