Non-canonical embeddings and a canonical torsion-free covering for Mori Dream Spaces

The talk will be divided in two parts, approximately 45 minutes each; the first part is intended to be an introduction to the more technical second part; there will be a brief break between the two parts.
In the first part of this talk I will recall a standard construction of an (almost)-canonical toric embedding of a (non-necessarily projective) Mori Dream Space (MDS), starting from its Cox ring. Moreover I will recall some notation about the GKZ-decomposition of the pseudo-effective and the moving cone of a MDS. In the second part we will see some obstruction to extending Hu-Keel birational geometric results to the non-projective setup. Then I will show how recent results, jointly obtained with L. Terracini for $\Q$-factorial complete toric varieties, can be easily extended to a general MDS, producing a projective embedding of every MDS of Picard number less than or equal to 2 and a canonical covering space, unramified in codimension 1, of a given MDS, which is still a MDS admitting a torsion-free class group.
In principle, an application of such a covering construction is that the Cox ring of a MDS, which is in general graded over a class group with non trivial torsion part, could be described in terms of the Cox ring of its canonical covering, which is now graded over a torsion-free class group. This is a work in progress.