without going back to the formalism of Deligne-Mumford.

In [@K2], the author explain how to use Berkovich space ideas in order to deduce properties of diverging sequences in $\rat_d$ and the vocabulary in [@A0] has been based on this idea. One of the goal of this work is also to explicit here the bridge between the two formalism in order to allow later to use this bridge the reverse way. This paper is based on some remarks sketched in [@A0].

[**A problem of normalization.**]{}

As we said, the space we are interested in, $\rat_d$, is a quotient space and it is usually convenient to use representatives instead of classes. The choice of representatives (often called normalizations) is usually a problem when one want to study the behavior in the boundary of the space. For example, it can happen that a sequence diverges in $\Rat_d$ and at the same time converges in $\rat_d$.

When we go to infinity we always have some critical points collapsing but the reverse can be false. Intuitively it is natural to think that one could understand the behavior of a diverging sequence by trying to understand how the critical points are collapsing together. That’s why it 