Specializations of symplectic and van Geemen--Sarti involutions on K3 surfaces

TL;DR

This paper investigates specializations of symplectic involutions on K3 surfaces, focusing on van Geemen--Sarti involutions, their Néron–Severi groups, and conditions under which the surface and its quotient are isomorphic or have isometric Néron–Severi groups.

Contribution

It identifies infinite families of K3 surfaces with symplectic involutions where the Néron–Severi groups of the surface and its quotient are isometric, and characterizes specializations of van Geemen--Sarti involutions with this property.

Findings

Infinite codimension 2 subfamilies with isometric Néron–Severi groups

Characterization of specializations where NS(X) ≅ NS(Y)

Existence of a 5-dimensional family with X ≅ Y and complex multiplication

Abstract

Given a symplectic involution $\iota$ on a K3 surface $X$, the desingularization $Y$ of $X/\iota$ is still a K3 surface, which in general has a different N\'eron--Severi group. Nevertheless, if the involution is induced by the translation by a 2-torsion section on an elliptic fibration (i.e. it is a van Geemen--Sarti involution) and the Picard number is minimal, the N\'eron--Severi groups of $X$ and $Y$ are known to be isometric. We first determine infinitely many codimension 2 subfamilies of projective K3 surfaces with a symplectic involution (not of van Geemen--Sarti type) whose generic members satisfy $NS(X)\simeq NS(Y)$. Then, we describe the cohomological action of a van Geemen--Sarti involution and we characterize specializations of K3 surfaces with a van Geemen--Sarti involution for which it is still true that $NS(X)\simeq NS(Y)$. There is a 5-dimensional family of K3 surfaces with van Geemen--Sarti involution for which $X\simeq Y$. The K3 surfaces in such a family admit complex multiplication, and we describe its cohomological action. We briefly discuss similar problems for order 3 symplectic automorphisms induced by a translation by a 3-torsion section on an elliptic fibration.