K3 surfaces with $\mathbb Z_2^2$ symplectic action

Abstract

Let $G$ be a finite abelian group which acts symplectically on a K3 surface. The Néron-Severi lattice of the projective K3 surfaces admitting $G$ symplectic action and with minimal Picard number was computed by Garbagnati and Sarti. We consider a four-dimensional family of projective K3 surfaces with $\mathbb {Z}_2^2$ symplectic action which do not fall into 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 Néron-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 $\mathbb {P}^2$.