On reflection maps from the $n$-space to the $n+1$-space

Abstract

We consider some problems about a reflected graph map germ $f$ from $({\mathbb{C}}^n,0)$ to $({\mathbb{C}}^{n+1},0)$. A reflected graph map is a particular case of a reflection map, which is defined using an embedding of ${\mathbb{C}}^n$ in ${\mathbb{C}}^p$ and then applying the action of a reflection group $G$ on ${\mathbb{C}}^p$. We present a description of the presentation matrix of $f_{\ast }{\mathcal{O}}_n$ as an ${\mathcal{O}}_{n+1}$-module via $f$ in terms of the action of the associated reflection group $G$. We also give a description for a defining equation of the image of $f$ in terms of the action of $G$. Finally, we provide an upper (and also a lower) bound for the multiplicity of the image of $f$ and some applications.