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

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In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of ℙ² 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.