The irreducible components of the moduli space of dihedral covers of algebraic curves

In this paper we introduce a new invariant for the action of a finite group $G$ on a compact complex curve of genus $g$. With the aid of this invariant we achieve the classification of the components of the moduli space of curves with an effective action by the dihedral group $D_n$. This invariant has been used in the meanwhile by the authors in order to extend the genus stabilization result of Livingston and Dunfield and Thurston to the ramified case. This new version contains an appendix clarifying the correspondence between the above components and the image loci in the moduli space M_g (classifying when two such components have the same image).Comment: 37 pages, final version appearing in 'Groups, Geometry and Dynamics

Cyclic and Abelian coverings of real varieties

We describe the birational and the biregular theory of cyclic and Abelian coverings between real varieties.Comment: 25 pages, dedicated to Slava (Viatcheslav) Kharlamov on the occasion of his 71-st birthday, final version to appear on Mathematische Annale

Topological equivalence of holomorphic foliation germs of rank 1 with isolated singularity in the Poincar/'e domain

We show that the topological equivalence class of holomorphic foliation germs of rank 1 with an isolated singularity of Poincaré type is determined by the topological equivalence class of the real intersection foliation of the (suitably normalized) foliation germ with a sphere centered in the singularity. We use this Reconstruction Theorem to completely classify topological equivalence classes of plane holomorphic foliation germs of Poincaré type and discuss a conjecture on the classification in dimension ≥3