We study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123-1158). 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.

Tiple Covers of K3 Surfaces / A. Garbagnati, M. Penegini. - In: NAGOYA MATHEMATICAL JOURNAL. - ISSN 0027-7630. - 248:(2022 Dec), pp. 939-979. [10.1017/nmj.2022.15]

Tiple Covers of K3 Surfaces

A. Garbagnati
Primo
;
M. Penegini
Ultimo
2022

Abstract

We study triple covers of K3 surfaces, following Miranda (1985, American Journal of Mathematics 107, 1123-1158). 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.
14E20; 14J28; 14J27; 14J29;
Settore MAT/03 - Geometria
dic-2022
giu-2022
https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/triple-covers-of-k3-surfaces/503DD5B819E0D2DB94AF8CA7156BBDC6
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/951717
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