Smooth double covers of K3 surfaces

TL;DR

This paper classifies the topological invariants and geometric properties of surfaces obtained as smooth double covers of K3 surfaces, providing explicit examples and analyzing their Hodge structures.

Contribution

It offers a classification of branch loci for smooth double covers of K3 surfaces and constructs explicit examples with various Kodaira dimensions.

Findings

Classified topological invariants of branch loci.

Constructed examples with Kodaira dimension 1 and 2.

Analyzed Hodge structure variations of the resulting surfaces.

Abstract

In this paper we classify the topological invariants of the possible branch loci of a smooth double cover $f:X\rightarrow Y$ of a K3 surface $Y$. We describe some geometric properties of $X$ which depend on the properties of the branch locus. We give explicit examples of surfaces $X$ with Kodaira dimension 1 and 2 obtained as double cover of K3 surfaces and we describe some of them as bidouble cover of rational surfaces. Then, we classify the K3 surfaces which admit smooth double covers $X$ satisfying certain conditions; under these conditions the surface $X$ is of general type, $h^{1,0}(X)=0$ and $h^{2,0}(X)=2$. We discuss the variation of the Hodge structure of $H^2(X,\mathbb{Z})$ for some of these surfaces $X$.