The Picard group of various families of $(\mathbb {Z}/2\mathbb {Z})^{4}$-invariant quartic K3 surfaces
Volume 186 / 2018
Acta Arithmetica 186 (2018), 61-86
MSC: Primary 14J28; Secondary 14C22, 14D05.
DOI: 10.4064/aa170718-7-3
Published online: 12 October 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.
