K3 Surfaces of Picard Rank One and Degree Two

Examples 1. A K3 surface of degree two is a double cover of P, ramified in a smooth sextic. K3 surfaces of degree four are smooth quartics in P. A K3 surface of degree six is a smooth complete intersection of a quadric and a cubic in P. And, finally, K3 surfaces of degree eight are smooth complete intersections of three quadrics in P. The Picard group of a K3 surface is isomorphic to Zn where n may range from 1 to 20. It is generally known that a generic K3 surface overC is of Picard rank one. This does, however, not yet imply that there exists a K3 surface over Q the geometric Picard rank of which is equal to one. The point is, genericity means that there are countably many exceptional subvarieties in moduli space. It seems that the first explicit examples of K3 surfaces of geometric Picard rank one have been constructed as late as in 2005 [vL]. All these examples are of degree four. The goal of this article is to provide explicit examples of K3 surfaces over Q which are of geometric Picard rank one and degree two.