Weakly--exceptional quotient singularities

A singularity is said to be weakly--exceptional if it has a unique purely log terminal blow up. In dimension $2$, V. Shokurov proved that weakly--exceptional quotient singularities are exactly those of types $D_{n}$, $E_{6}$, $E_{7}$, $E_{8}$. This paper classifies the weakly--exceptional quotient singularities in dimensions $3$ and $4$

G-birational rigidity of the projective plane

Given a surface S and a finite group G of automorphisms of S, consider the birational maps S⤏ S′ that commute with the action of G. This leads to the notion of a G-minimal variety. A natural question arises: for a fixed group G, is there a birational G-map between two different G-minimal surfaces? If no such map exists, the surface is said to be G-birationally rigid. This paper determines the G-rigidity of the projective plane for every finite subgroup G⊂ PGL 3(C). © 2018, Springer International Publishing AG, part of Springer Nature.11Nscopu