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 \(H^G_n(Y)\) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary \(G\) we prove that \(H^G_n(Y)\) is a coarse moduli variety. For families of pointed Galois covers of \((Y,y_0)\) we prove that the Hurwitz space \(H^G_n(Y,y_0)\) 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.