The Picard Group of Various Families of $(\mathbb{Z}/2\mathbb{Z})^{4}$-invariant Quartic K3 Surfaces

TL;DR

This paper investigates the Picard groups of quartic K3 surfaces invariant under a specific group action, describing their geometric configurations and lattice structures, including lines and conics, across various families.

Contribution

It provides explicit descriptions of the Picard groups for families of invariant quartic K3 surfaces, detailing their generators and lattice structures, which was previously not fully understood.

Findings

Picard groups generated by lines and conics on the surfaces

Families contain surfaces with 8, 16, 24, or 32 lines

Identified Picard group structures as lattices for general members

Abstract

The subject of this paper is the study of various families of quartic K3 surfaces which are invariant under a certain $(\mathbb{Z}/2\mathbb{Z})^{4}$ action. In particular, we describe families whose general member contains $8,16,24$ or $32$ lines as well as the $320$ conics found by Eklund (some of which degenerate into the mentioned lines). The second half of this paper is dedicated to finding the Picard group of a general member of each of these families, and describing it as a lattice. It turns out that for each family the Picard group of a very general surface is generated by the lines and conics lying on said surface.