Embedding non-projective Mori dream space

This paper is devoted to extend some Hu–Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, Q-factorial algebraic varieties with finitely generated class group and Cox ring, here called weak Mori dream spaces (wMDS), are considered. Conditions guaranteeing the existence of a neat embedding of a (completion of a) wMDS into a complete toric variety are studied, showing that, on the one hand, those which are complete and admitting low Picard number are always projective, hence Mori dream spaces in the sense of Hu–Keel. On the other hand, an example of a wMDS that does not admit any neat embedded sharp completion (i.e. Picard number preserving) into a complete toric variety is given, on the contrary of what Hu and Keel exhibited for a MDS. Moreover, termination of the Mori minimal model program for every divisor and a classification of rational contractions for a complete wMDS are studied, obtaining analogous conclusions as for a MDS. Finally, we give a characterization of wMDS arising from a small Q-factorial modification of a projective weak Q-Fano variety.