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

Given a smooth, projective curve Y , a point y₀∈ Y , a positive integer n , and a transitive subgroup G of the symmetric group Sd , we study smooth, proper families, parameterized by algebraic varieties, of pointed degree d covers of (Y,y₀) , (X,x₀)→ (Y,y₀) , branched in n points of Y y₀ , 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,y₀) 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.