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Some geometric aspects of Puiseux surfaces. (English) Zbl 1061.14004

Essentially, a Puiseux surface is an embedded algebroid surface \(S=\) \(\operatorname{Spec}{\mathbb C}[[X,Y,Z]]/(F)\), where \(F\) is a polynomial in \(Z\) whose coefficients are series in \({\mathbb C}[[X,Y]]\) and the roots of \(F\) are Puiseux series of the same order in the indeterminates \(X\) and \(Y\). Several authors, H.W.E. Jung [J. für Math. 133, 289–314 (1908; JFM 39.0493.01)], J. Lipman [Am. J. Math. 87, 874–898 (1965; Zbl 0146.17301)], Y.-N. Gau [Topology 25, 495–519 (1986; Zbl 0606.32008)], and others have studied a particular case, the so called quasi-ordinary Puiseux surfaces, in relation to the problem of resolution of surface singularities.
This paper, which is part of the author’s doctoral thesis, is devoted to study to which extent the known results of the quasi-ordinary case are true in general. Section 2 shows how are equations for the tangent cone, properties concerning the equi-multiple locus are given in Section 3 and finally, in Section 4, the author indicates which are, in his opinion, the more interesting unsettled questions for these surfaces.

MSC:

14B05 Singularities in algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
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References:

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