Van Geemen--Sarti involutions and elliptic fibrations on K3 surfaces double cover of $\mathbb{P}^2$

TL;DR

This paper classifies elliptic fibrations on certain K3 surfaces, describes equations for many, and details the associated van Geemen--Sarti involutions and their induced 2-isogenies between K3 surfaces.

Contribution

It provides a comprehensive classification of elliptic fibrations and van Geemen--Sarti involutions on K3 surfaces that are double covers of blow-ups of projective planes.

Findings

Classification of elliptic fibrations on these K3 surfaces

Explicit equations for many elliptic fibrations

Description of 2-isogenies induced by involutions

Abstract

In this paper we classify the elliptic fibrations on K3 surfaces which are the double cover of a blow up of $\mathbb{P}^2$ branched along rational curves and we give equations for many of these elliptic fibrations. Thus we obtain a classification of the van Geemen--Sarti involutions (which are symplectic involutions induced by a translation by a 2-torsion section on an elliptic fibration) on such a surface. Each van Geemen--Sarti involution induces a 2-isogeny between two K3 surfaces, which is described in this paper.