Smooth models of singular $K3$-surfaces

TL;DR

This paper classifies smooth models of singular K3-surfaces with small discriminant, revealing a correlation between discriminant size and lines, and constructs examples with many lines and models.

Contribution

It provides a classification of smooth spatial models of singular K3-surfaces of small discriminant and explores their geometric properties.

Findings

Fermat quartic has exactly three smooth spatial models

Correlation between discriminant and number of lines in models

Constructed K3-quartic with 52 lines and singular points

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.