Equivariant completions of toric contraction morphisms

We treat equivariant completions of toric contraction morphisms as an application of the toric Mori theory. For this purpose, we generalize the toric Mori theory for non-$\mathbb{Q}$-factorial toric varieties. So, our theory seems to be quite different from Reid’s original combinatorial toric Mori theory. We also explain various examples of non-$\mathbb{Q}$-factorial contractions, which imply that the $\mathbb{Q}$-factoriality plays an important role in the Minimal Model Program. Thus, this paper completes the foundations of the toric Mori theory and show us a new aspect of the Minimal Model Program.