# on rational singularities

Let a cartesian diagram

Let $X'\rightarrow X$ be a rational resolution of singularities of $k$-schemes of finite type and $Y$ a closed subscheme.

Let $Y'\rightarrow Y$ be the base change to $Y$, we assume that $Y'$ is still smooth, is it still a rational resolution of singularities?

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Of course not. For example, let $X$ be a quadratic cone, $X'$ its blowup, and $Y$ the vertex of the cone. –  Sasha Apr 13 at 10:40