This paper is devoted to extend some Hu–Keel results on Mori dream spaces (MDS) beyond the projective setup. Namely, $$\mathbb {Q}$$ 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 $$\mathbb {Q}$$ Q -factorial modification of a projective weak $$\mathbb {Q}$$ Q -Fano variety.

中文翻译：

---

嵌入非投射 Mori 梦想空间

本文致力于扩展 Mori 梦想空间 (MDS) 上的一些 Hu-Keel 结果超出投影设置。即，$$\mathbb {Q}$$ Q -阶乘代数变体具有有限生成的类群和 Cox 环，这里称为弱 Mori 梦空间 (wMDS)。研究了保证 (a) wMDS 的整齐嵌入到一个完整的复曲面变体中的条件，表明一方面，那些完整的和承认低皮卡德数的总是投射的，因此 Mori 梦想空间胡-龙骨的感觉。另一方面，给出了一个 wMDS 的例子，它不允许任何整齐的嵌入尖锐完成（即保留 Picard 数）到完整的复曲面变体中，与 Hu 和 Keel 为 MDS 展示的相反。而且，研究了每个除数的 Mori 最小模型程序的终止和完整 wMDS 的合理收缩分类，获得与 MDS 类似的结论。最后，我们给出了由投影弱 $$\mathbb {Q}$$ Q -Fano 变体的小 $$\mathbb {Q}$$ Q -因子修改产生的 wMDS 的表征。