For simplicity, we work over $\mathbb{C}$, and every variety is projective.
Let $f: X \to Y$ be a flat morphism between $X$ and $Y$ with connected fibers. Here $X$ has at worst terminal singularities, $Y$ is smooth, and a general fiber $F$ is also smooth. Let $Y' \to Y$ be a finite (flat) morphism from a smooth variety $Y'$ to $Y$ and $X'=X \times_{Y'} Y$. My question is: when does $X'$ have at worst rational singularities?
I think certain assumption must be put on $f$. The only case I know about this question is when $\dim X = 2$, $\dim Y=1$, and all fibers of $f$ are semi-stable curves. However, when $\dim Y > 1$, I do not know what assumption is needed for relatively dimension one case or in general.
Thanks!
