# fearful of defining equivalent germs for non isolated singularities

Two power series $G(x_1, \ldots, x_n)$ and $F(x_1, \ldots, x_n)$ are equivalent over $\mathbb{C}$ if there is an automorphism of the ring $\mathbb{C}[[x_1, \ldots, x_n]]$ given by $x_1 \to \phi(x_1, \ldots, x_n)$ and an invertible element $$u(x_1, \ldots, x_n) \in \mathbb{C}[[x_1, \ldots, x_n]]$$ such that $$u(x_1, ...x_n)G(x_1, \ldots, x_n) =F(\phi_1, \ldots, \phi_n)$$ We can use this to say that two singularities given by $(F=0)$ and $(G=0)$ are equivalent if their equations are equivalent on the previous sense (see [1]).

Example ([2]): The quartic plane curve $(xz+y^2)^2+x^4$ has an $A_7 =x^2+y^8$ singularity at $(x=y=0)$.

I am quite comfortable with this sort of games for isolated singularities..However, for non isolated singularities, I do get concerned about things making sense...

What are the technical details and/or traps that I should look ? Is there any warning... counterexample.. reference?

Thanks!

[1] Real Algebraic Threefolds I by Kollar http://arxiv.org/pdf/alg-geom/9712004v1.pdf

[2] Wall, C. T. C. "Geometry of quartic curves." Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 117. No. 3. [Cambridge, Eng.] Cambridge Philosophical Society., 1995.

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Équivalence classes are simply the orbits for the action of a group on power series (which changes variables according to automorphisms, and multiplies by invertible power series), and this covers all cases. –  Mariano Suárez-Alvarez Feb 28 '14 at 1:24
Should your automorphism be written $x_i \mapsto \phi_i(x_1,\ldots,x_n)$? –  S. Carnahan Mar 1 '14 at 14:47