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You probably meant $\pi: X\to Y$ and not $\pi:Y\to X$. That way, any singularity that appears on a scheme of finite type over a field can be mapped to a smooth variety in a finite way. (This claim is implicit in VA's and Torsten Ekedahl's comments above).

A fairly general criterion for a singularity to be rational is given in my paper A characterization of rational singularities. In particular it covers your case in any dimension.