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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?


[1] Real Algebraic Threefolds I by Kollar

[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

1 Answer 1

The standard definition of contact equivalence, also called V-equivalence, or K-equivalence, uses convergent power series. This is indeed the same as the zero sets (counted with multiplicity if the function f has multiple factors) being isomorphic. In the same way your formal equivalence is the same as formal isomorphism of the zero sets.

The example you give suggests that you want to conclude that a given singularity is isomorphic to some known non-isolated one. In the classification of say singularities with 1-dimensional singular locus one fixes the transverse singularities and has then again a sort of finite determinacy, so again it suffices to work with formal power series.

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