Elliptic fibrations on K3 surfaces: classifications, equations and specializations

The K3 surfaces are the unique surfaces which can admit more than one elliptic fibration. So, it is a natural problem to classify all the elliptic fibrations defined on the K3 surfaces in a specific family. There are two different approaches to this problem, one was introduced by Nishiyama and it is more lattice theoretic, the other was introduced by Oguisoand it is more geometric. In this talk we will present the second one, applying it to classify the elliptic fibrations on K3 surfaces which are generic in the families of K3 surfaces admitting a non-symplectic involution with prescribed fixed locus. Then, we will concentrate ourselves on K3 surfaces $X$ which are obtained considering a base change of order 2 on an rational elliptic surface $R$. We will show that a lot of the elliptic fibrations on $X$ are obtained by conic bundles on $R$. This allows us to find their equations by a very simple algorithm and to construct some easy specializations.