The Picard lattice of a family of double covers of P^2

A K3 surface is a smooth projective surface with trivial canonical divisor and trivial first cohomology group. A double cover of P^2 ramified along a smooth sextic curve is an example of K3 surface. To every K3 surface it is possible to associate a lattice, called Picard (or Néron-Severi) lattice. The Picard lattice encodes important information about the arithmetic and the geometry of the surface, and it is often not easy to compute. In this talk I consider a 1-dimensional family of K3 surfaces that are double covers of P^2 ramified along an almost diagonal sextic, and I am going to show how we computed the Picard lattice of the generic member of the family. This is a joint work with  Florian Bouyer, Edgar Costa, Chris Nicholls, Isabel Vogt, and Mckenzie West.