We study algebraic K3 surfaces (defined over the complex number field) with a symplectic automorphism of prime order. In particular we consider the action of the automorphism on the second cohomology with integer coefficients (by a result of Nikulin this action is independent on the choice of the K3 surface). With the help of elliptic fibrations we determine the invariant sublattice and its perpendicular complement, and show that the latter coincides with the Coxeter-Todd lattice in the case of automorphism of order three.In the paper [Ni1] Nikulin studies finite abelian groups G acting symplectically (i.e. G |H 2,0 (X,C) = id |H 2,0 (X,C) ) on K3 surfaces (defined over C). One of his main result is that the action induced by G on the cohomology group H 2 (X, Z) is unique up to isometry. In [Ni1] all abelian finite groups of automorphisms of a K3 surface acting symplectically are classified. Later Mukai in [Mu] extends the study to the non-abelian case. Here we consider only abelian groups of prime order p which, by Nikulin, are isomorphic to Z/pZ for p = 2, 3, 5, 7.
Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice U 3 ⊕ E 8 (−1) 2 depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.
Abstract. The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a carefull analysis of the divisors contained in the Picard lattice we study their projective models, giving necessary and sufficient conditions to have an even set. Moreover we investigate their relation with K3 surfaces with a Nikulin involution. IntroductionIt is a classical problem in algebraic geometry to determine when a set of (−2)-rational curves on a surface is even. This means the following: let L 1 , . . . , L N be rational curves on a surface X then they form an even set if there is δ ∈ P ic(X) such thatThis is equivalent to the existence of a double cover of X branched on L 1 + . . . + L n . This problem is related to the study of even sets of nodes, in fact a set of nodes is even if the (−2)-rational curves in the minimal resolution are an even set. In particular the study of even sets on surfaces plays an important role in determining the maximal number of nodes a surface can have (cf. e.g. [Be], [JR]). Here we restrict our attention to K3 surfaces. In a famous paper of 1975 [N1] Nikulin shows that an even set of disjoint rational curves (resp. of distinct nodes) on a K3 surface contains 0, 8 or 16 rational curves (nodes). If the even set on the K3 surface X is made up by sixteen rational curves, the surface covering X is birational to a complex torus A and X is the Kummer surface of A. This situation is studied by Nikulin in [N1]. If the even set on X is made up by eight rational curves then the surface covering X is also a K3 surface. There are some more general results about even sets of curves not necessarily disjoint. More recently in [B1] Barth studies the case of even sets of rational curves on quartic surfaces (i.e. K3 surfaces in P 3 ) also in the case that the curves meet each other, he finds sets containing six or ten lines too. In the paper [B2] he discusses some particular even sets of disjoint lines and nodes on K3 surfaces whose projective models are a double cover of the plane, a quartic in P 3 or a double cover of the quadric P 1 × P 1 , and he gives necessary and sufficient conditions to have an even set. Our purpose is to study algebraic K3 surfaces admitting an even set of eight disjoint rational curves. We investigate their Picard lattices, moduli spaces and projective models. The minimal possible Picard number is nine, and we restrict our study to the surfaces with this Picard number. The techniques used by Barth in his article are mostly geometric, here we investigate first the Picard lattices of the K3 surfaces and the ampleness of certain divisors, then we study the projective models. We find again the cases studied by Barth and we discuss many new cases, with a special attention to complete intersections. We give also an explicit
In the first part of this paper we give a survey of classical results on Kummer surfaces with Picard number 17 from the point of view of lattice theory. We prove ampleness properties for certain divisors on Kummer surfaces and we use them to describe projective models of Kummer surfaces of (1, d)polarized Abelian surfaces for d = 1, 2, 3. As a consequence we prove that in these cases the Néron-Severi group can be generated by lines. In the second part of the paper we use Kummer surfaces to obtain results on K3 surfaces with a symplectic action of the group (Z/2Z) 4 . In particular we describe the possible Néron-Severi groups of the latter in the case that the Picard number is 16, which is the minimal possible. We describe also the Néron-Severi groups of the minimal resolution of the quotient surfaces which have 15 nodes. We extend certain classical results on Kummer surfaces to these families.
We observe that an interesting method to produce non-complete intersection subvarieties, the generalized complete intersections from L. Anderson and coworkers, can be understood and made explicit by using standard Cech cohomology machinery. We include a worked example of a generalized complete intersection Calabi-Yau threefold.
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