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 S₃ 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 explicitly a family parameterized by a Hurwitz space and prove that it is universal. We use classical tools of algebraic topology and of complex algebraic geometry.