In this short note, we show that a construction by Ottem provides an example of a rationally connected variety that is not birationally equivalent to a Mori dream space with terminal singularities.

A singularity is said to be weakly exceptional if it has a unique purely log terminal blow-up. This is a natural generalization of the surface singularities of types Dn, E6, E7 and E8. Since this idea was introduced, quotient singularities of this type have been classified in dimensions up to at most 4. This note extends that classification to dimension 5.

The Nagata automorphism is a kind of complicated automorphism on the affine 3-space . For a long time, it remained unknown whether or not the Nagata automorphism is tame until Shestakov and Umirbaev at last proved that it is not tame in 2004, by purely algebraic methods (e.g. Poisson algebra). In this paper, we consider a certain necessary condition for a given automorphism on to be tame from the point of view of the Sarkisov program established by Corti. Furthermore, by using it, we shall give a new algebro-geometric proof of the non-tameness of the Nagata automorphism.