Embedding non-projective Mori Dream Spaces

TL;DR

This paper extends results on Mori dream spaces to a broader class called weak Mori dream spaces, exploring their embeddings, properties, and the Mori minimal model program, revealing both similarities and key differences from classical Mori dream spaces.

Contribution

It introduces weak Mori dream spaces, studies their embeddings into toric varieties, and analyzes their minimal model program and rational contractions, extending Mori dream space theory.

Findings

Complete weak MDS with low Picard number are projective.

Existence of weak MDS without neat sharp completions.

Termination of Mori MMP for all divisors on weak MDS.

Abstract

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 \emph{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 does not admitting any neat embedded \emph{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 (MMP) 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 a wMDS arising from a small $\Q$-factorial modification of a projective weak $\Q$-Fano variety.