Universal flattening of Frobenius

For a variety $X$ of positive characteristic and a non-negative integer $e$, we define its $e$-th F-blowup to be the universal flattening of the $e$-iterated Frobenius of $X$. Thus we have the sequence (a set labeled by non-negative integers) of blowups of $X$. Under some condition, the sequence stabilizes and leads to a nice (for instance, minimal or crepant) resolution. For tame quotient singularities, the sequence leads to the $G$-Hilbert scheme.