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 $U\oplus ⟨12⟩$, the Verrill's pencil with transcendental lattice $U\oplus ⟨6⟩$ and another pencil linked to Verrill's pencil with transcendental lattice $U\oplus ⟨24⟩$. Many corollaries are deduced. For example, some singular K3 surfaces belong to different pencils or are Kummer surfaces of K3 surfaces of another pencil.

arXiv comment14 pages