K3 surfaces with $\mathbb{Z}_2^2$ symplectic action

TL;DR

This paper constructs and analyzes a specific family of K3 surfaces with a $Z_2^2$ symplectic action, revealing their lattice structure, relation to covers of the projective plane, and specialization to Kummer surfaces.

Contribution

It introduces a new 4-dimensional family of K3 surfaces with $Z_2^2$ symplectic action, detailing their Néron-Severi lattice and connections to branched covers and Kummer surfaces.

Findings

Néron-Severi lattice generated by 24 rational curves

Surfaces specialize to Kummer surfaces of product of elliptic curves

Relation to $Z_2$-covers and Hirzebruch-Kummer coverings

Abstract

Let $G$ be a finite abelian group which acts symplectically on a K3 surface. The N\'eron-Severi lattice of the projective K3 surfaces admitting $G$ symplectic action and with minimal Picard number is computed by Garbagnati and Sarti. We consider a $4$-dimensional family of projective K3 surfaces with $\mathbb{Z}_2^2$ symplectic action which do not fall in the above cases. If $X$ is one of these K3 surfaces, then it arises as the minimal resolution of a specific $\mathbb{Z}_2^3$-cover of $\mathbb{P}^2$ branched along six general lines. We show that the N\'eron-Severi lattice of $X$ with minimal Picard number is generated by $24$ smooth rational curves, and that $X$ specializes to the Kummer surface $\textrm{Km}(E_i\times E_i)$. We relate $X$ to the K3 surfaces given by the minimal resolution of the $\mathbb{Z}_2$-cover of $\mathbb{P}^2$ branched along six general lines, and the corresponding Hirzebruch-Kummer covering of exponent $2$ of $\mathbb{P}^2$.