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

Milena Barbosa Gama, Otoniel Nogueira da Silva

TL;DR

This work develops a symmetry-driven framework for reflected graph map germs $f=(w,h)$ from $\mathbb{C}^n$ to $\mathbb{C}^{n+1}$ via a finite reflection group $G$. It provides an explicit ${\mathcal O}_{n+1}$-presentation matrix for $f_*\mathcal{O}_n$ in terms of the $G$-action on $h$, and derives a determinantal defining equation $F(\mathbf X,Z)$ of the image, with a factorization $F(\mathbf w,Z)=\prod_{k=1}^d (Z-g_k\bullet h)$. The paper also establishes general multiplicity bounds in terms of the degrees of $G$ and applies the theory to concrete cases, including double point spaces, dihedral map germs with explicit computations, and a quasihomogeneous nonexistence result for corank-2 reflection maps with distinct weights. Collectively, the results offer a practical, group-theoretic approach to analyzing the geometry and singularities of reflected graphs and their images.

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.

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

TL;DR

This work develops a symmetry-driven framework for reflected graph map germs from to via a finite reflection group . It provides an explicit -presentation matrix for in terms of the -action on , and derives a determinantal defining equation of the image, with a factorization . The paper also establishes general multiplicity bounds in terms of the degrees of and applies the theory to concrete cases, including double point spaces, dihedral map germs with explicit computations, and a quasihomogeneous nonexistence result for corank-2 reflection maps with distinct weights. Collectively, the results offer a practical, group-theoretic approach to analyzing the geometry and singularities of reflected graphs and their images.

Abstract

In this work we consider some problems about a reflected graph map germ from to . A reflected graph map is a particular case of a reflection map, which is defined using an embedding of in and then applying the action of a reflection group on . In this work, we present a description of the presentation matrix of as an -module via in terms of the action of the associated reflection group . We also give a description for a defining equation of the image of in terms of the action of . Finally, we provide an upper (and also a lower) bound for the multiplicity of the image of and some applications.
Paper Structure (10 sections, 16 theorems, 116 equations, 4 figures)

This paper contains 10 sections, 16 theorems, 116 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reflection groups and reflection maps
  4. The presentation matrix of reflection maps
  5. The image of a reflected graph map
  6. The multiplicity of a reflection map germ
  7. Some applications and examples
  8. Double point spaces
  9. Dihedral map germs
  10. Quasihomogeneous reflection maps

Key Result

Theorem 1.1

A defining equation of the image of $f(\textit{x})=(w(\textit{x}),h(\textit{x}))$ (given by the $0$-Fitting ideal of the presentation matrix of$f_*{\cal O}_n$) is given by the following alternating sum

Figures (4)

  • Figure 1: The $C_5$ singularity viewed as a reflection map
  • Figure 2: The reflection map $f(x,y)=(x^2,y^2,x^3+y^3+xy)$ (real points).
  • Figure 3: Illustration of a reflection map
  • Figure 4: Orbit map of a reflection group.

Theorems & Definitions (35)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4
  • Remark 2.5
  • Example 2.6
  • Theorem 2.7
  • Remark 2.8
  • Proposition 2.9
  • ...and 25 more