## The Picard group of various families of $(\mathbb {Z}/2\mathbb {Z})^{4}$-invariant quartic K3 surfaces

### Volume 186 / 2018

#### 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 (2010) (some of which degenerate to the above 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 the surface.