Hurwitz moduli varieties parameterizing pointed covers of an algebraic curve with a fixed monodromy group

Given a smooth, projective curve $Y$, a point $y_0\in Y$, a positive integer $n$, and a transitive subgroup $G$ of the symmetric group $S_d$ we study smooth, proper families, parameterized by algebraic varieties, of pointed degree $d$ covers of $(Y,y_0)$, $(X,x_0)\to (Y,y_0)$, branched in $n$ points of $Y\setminus y_0$, whose monodromy group equals $G$. We construct a Hurwitz space $H$, an algebraic variety whose points are in bijective correspondence with the equivalence classes of pointed covers of $(Y,y_0)$ of this type. We construct explicitly a family parameterized by $H$, whose fibers belong to the corresponding equivalence classes, and prove that it is universal. We use classical tools of algebraic topology and of complex algebraic geometry.