The K3 surfaces are regular surfaces which admit an holomorphic symplectic form. We will consider finite order symplectic automorphisms: the minimal resolution of the quotient of a K3 surface by such an automorphism is another K3 surface. This creates a relation between families (a priori different) of K3 surfaces, which can be described in terms of lattice polarized K3 surfaces. We will review the classical results for the symplectic involutions and we describe the more recent generalizations for the order 3 automorphisms. Then, we will concentrate on specific subsets of K3 surfaces admitting symplectic involutions, characterized by the following property: the K3 surface which admits the symplectic automorphism and the one which is the desingularization of its quotient (by the symplectic automorphisms) lie in the same family of K3 surface. This property can be characterized in a lattice theoretically way, but it is not equivalent to be polarized with a prescribed lattice.

This talk deals with Cox rings of Calabi-Yau varieties X which are general anticanonical hypersurfaces in smooth toric Fano varieties Z. We present two complementary results, formulated in terms of primitive pairs of the anticanonical polytope of Z. The first gives combinatorial conditions ensuring that X is a Mori dream space and provides an explicit presentation of its Cox ring. The second shows that certain relations among primitive pairs force Bir(X) to be infinite, hence X is not a Mori dream space. As an application, we show that for Calabi-Yau hypersurfaces in dimensions two and three, either the Cox ring is finitely generated or the birational automorphism group is infinite. In the K3 case, where the Mori dream classification was already known via lattice theory, our approach gives a combinatorial interpretation together with explicit Cox ring presentations in the Mori dream cases. This is joint work with Antonio Laface and Luca Ugaglia.