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 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 Néron-Severi lattice of $X$ with minimal Picard number is generated by $24$ smooth rational curves, and that $X$ specializes to the Kummer surface $\text{Km}(E_i\times E_i)$. 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 ${\mathbb{P}}^2$.

arXiv comment24 pages, 6 figures. Final version with minor corrections and additions. To appear in the Rocky Mountain Journal of Mathematics

References in corpus (1)

• Periods of an arrangement of six lines and Campedelli surfaces