In this work 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 $\mathbb{ C}^{ p}$. In this work, we present a description of the presentation matrix of $f_*{ \cal O}_n$ as an ${ \cal O}_{ n+1}$-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.
Categories: math.AG math.CV