Triple covers of K3 surfaces

TL;DR

This paper investigates triple covers of K3 surfaces, classifying Galois and non-Galois cases, analyzing their geometric properties, and providing explicit constructions across various Kodaira dimensions.

Contribution

It offers a comprehensive classification of Galois triple covers of K3 surfaces, introduces criteria for non-Galois covers, and constructs examples with diverse geometric features.

Findings

Classified Galois triple covers and computed their invariants.

Constructed non-Galois triple covers using specific vector bundles.

Provided examples across all admissible Kodaira dimensions.

Abstract

We study triple covers of K3 surfaces, following Miranda's theory of triple covers. We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vector bundles on a K3 surface $S$ which determine a non-Galois triple cover of $S$. The examples presented are in any admissible Kodaira dimension and in particular we provide the constructions of irregular covers of K3 surfaces and of surfaces with geometrical genus equal to 2 whose transcendental Hodge structure splits in the sum of two Hodge structures of K3 type.