On a family of K3 surfaces with $\mathcal{S}_4$ symmetry

TL;DR

This paper constructs and analyzes a family of K3 surfaces with symmetric group S4 symmetry, revealing their Picard ranks, symplectic actions, and modularity properties through Picard-Fuchs equations.

Contribution

It introduces a new one-parameter family of K3 surfaces with S4 symmetry and extends Picard-Fuchs computations to semi-ample hypersurfaces, linking to Mirror Moonshine.

Findings

Members have Picard rank 19

Symmetric group acts symplectically on the surfaces

Picard-Fuchs equations show modularity properties

Abstract

The largest group which occurs as the rotational symmetries of a three-dimensional reflexive polytope is the symmetric group on four elements. There are three pairs of three-dimensional reflexive polytopes with this symmetry group, up to isomorphism. We identify a natural one-parameter family of K3 surfaces corresponding to each of these pairs, show that the symmetric group on four elements acts symplectically on members of these families, and show that a general K3 surface in each family has Picard rank 19. The properties of two of these families have been analyzed in the literature using other methods. We compute the Picard-Fuchs equation for the third Picard rank 19 family by extending the Griffiths-Dwork technique for computing Picard-Fuchs equations to the case of semi-ample hypersurfaces in toric varieties. The holomorphic solutions to our Picard-Fuchs equation exhibit modularity properties known as "Mirror Moonshine"; we relate these properties to the geometric structure of our family.