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Irreducible components of Hurwitz spaces parameterizing Galois coverings of curves of positive genus

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Let Y be a smooth, projective, irreducible complex curve. A G-covering p : C → Y is a Galois covering, where C is a smooth, projective, irreducible curve and an isomorphism G ∼ −→ Aut(C/Y ) is fixed. Two G-coverings are equivalent if there is a G-equivariant isomorphism between them. We are concerned with the Hurwitz spaces H n (Y ) and H G n (Y, y0). The first one parameterizes Gequivalence classes of G-coverings of Y branched in n points. The second one, given a point y0 ∈ Y , parameterizes G-equivalence classes of pairs [p : C → Y, z0], where p : C → Y is a G-covering unramified at y0 and z0 ∈ p (y0). When G = Sd one can equivalently consider coverings f : X → Y of degree d with full monodromy group Sd. The Hurwitz spaces are smooth algebraic varieties and associating to a covering its branch divisor yields finite etale morphisms H n (Y ) → Y (n) − ∆ and H n (Y ) → (Y − y0) (n) − ∆, where ∆ is the codimension one subvariety of Y (n) whose points correspond to effective non simple divisors of Y . The main result of the present paper is the explicit calculation of the monodromy action of the fundamental groups of Y (n) −∆ and (Y − y0) (n) −∆ on the fibers of the above topological coverings (see Theorem 2.8 and Theorem 2.10). The connected (=irreducible) components of H n (Y ) and H G n (Y, y0) are in one-to-one correspondence with the orbits of these monodromy actions. The case Y = P, G = Sd is classical. Hurwitz, using results of Clebsch [Cl], proved in [Hu] the connectedness of the space which parameterizes equivalence classes of simple coverings of P branched in n points (see [Vo2, Lemma 10.15] for a modern account). The Hurwitz spaces of Galois coverings of P were first introduced and studied by Fried in [Fr] in connection with the inverse Galois problem. Fried and Volklein prove in [FV] that H n (P ) has a structure of algebraic variety over Q and if furthermore the center of G is trivial they relate the solution of the inverse Galois problem to the existence of Q-rational points of H n (P ). They also address the problem of determining the connected components of the complex variety H n (P ). Berstein and Edmonds study in [BE] the Hurwitz spaces of simply branched coverings X → Y and address the problem of its connectedness in relation to the topological classification of the generic branched coverings between