{"title":"On reflection maps from n-space to n+1-space","authors":"Milena Barbosa Gama, Otoniel Nogueira da Silva","doi":"arxiv-2408.11669","DOIUrl":null,"url":null,"abstract":"In this work we consider some problems about a reflected graph map germ $f$\nfrom $(\\mathbb{C}^n,0)$ to $(\\mathbb{C}^{n+1},0)$. A reflected graph map is a\nparticular case of a reflection map, which is defined using an embedding of\n$\\mathbb{C}^n$ in $\\mathbb{C}^{p}$ and then applying the action of a reflection\ngroup $G$ on $\\mathbb{C}^{p}$. In this work, we present a description of the\npresentation matrix of $f_*{\\cal O}_n$ as an ${\\cal O}_{n+1}$-module via $f$ in\nterms of the action of the associated reflection group $G$. We also give a\ndescription for a defining equation of the image of $f$ in terms of the action\nof $G$. Finally, we present an upper (and also a lower) bound for the\nmultiplicity of the image of $f$ and some applications.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"8 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.11669","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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 present an upper (and also a lower) bound for the
multiplicity of the image of $f$ and some applications.