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On the analytic equivalence of curves. (English) Zbl 0612.14020
Let R be the algebra $$k[[X_ 1,...,X_ N]]$$ over an algebraically closed field k, $$(k^ N,0)$$ the k-scheme Spec(R), and $$X=Spec(R/I)$$ a curve of $$(k^ N,0)$$ where I is an ideal of R. The n-th truncation $$\bar X{}_ n$$ of X is the closed zero-dimensional subscheme of $$(k^ N,0)$$ given by $$\bar X{}_ n=Spec(R/I+(X_ 1,...,X_ N)^ n)$$. Then, there exists an integer t(X) such that the analytic type of X is determined by its n-truncations $$\bar X{}_ n$$ for all $$n\geq t(X)$$. This fact is shown by H. Hironaka [Notes of the Woods Hole Seminar in Algebraic Geometric (1964); see also Arithmetical algebraic Geom., Proc. Conf. Purdue Univ. 1963, 153-200 (1965; Zbl 0147.205)] for more general subschemes X, not necessarily curves. In the above curve case, the author gives the explicit value of t(X) in terms of the Milnor number of X in this paper.
Reviewer: S.Koizumi

##### MSC:
 14H05 Algebraic functions and function fields in algebraic geometry
##### Keywords:
regularity index; truncation; analytic type; Milnor number
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##### References:
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