algebraic geometry. There is a natural compactification of $\Mod_X$ called the Deligne-Mumford compactification.

[**Another point of view on holomorphic families.**]{}

Let’s go back to our initial problem but with a different point of view. Consider an holomorphic family $$F_t(z):=\frac{a_d(t)z^d+\ldots+a_1(t)z+a_0(t)}{b_d(t)z^d+\ldots+b_1(t)z+b_0(t)},\quad t\in D({0,1}).$$ For the study of the boundary of $\rat_d$, we will consider that $f_t\in\Rat_d$ if and only if $t\neq0$.

By holomorphic family we mean that the $a_i(t)$ and $b_j(t)$ are Laurent series in $t$ without essential singularities. The other idea is to consider that instead of having a family of rational map with coefficients in $\C$, we can think of $F_t$ as only one rational map whose coefficients are Laurent series without essential singularities or in another field that contains them. We denote by $\LL$ a field that contains these series and that will be specified when we will really need it. The only thing that the reader should keep in mind is that even if this field is really bigger than what we are used to see in holomorphic dynamics, the only interesting property for us is that it our series.

Thus what we will be considering is a rational map

$$F_t(z):=\frac{a_dz^d+\ldots+a_0}{b_dz^d+\ldots+b_0}\in \LL(z).$$

The field $\LL$ has the particularity 