Images of maps from $(\mathbb{C}^2,0)$ to $(\mathbb{C}^n,0)$

TL;DR

This work analyzes germs $F:(\mathbb{C}^2,0)\to(\mathbb{C}^n,0)$ with finite map behavior and studies the image singularities $(X,0)$. By leveraging 3-manifold topology of the link $L$ of $(X,0)$, it proves an equivalence: when $(X,0)$ is normal, it is a quotient of $(\mathbb{C}^2,0)$ by a finite subgroup of $U(2)$ and $\pi_1(L)$ is finite; this yields a complete picture of possible topologies, notably simple hypersurface singularities $A_n$, $D_n$, $E_6$, $E_7$, $E_8$. The paper then treats the non-normal case via normalization, providing an algorithm to decide when a singularity is an image of a finite map and describing how to construct the corresponding $F$ by composing a quotient map with the normalization. It culminates in explicit invariant-theory constructions for cyclic and binary polyhedral groups (and their products with cyclic groups), deriving concrete equations for the image singularities and detailing the corresponding embedding dimensions and multiplicities. Overall, the results classify topological types of quotient-like image singularities and furnish concrete maps realizing these topologies, with special emphasis on the simple singularities and their 3-manifold topologies.

Abstract

Let $F:(\mathbb{C}^2,0)\to (\mathbb{C}^n,0)$ be the germ of a finite map and $(X,0)$ be its image. We will in this article using the topology of the link show that $(X,0)$ has to be a quotient singularity if it is normal and describe the possible topological types. Including a discussion of the groups and examples of how to construct a map to a given topology. We will also discuss the case when the image is not normal.

Theorems & Definitions (20)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2
• Corollary 2.3
• Definition 2.4
• Definition 2.5
• Proposition 2.6
• Theorem 2.7: JSJ-Decomposition
• Definition 2.8
• Lemma 3.1