Van Geemen-Sarti involutions and elliptic fibrations on $K3$ surfaces double cover of $\mathbb{P}^2$

Abstract

In this paper we classify the elliptic fibrations on $K3$ surfaces which are the double cover of a blow up of $\mathbb{P}^2$ 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.