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There are multiple definitions of when points $P$ and $Q$ on schemes $X$ and $Y$ are equisingular. One can require that either $P$ and $Q$ have isomorphic analytic (i.e. complex analytic), or formal, or étale neighborhoods. My main question is how are these three definition related. I know that for simple singularities they are equivalent, and for difficult ones I guess they are non-equivalent. Does anyone know precise statements, references, proofs, examples when these are non-equivalent notions?

The analytic and the étale case is a little bit subtler. One can ask for the adequate local rings to be isomorphic and also for isomorphic analytic/étale neighborhoods, as above. Are these two approaches equivalent?

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up vote 13 down vote accepted

All these notions are equivalent; Mike Artin proved that. For the étale topology this is in "Algebraic approximation of structures over complete local rings", Inst. Hautes Études Sci. Publ. Math. No. 36 1969, 23–58; the analytic case it treated in "On the solutions of analytic equations", Invent. Math. 5 1968, 277–291.

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None of these notions is the usual definition of equisingular. For example, any choice of four distinct lines though some given point in the plane is usually considered equisingular to any other, however, the analytic isomorphism class depends on the cross-ratio of the slopes.

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I think Zsolt meant "locally isomorphic". –  Sándor Kovács Jan 10 '11 at 7:15
    
I am not sure I understand this answer perfectly. Are they non-isomorphic analytically locally at the point of intersection? What about formal isomorphism at the same point? –  Zsolt Patakfalvi Jan 11 '11 at 16:42
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They are non-isomorphic even at the formal level --- look at m^4/m^5. The quartic form gives you the cross-ratio (well, essentially) that Vivek mentioned. –  Ravi Vakil Aug 29 '11 at 18:30
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