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.