Singularities of Cox rings and varieties of Fano/Calabi-Yau type

A smooth hypersurface of a projective space is either Fano/Calabi-Yau/canonically polarized depending on its degree. These three cases precisely correspond to the cases where the singularity of the cone over the hypersurface is log terminal/(strictly) log canonical/worse than log canonical. I will give a generalization of this phenomena to Mori dream spaces, a class of varieties which have very nice properties from the point of view of the minimal model theory. In this generality, the cone over the hypersurface will be generalized to the spectrum of the Cox ring of the Mori dream space. This is a joint work with Yujiro Kawamata.