Let $f$ be a polynomial in $k$ complex variables, and suppose the affine variety $V$ given by $f = 0$ has an isolated singularity at the origin, but is otherwise smooth. Now assume that some cyclic group $G \cong \mathbb{Z}_n$ acts on $\mathbb{C}^k$ in such a way that $V$ is invariant, and the origin is the unique fixed point. Are there simple conditions which determine whether $V/G$ can be smoothed?

An easy-to-read reference on smoothing singularities would also be appreciated.


I am interested in quotients of the threefold node, described in $\mathbb{C^4}$ by any non-degenerate homogeneous quadratic polynomial $f$, by cyclic subgroups of $GL(4,\mathbb{C}$). If $f$ is invariant, then by choice of coordinates it can be put in the form $f = x_1 x_4 - x_2 x_3$, $G$ acts on the whole family given by $f_t = f + t$, and this suffices to smooth the quotient. If $f$ is not invariant, then I'm not sure what happens, as there is generally no equivariant smoothing of the node (consider $(x_1, x_2, x_3, x_4) \to (i x_1, i x_2, i x_3, -i x_4)$ and $f \to -f$).

(Edit: No bites, a day later. Is the question unclear?)

  • $\begingroup$ I'm a bit confused. In the second example you give, is it also a smoothing by itself as the quotient will identify $f_t$ and $f_{-t}$ as one smooth fiber for $t\neq 0$? $\endgroup$
    – CYXU
    Commented Mar 9, 2013 at 13:13
  • $\begingroup$ I see what you mean, but that seems strange: the smooth fibres are exactly the same as those for the deformations of the 'upstairs' geometry. $\endgroup$ Commented Mar 11, 2013 at 16:27


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