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Transversal parameters and tangential flatness

Published online by Cambridge University Press:  24 October 2008

U. Orbanz
U. Orbanz
Affiliation:
Mathematisches Institut, Universität zu Köln, 5000 Köln 41, West Germany
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Let X be an algebraic variety over a field k, z a point of X and Let t = {t1..., ts} be part of a system of parameters of R. Then these parameters are algebraically independent over k, and therefore they define (near z) a projection f: X → As(k) to an s-dimensional affine space over k, sending z to the origin 0∊s(k). In general, the Hilbert function (resp. multiplicity) of will be worse than that of , and we will call the system t transversal, if the Hilbert function (resp. multiplicity) of R and R/tR agree. This gives two notions of transversality: one for Hilbert functions (H-transversal), and a weaker one for multiplicities (e-transversal). For s = dim R we recover the notion of a transversal system of parameters introduced by Zariski for studying equisingularity problems (see e.g. [17]). In the above set-up the numerical characters are defined with respect to the maximal ideal of R, but we will consider this problem for arbitrary ideals I using generalized Hilbert functions of type H[x, I, R] (see [7] and [10]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 98 , Issue 1 , July 1985 , pp. 37 - 49
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

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