Let $k$ be an algebraically closed field. Let $f:X \to Y$ be a morphism of $k$-algebraic schemes, that is separated schemes of finite type over $k$.
1) $Y$ is a smooth variety (that is, it is integral too) over $k$ resp. regular.
2) All fibers $X_y$ for $y$ a closed point of $Y$ are isomorphic to a smooth/regular variety $F$.
Is it true then, that $X$ is a smooth/regular variety? If not in general, is it true if $F$ is a group variety over $k$?
It would be enough of course, to show, that $X$ is smooth over $Y$ or even only that $f$ is flat, or that $X$ is Cohen-Macaulay but at the moment I find no way to prove even this.