Higher Chow cycles on some K3 surfaces with involution

TL;DR

This paper constructs explicit higher Chow cycles on certain K3 surfaces with involution, demonstrating their indecomposability for a wide range of Picard ranks, and introduces new examples in the middle Picard rank.

Contribution

It provides the first explicit families of higher Chow cycles on lattice-polarized K3 surfaces with middle Picard rank, using a degeneration method for proof.

Findings

Explicit higher Chow cycles constructed for 2<r<18

Indecomposability proven for very general members

First such examples in middle Picard rank

Abstract

We construct, for each 2<r<18, 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.