Let the cyclic group on $n$ elements, $C_n$, act on $\mathbb A^n$ by permuting the co-ordinates (over a field $k$). If $n \neq 0 \in k$, we can resolve the singularities of $X = \mathbb A^n/C_n$ by toric methods since the action can be linearized.

What about the case where $n = 0 \in k$? Are the exceptional divisors still projective spaces and given by blowing up along the singular locus? Can we blow up $\mathbb A^n$ first along some nice locus and then quotient by $C_n$ to find the resolution?



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