Transcendental Lattices of Certain Singular K3 Surfaces

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 \(U \oplus \langle 12 \rangle \) , 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.