Transcendental lattices of certain singular K3 surfaces

TL;DR

This paper computes the transcendental lattices of specific singular K3 surfaces from three different pencils, revealing relationships between these surfaces and their classifications.

Contribution

It provides explicit calculations of transcendental lattices for singular K3 surfaces in three pencils, establishing connections between different surface families.

Findings

Identified transcendental lattices for three K3 surface pencils.

Discovered some singular K3 surfaces belong to multiple pencils.

Showed certain K3 surfaces are Kummer surfaces of others.

Abstract

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