RELATIVE DIHEDRAL GROUP ACTIONS ON RATIONAL ELLIPTIC SURFACES

Abstract

In this paper we give an explicit method for constructing every rational elliptic surface that has a four-torsion section. We use this construction to study the moduli space of birational equivalence classes of Galois covers that arise from rational elliptic surfaces with a relative dihedral group action. As a conclusion we find that the moduli space for such Galois covers whose Galois groups are the dihedral group of order eight is a nodal rational curve.