The branching orders of a quasi-ordinary projection
type: Item
instance of: scholarly article
title: The branching orders of a quasi-ordinary projection
zbMATH Open document ID: 0847.32044
DOI: 10.1007/BF02572617
author: Kyungho Oh
published in: Mathematische Zeitschrift
publication date: 1996-10-15T00:00:00Z
full work available at URL: https://eudml.org/doc/174812
review text: Let \((X, x_0) \subset (\mathbb{C}^{d + 1}, 0)\) be a germ of an irreducible complex analytic hypersurface. The germ \((X, x_0)\) is called quasi-ordinary if, after a suitable coordinate change, the projection on the first \(d\)-coordinates \[ \pi : (X, x_0) \to (\mathbb{C}^d, 0),\;(t_1, \ldots, t_d, z) \mapsto (t_1, \dots, t_d) \tag{1} \] induces a branched covering and its discriminant locus \(D\) is contained in the coordinate hyperplanes of \(\mathbb{C}^d\). The map \(\pi\) is called the quasi-ordinary projection. Let \(x \in X\) be a point on \(X\), regarding \(X\) as a representative of the germ \((X,x)\). By definition, the germ \((X,x)\) is also quasi-ordinary and the map \( \pi\) in (1) induces an unbranched covering \[ \pi : U(x) - \pi^{-1} (D) \to \pi \bigl( U(x) \bigr) - D \] for a sufficiently small polydisk neighborhood \(U(x)\) of \(x\) in \(X\). The covering degree of this map is the same for any sufficiently small neighborhood \(U(x)\) and is called the branching order \(o(x,X)\) at \(x \in X\). Let \(Z_i \subset X\) be the zero-set of \(t_i\) and \(m_i\) be the brancing order at a generic point of \(Z_i\). After relabelling the \(t_i\), we may assume that \[ m_d \leq m_{d - 1} \leq \cdots \leq m_1. \] It turns out that the set \(\{m_i |i > 1\}\), which depends on a priori the choice of the projection \(\pi\), is indeed a topological invariant of the germ \((X, x_0)\) itself. This intriguing fact was proved by Lipman. In this paper, we give a short proof of this fact using our previous results. Also a topological characterization of the inverse image \(\pi^{-1} (\text{Sing} (D))\) of the singular locus of the discriminant \(D\) is provided.
Mathematics Subject Classification ID: 32S50
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 856567
zbMATH Keywords: quasi-ordinary projection
zbMATH Keywords: branching order
MaRDI profile type: Publication
cites work: On the Ramification of Algebraic Functions
cites work: Local homological properties of analytic sets
cites work: Q5615527
cites work: The homology groups of the links of quasi-ordinary singularities
cites work: Topological Types of Quasi-Ordinary Singularities
OpenAlex ID: W2316609798
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description: scientific article; zbMATH DE number 856567