A note on simple $K3$ singularities and families of weighted $K3$ surfaces

Abstract: We discuss properties of the Seifert form for simple $K3$ singularities, and of the Picard lattices of families of weighted $K3$ surfaces. We study a collection $\mathcal{M}{(\rho,\,\delta)}$ of $K3$ surfaces polarized by their Picard lattices that are in the set $\mathcal{L}{(\rho,\,\delta)}$ of certain lattices. We also report a numerical formula that relates the Seifert form for the singularities and the Picard lattice of the family.