Hurwitz spaces and liftings to the Valentiner group

Abstract

Abstract. We study the components of the Hurwitz scheme of ramified coverings of P 1 with monodromy given by the alternating group A6 and elements in the conjugacy class of product of two disjoint cycles. In order to detect the connected components of the Hurwitz scheme, inspired by the case of the spin structures studied by Fried for the 3-cycles, we use as invariant the lifting to the Valentiner group, triple covering of A6. We prove that the Hurwitz scheme has two irreducible components when the genus of the covering is greater than zero, in accordance with the asymptotic solution found by Bogomolov and Kulikov.

Hyperelliptic odd coverings

“…It follows from the irreducibility that all the coverings of H odd can be obtained in this way. We remark that this can be generalised to higher degrees and different ramification type by using an approach similar to [17] to compute the number of irreducible components of the Hurwitz spaces. It would be interesting to generalise this more explicit computation to hyperelliptic curves, and counting the number of connected components of H HOC g .…”