Motivic cycles on K3 double covers of del Pezzo surfaces

TL;DR

This paper constructs motivic cohomology cycles on K3 surfaces that are double covers of del Pezzo surfaces, generalizing previous work on fourfold covers of the projective plane, advancing understanding of algebraic cycles on complex surfaces.

Contribution

It introduces a new method to construct motivic cohomology cycles on K3 surfaces arising from double covers of del Pezzo surfaces, extending prior results to a broader class of surfaces.

Findings

Constructed explicit motivic cohomology cycles on K3 double covers.

Generalized previous constructions from fourfold covers of P^2 to del Pezzo base surfaces.

Provides tools for studying algebraic cycles on complex surfaces.

Abstract

We construct motivic cohomology cycles in the group $H^3_{\mathcal M}(Z,{\mathbb Q}(2))$ where $Z$ is a K3 surface obtained as a double cover of a del Pezzo surface $X$ branched at a curve in $|-2K_X|$. The construction uses (-1) curves on the del Pezzo and is a generalization of a recent pre-print of Ken Sato arXiv: 2408.09102 where he considers the case of fourfold covers of ${\mathbb P}^2$ branched at a quartic curve.