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

Abstract

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.

Final version. Manuscript accepted for publication in Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl