Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Given a smooth, projective curve \ (Y\), a finite group \ (G\) and a positive integer n we study smooth, proper families \ (X Y S S\) of Galois covers of \ (Y\) with Galois group isomorphic to G branched in \ (n\) points, parameterized by algebraic varieties \ (S\). When \ (G\) is with trivial center we prove that the Hurwitz space \ (HGₙ (Y) \) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary \ (G\) we prove that \ (HGₙ (Y) \) is a coarse moduli variety. For families of pointed Galois covers of \ ( (Y, y₀) \) we prove that the Hurwitz space \ (HGₙ (Y, y₀) \) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group \ (G\). We use classical tools of algebraic topology and of complex algebraic geometry.