Canad. Math. Bull. 49(2006), 592-608
Printed: Dec 2006
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.
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