On the zero-dimensional strata of the Fulton-MacPherson conpactification

Question

Let $\operatorname{Conf}_n(\mathbb{R})$ be the configuration space of $n$ marked points on the real line. What is the difference between $\operatorname{Conf}_n(\mathbb{R})$ and the locus of zero-dimensional strata $\overline{\operatorname{Conf}}_n^0(\mathbb{R})$ of the Fulton-MacPherson compactification $\overline{\operatorname{Conf}}_n(\mathbb{R})$?

Answer 1

The zero-dimensional stratum is the quotient of $\mathrm{Conf}_n(\mathbb{R})$ under the action of the group $\mathbb{R}_+ \rtimes \mathbb{R}$ of positive rescalings and translations. So for example with $n=1$ you get a singleton, with $n=2$ you get a two-point space, with $n=3$ you get six closed segments, etc.