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In this paper we classify the elliptic fibrations on K3 surfaces which arethe double cover of a blow up of $\mathbb{P}^2$ branched along rational curvesand we give equations for many of these elliptic fibrations. Thus we obtain aclassification of the van Geemen--Sarti involutions (which are symplecticinvolutions induced by a translation by a 2-torsion section on an ellipticfibration) on such a surface. Each van Geemen--Sarti involution induces a2-isogeny between two K3 surfaces, which is described in this paper.

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