Transcendental lattices of certain singular K3 surfaces

Abstract

We compute the transcendental lattices of the singular K3 surfaces belonging to three pencils of K3 surfaces, namely the Apéry-Fermi pencil with transcendental lattice U ⊕ 12 , the Verrill's pencil with transcendental lattice U ⊕ 6 and another pencil linked to Verrill's pencil with transcendental lattice U ⊕ 24 . Many corollaries are deduced. For example, some singular K3 surfaces belong to different pencils or are Kummer surfaces of K3 surfaces of another pencil.