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

We study the analytic structure of the double and triple point spaces $M_2(f)$ and $M_3(f)$ of finite multi-germs $f\text{\hspace{0.17em}⁣}:(X,S)\to ({\mathbb{C}}^{n+1},0)$, 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.