On singularities of folding maps and augmentations. (English) Zbl 0939.58037
The author studies the properties of two different types of map germs of the form \(f: (\mathbb{C}^n, S) \rightarrow (\mathbb{C}^p, 0)\), where \(S\) is a finite set of points of \(\mathbb{C}^n\). The first is the fold type map, which is a map of the form:
\[
(x_1, \dots, x_{n-1}, y)\mapsto (x_1, \dots, x_{n-1}, y^2, yh_1(x,y^2), \dots, yh_{p-n} (x, y^2))
\]
where \(h_i, i=1, \dots, p-n\), are holomorphic functions. The second type of map is the augmentation of a map germ.
The main result for folding maps in this paper is that for \(f\) finitely \(A\)-determined, with \(p< 2n-1\), then \[ A_e\text{-codimension} (f)\leq \mu _I(f). \] If \(p=n+1\)and \(f\) is quasihomogeneous, then this is an equality.
The main result for augmentations is that if the deformation \(F\) of \(f\) is stable, then \[ \tau (g). A_e\text{-codimension} (f)\leq A_e\text{-codimension} (A_{F,g}(f)) \] where \(\tau (g)\) denotes the Tjurina number of \(g\). If \(F\) is in the nice dimensions or \(g\) is quasihomogeneous, then there is equality.
The main result for folding maps in this paper is that for \(f\) finitely \(A\)-determined, with \(p< 2n-1\), then \[ A_e\text{-codimension} (f)\leq \mu _I(f). \] If \(p=n+1\)and \(f\) is quasihomogeneous, then this is an equality.
The main result for augmentations is that if the deformation \(F\) of \(f\) is stable, then \[ \tau (g). A_e\text{-codimension} (f)\leq A_e\text{-codimension} (A_{F,g}(f)) \] where \(\tau (g)\) denotes the Tjurina number of \(g\). If \(F\) is in the nice dimensions or \(g\) is quasihomogeneous, then there is equality.
Reviewer: Jiancheng Zou (Beijing)
MSC:
58K40 | Classification; finite determinacy of map germs |
58K60 | Deformation of singularities |
14B07 | Deformations of singularities |
32S30 | Deformations of complex singularities; vanishing cycles |