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

TL;DR

This paper studies reflection maps from complex n-space to (n+1)-space, describing their algebraic structure, image equations, and multiplicity bounds, with applications in singularity theory.

Contribution

It provides a detailed description of the module presentation and image equations for reflection maps, advancing understanding of their algebraic and geometric properties.

Findings

Presentation matrix of $f_*{ m O}_n$ explicitly described

Defining equations of the image derived from group action

Bounds established for the multiplicity of the image

Abstract

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.