Higher Chow Cycles on Some <i>K</i>3 Surfaces with Involution

Abstract We construct, for each $3\leq r \leq 17$, an explicit family of higher Chow cycles of type $(2, 1)$ on a family of lattice-polarized $K3$ surfaces of generic Picard rank $r$, and prove that the indecomposable part of this cycle is non-torsion for very general members of the family. These are the first explicit examples of such families in middle Picard rank. Our construction is based on singular double plane model of $K3$ surfaces, and the proof of indecomposability is done by a degeneration method.