A comprehensive approach to the moduli space of quasi-homogeneous singularities

We study the relationship between singular holomorphic foliations in $({\mathbb{C}}^2,0)$ and their separatrices. Under mild conditions we describe a complete set of analytic invariants characterizing foliations with quasi-homogeneous separatrices. Further, we give the full moduli space of quasi-homogeneous plane curves. This paper has an expository character in order to make it accessible also to non-specialists.