Higher Chow cycles on K3 surfaces attached to plane quartics

TL;DR

This paper constructs explicit higher Chow cycles on K3 surfaces derived from plane quartics, demonstrating their role in generating a rank 2 subgroup in the indecomposable higher Chow group for general cases.

Contribution

It provides an explicit construction of higher Chow cycles on K3 surfaces related to plane quartics using bitangents, and proves their significance in the higher Chow group.

Findings

Higher Chow cycles generate a rank 2 subgroup in the indecomposable part.

Construction uses pairs of bitangents of quartic curves.

Regulator map computations confirm the subgroup's rank.

Abstract

In this paper, we give an explicit construction of higher Chow cycles of type $(2,1)$ on $K3$ surfaces obtained as quadruple coverings of the projective plane ramified along smooth quartics. The construction uses a pair of bitangents of the quartics. We prove that the higher Chow cycles generate a rank 2 subgroup in the indecomposable part of the higher Chow group for very general members, by using a specialization argument and an explicit computation of the regulator map.