  Interest in Conformal Field Theories and Quantum Field Theory lead physicists
to consider configuration spaces of marked points on the complex projective
line, $Conf_{0,d}(\mathbb{P})$. In this paper, a real semi-algebraic
stratification of $Conf_{0,d}(\mathbb{C})$, invariant under Coxeter-Weyl group
is constructed, using the natural relation of this configuration space with the
space $Dpol_d$ of complex monic degree $d>0$ polynomials in one variable with
simple roots. This decomposition relies on subsets of $Dpol_d$ forming a good
cover in the sense of Cech of $Dpol_d$ and such that each piece of the
decomposition is a set of polynomials, indexed by a decorated graph reminiscent
of Grothendieck's dessins d'enfant. This decomposition in Coxeter-Weyl chambers
brings into light a very deep interaction between the real locus of the moduli
space $\overline{\mathcal{M}}_{0,d}(\mathbb{R})$ and the complex one
$\overline{\mathcal{M}}_{0,d}(\mathbb{C})$. Using this decomposition, the
existence of geometric invariants of those configuration spaces has been shown.
Many examples are provided. Applications of these results in braid theory are
discussed, namely for the braid operad.
