On the analytic structure of double and triple points in the target of finite holomorphic multi-germs

TL;DR

This paper investigates the analytic structure of double and triple point spaces in finite holomorphic multi-germs, establishing Cohen-Macaulay properties and explicit ideal descriptions under certain conditions.

Contribution

It extends known results from mono-germs to multi-germs, providing explicit ideal formulas and Cohen-Macaulay conditions for double and triple point spaces.

Findings

Double and triple point spaces are Cohen-Macaulay under certain conditions.

Explicit defining ideals for these spaces are derived.

Results generalize mono-germ cases to multi-germs.

Abstract

We study the analytic structure of the double and triple point spaces $M_2(f)$ and $M_3(f)$ of finite multi-germs $f\colon (X,S)\to(\mathbb{C}^{n+1},0)$, based on results of Mond and Pellikaan for the mono-germ case. We show that these spaces are Cohen-Macaulay, provided that certain dimensional conditions are satisfied, and give explicit expressions for their defining ideals in terms of those of their mono-germ branches.