# On the geometry of singular K3 surfaces with discriminant 3, 4 and 7

## TL;DR

This paper constructs and characterizes singular K3 surfaces with specific discriminants as double covers of the projective plane, exploring their moduli, automorphisms, and period mappings.

## Contribution

It introduces a new geometric construction for singular K3 surfaces with discriminants 3 and 4 and generalizes this approach to a moduli space of such surfaces.

## Key findings

- Constructed singular K3 surfaces as double covers over the projective plane.
- Identified conditions for automorphism groups of these K3 surfaces.
- Characterized surfaces via singular fiber configurations and period mappings.

## Abstract

We give construction of singular K3 surfaces with discriminant 3 and 4 as double coverings over the projective plane. Focusing on the similarities in their branching loci, we can generalize this construction, and obtain a three dimensional moduli space of certain K3 surfaces which admit infinite automorphism groups. Moreover, we show that these K3 surfaces are characterized in terms of the configuration of the singular fibres and a global section, and also in terms of periods.