Join operation and ${\mathcal A}$-finite map-germs

TL;DR

This paper introduces elementary join map-germs to generate new ${\mathcal A}$-finite map-germs, providing a general form for monomial maps from complex spaces, using invariants like delta and curve invariants.

Contribution

It defines elementary join map-germs and describes a general form of ${\mathcal A}$-finite monomial maps, expanding understanding of map-germ classifications.

Findings

Defined elementary join map-germs for constructing ${\mathcal A}$-finite maps

Provided a general form for monomial ${\mathcal A}$-finite maps in complex spaces

Utilized delta invariant and curve invariants as main tools

Abstract

In this work we define some map-germs, called elementary joins, for the purpose of producing new ${\mathcal A}$-finite map-germs from them. In particular, we describe a general form of an ${\mathcal A}$-finite monomial map from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{p},0)$ for $p\geq 2n$ of any corank in terms of elementary join maps. Our main tools are the delta invariant and some invariants of curves.