Some Preliminaries for Families of Cyclic Covers

In this chapter we collect the remaining preparations for the computations concerning the VHS of our families π : C → P _n of cyclic covering of P¹, which we construct in this chapter.

Let V denote the VHS of the family X → Y of curves and Mon⁰(V) denote the identity component of the Zariski closure of the monodromy group of V. In Section 3.1 we introduce the generic Hodge group Hg(V), which is the maximum of the Hodge groups of all occurring Hodge structures in V. Moreover Hg(V) coincides with the Hodge groups of the Hodge structures in V over the complement of a unification of countably many submanifolds of Y . Our families π : C → Pn are constructed in Section 3.2. We will also make some general remarks about the monodromy representation of V including the fact that the Galois group action yields an eigenspace decomposition in Section 3.2. In Section 3.3 we make some explicit computations of the monodromy representations of these eigenspaces. These computations are motivated from the fact that Mon⁰(V) is a normal subgroup of the derived group Hg^der(V) of the generic Hodge group! as we see in Section 3.1.