Group Actions, Cyclic Coverings and Families of K3-surfaces

In this paper we describe six pencils of K3-surfaces which have large Picardnumber (15 ≤ ρ ≤ 20) and contain precisely five singular fibers: four have A-D-E singularities and one is non-reduced. In particular we describe these surfaces as cyclic coverings of the K3-surfaces of [BS]. In many cases using this description and latticetheory we are able to compute the exact Picard-number and to describe explicitly the Picard-lattices. 0. Introduction In the last years using different methods (toric geometry, mirror symmetry, etc.) have been constructed and studied many families of K3-surfaces with large Picard-Number and small number of singular fibers (see e.g. [D], [VY] and [Be]). In these notes using group actions and cyclic coverings we describe some new one. In [BS] the authors describe three pencils of K3-surfaces where the generic surface has Picard-number 19 and the pencils contain four singular fibers with singularities of A−D − E type and one non-reduced fiber. The families arise as minimal resolutions of quotients Xλ/G were G is a subgroup of SO(4) containing the Heisenberg group and {Xλ}λ∈P1 is a G-invariant pencil of surfaces in P3, the latter are described in [S1]. The groups G are the so-called bi-polyhedral groups: T×T , O ×O, I × I, where T , O and I denote the tetrahedral group, the octahedral group and the icosahedral group in SO(3) (we recall their precise definition in the first Section), the pencils X6 λ, X 8 λ, X 12 λ of T × T−, O × O−, resp. I × I−invariant surfaces, have degrees 6, 8, resp. 12. In these notes we consider some special normal subgroup H of T × T and O×O (I × I is simple) so that the minimal resolutions of the quotients are pencils of K3surfaces, these have then five singular fibers (one is non-reduced) and large Picard-number. In Section 2 we describe all the subgroupsH with this property. In the Sections 3, 4, 5 and 6 we describe six new one-dimensional families of K3-surfaces: the generic surface of two of the pencils has Picard-number ρ = 19 and there are four singular surfaces with ρ = 20 as in the case of the families X6 λ/T ×T or X 8 λ/O×O of [BS]. The other four pencils seem to have smaller Picard-number. We give in any case a lower bound for ρ and in each case except one we could identify surfaces with ρ = 20. The methods which we use in these Sections are essentially the same as in [BS]. In the Sections 7 and 9, we describe the K3surfaces as cyclic coverings of the K3-surfaces of [BS], more precisely let Yλ,T×T and Yλ,O×O denote the minimal resolutions of the quotients X6 λ/T ×T and X 8 λ/O×O, then taking two special 3-divisible classes of rational curves in the Neron-Severi group NS(Yλ,T×T ) and two special 2-divisible classes of rational curves in the Neron-Severi group NS(Yλ,O×O) one can construct cyclic coverings (two for each surface) which correspond (up to contract some (-1)-curves) to the minimal resolutions of the surfaces X6 λ/H and X 8 λ/H ′ for some normal subgroup H of T × T and H ′ of O × O. By doing cyclic coverings of the latter, Date: February 1, 2008. 1 2 ALESSANDRA SARTI one obtains more K3-surfaces. In Section 8 and 9 by using these descriptions and the results of [BS], Section 6, we compute explicitly the Picard-lattice of the K3-surfaces of two of the families, more precisely of those where the generic surface has ρ = 19 and of the four special surfaces with ρ = 20 contained in these families. We compute also the Picard-lattice of the surfaces with ρ = 20 in the other pencils. I would like to thank W. Barth for introducing me to cyclic coverings and for many useful discussions. 1. Notations and preliminaries There are two classical 2 : 1 coverings: ρ : SU(2) → SO(3) and σ : SU(2)× SU(2) → SO(4). Let Gi ⊂ SO(3), i = 1, 2 denote the polyhedral group T or O. We consider the binary group G̃i := ρ −1(Gi) ⊂ SU(2) and the σ-image: σ(G̃1 × G̃2) ⊂ SO(4), which by abuse of notation we denote with G1 ×G2 and we denote an element of SU(2)× SU(2) and its image in SO(4) by (p1, p2). These groups have been studied in [S1], there the group T ×T is called G6 and the group O × O is called G8. We will denote by X 6 λ = s6 + λq 3 and by X8 λ = s8 + λq 4 the pencils of T × T and of O ×O-invariant surfaces in P3, which are described in [S1], s6 denotes a T×T -invariant homogeneous polynomial of degree six and s8 denotes an O×O-invariant homogeneous polynomial of degree eight, q := x0+x 2 1+x 2 2+x 2 3 is the equation of the quadric P1 × P1 in P3. The base locus of the pencils X n λ are 2n lines (n = 6, 8) on the quadric, n in each ruling. These pencils contain exactly four nodal surfaces. We recall the value of the λi and the number of nodes on Xλi in the table below (cf. [S1]): n = 6 n = 8 λ1 λ2 λ3 λ4 λ1 λ2 λ3 λ4 −1 − 3 − 7 12 − 1 4 −1 − 3 4 − 9 16 − 5 9 12 48 48 12 24 72 144 96 We recall also the matrix: