It is well known that there is a line through 2 points in the plane, a conic through 5 points and nodal cubics through 8 points. In general there are rational curves of degree d through 3d-1 points in a plane. We will show how the existence of these rational curves is related to the existence of higher Chow  cycles in families of Kummer surfaces of Abelian surfaces and more generally certain K3 surfaces arising from Del Pezzo surfaces.

As an application we show how these cycles can be used to prove algebraicity of values of higher Green’s functions at CM points along the lines of a conjecture (now a theorem of Li)  of Gross-Zagier and Gross-Kohnen-Zagier.