Van Geemen-Sarti involutions and elliptic fibrations on K3 surfaces double cover of P2

In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of P^2 branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen–Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen–Sarti involution induces a 2-isogeny between two K3 surfaces, which is described in this paper.