Smooth models of singular K3-surfaces

Abstract

We show that the classical Fermat quartic has exactly three smooth spatial models. As a generalization, we give a classification of smooth spatial (as well as some other) models of singular K3-surfaces of small discriminant. As a by-product, we observe a correlation (up to a certain limit) between the discriminant of a singular K3-surface and the number of lines in its models. We also construct a K3-quartic surface with 52 lines and singular points, as well as a few other examples with many lines or models.