Hurwitz spaces and liftings to the Valentiner group
Riccardo Moschetti (University of Stavanger)
Aula Beltrami - Wednesday, July 20, 2016 h.16:00
Abstract. The symmetric and the alternating groups are the unique possible examples of monodromy if we are in the case of an indecomposable cover X -> P^1 with X a generic
complex curve of genus greater than 3. Fried proved that the spin structure determines the irreducible components of the Hurwitz space of coverings of P^1 of degree n
branched on r points with r greater or equal than n greater or equal than 5, and monodromy given by the conjugacy class of 3-cycles in A_n. The crucial point consists on
the construction of the so-called lifting invariant, described algebraically by using a lifting to the double cover of A_n. This strategy does not work if one considers elements
with even order. The cases of alternating groups of order six and seven are exceptional because they are the only two examples of alternating groups that admit a covering of
degree three. This makes it possible to construct lifting invariants in case of ramification of order two and four.
We study the components of the Hurwitz scheme of ramified coverings of P^1 with monodromy given by the alternating group A_6 and elements in the conjugacy
class of product of two disjoint cycles. In order to detect the connected components of the Hurwitz scheme, we use as invariant the lifting to the Valentiner group, triple covering of A_6.
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