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.
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