On a conjecture of Fujino and Sato

TL;DR

This paper provides new, simplified proofs for results on non-projective $Q$-factorial toric varieties and extends the concepts to weak Mori Dream Spaces, emphasizing the role of secondary fans and wall-crossing.

Contribution

It offers short, conceptual proofs of Fujino--Sato's results without dimension restrictions and generalizes the framework to weak Mori Dream Spaces using secondary fan decompositions.

Findings

Short proofs for Fujino--Sato's results avoiding dimension restrictions

Extension of the framework to weak Mori Dream Spaces

Highlighting the role of secondary fans and wall-crossing

Abstract

We revisit results of Fujino--Sato on complete non-projective $\mathbb Q$-factorial toric varieties and their conjectural factorization by flips. We show that their main results admit short conceptual proofs, avoiding any restriction on the dimension and the Picard number, from the general theory of Cox rings and Mori Dream Spaces, once one organizes small $\mathbb Q$-factorial modifications via the GKZ (secondary fan) decomposition of the moving cone. Moreover, we extend this viewpoint beyond the toric case by proving an analogous statement for complete $\mathbb Q$-factorial weak Mori Dream Spaces: any non-projective such variety admits a divisor $D$ and a $D$-flip to a (projective) Mori Dream Space. Our approach highlights the role of chambers and wall-crossing in the secondary fan as a unifying framework for these constructions.