Étale coverings in codimension 1 with applications to Mori Dream Spaces

The present paper is devoted to developing relations between Galois étale coverings in codimension 1 and étale fundamental groups in codimension 1 of algebraic varieties, aimed to studying the topology of Mori dream spaces. In particular, the universal étale covering in codimension 1 of a non-degenerate toric variety and a canonical Galois étale covering in codimension 1 of a Mori dream space (MDS) are exhibited. Sufficient conditions for the latter being either still a MDS or the universal étale covering in codimension 1 are given. As an application, a canonical toric embedding of K3 universal coverings, of Enriques surfaces which are Mori dream, is described.