On invariants of a map germ from n-space to 2n-space

TL;DR

This paper studies invariants of map germs from n-space to 2n-space, providing formulas for double point counts, invariants for quasihomogeneous cases, and conditions for Whitney equisingularity, with applications to Euler obstruction calculations.

Contribution

It introduces a method to compute the number of double points using Mond's ideal, and links invariants to Whitney equisingularity and Euler obstruction for map germs.

Findings

Double point number d(f) equals the length of the local ring of D^2(f).

Formulas for d(f) in quasihomogeneous cases using weights and degrees.

Set of invariants characterizing Whitney equisingularity of unfoldings.

Abstract

We consider $\mathcal{A}$-finite map germs $f$ from $(\mathbb{C}^n,0)$ to $(\mathbb{C}^{2n},0)$. First, we show that the number of double points that appears in a stabilization of $f$, denoted by $d(f)$, can be calculated as the length of the local ring of the double point set $D^2(f)$ of $f$, given by the Mond's ideal. In the case where $n\leq 3$ and $f$ is quasihomogeneous, we also present a formula to calculate $d(f)$ in terms of the weights and degrees of $f$. Finally, we consider an unfolding $F(x,t) = (f_t(x),t)$ of $f$ and we find a set of invariants whose constancy in the family $f_t$ is equivalent to the Whitney equisingularity of $F$. As an application, we present a formula to calculate the Euler obstruction of the image of $f$.