The branching orders of a quasi-ordinary projection
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.
scientific article; zbMATH DE number 856567