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.

Reorganized, new results cited. Final version accepted for publication

References in corpus (2)

• Lines on smooth polarized $K3$-surfaces
• Lines in supersingular quartics

Cited by in corpus (5)

• Lines on smooth polarized $K3$-surfaces
• Symmetries and equations of smooth quartic surfaces with many lines
• Conics in sextic $K3$-surfaces in ${\mathbb{P}}^4$
• Lines on K3-quartics via triangular sets
• On a smooth quartic surface containing 56 lines which is isomorphic as a K3 surface to the Fermat quartic