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# Group Actions, Cyclic Coverings and Families of K3-Surfaces

Published:2006-12-01
Printed: Dec 2006
• Alessandra Sarti
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## Abstract

In this paper we describe six pencils of $K3$-surfaces which have large Picard number ($\rho=19,20$) and each contains precisely five special fibers: four have A-D-E singularities and one is non-reduced. In particular, we characterize these surfaces as cyclic coverings of some $K3$-surfaces described in a recent paper by Barth and the author. In many cases, using 3-divisible sets, resp., 2-divisible sets, of rational curves and lattice theory, we describe explicitly the Picard lattices.
 MSC Classifications: 14J28 - $K3$ surfaces and Enriques surfaces 14L30 - Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 14E20 - Coverings [See also 14H30] 14C22 - Picard groups