Delta-components of moduli spaces of covers

Delta-components of moduli spaces of covers. We are interested in irreducible components of moduli spaces of G-covers and their fields of definition. Our results allow us to build, for any finite group, such components defined over Q. Our method provides a large scope for the ramification type of the covers. These components are obtained by deforming certain covers of the boundary of the moduli spaces. Finally, these components are also compatible in a tower of Hurwitz spaces; we get projective systems of components of the modular tower.