Exceptional quotient singularities

Abstract

A singularity is said to be exceptional (in the sense of V. Shokurov), if for any log canonical boundary, there is at most one exceptional divisor of discrepancy -1. This notion is important for the inductive treatment of log canonical singularities. The exceptional singularities of dimension 2 are known: they belong to types E 6 , E 7 , E 8 after Brieskorn. In our previous paper, it was proved that the quotient singularity defined by Klein's simple group in its 3-dimensional representation is exceptional. In the present paper, the classification of all the three-dimensional exceptional quotient singularities is obtained. The main lemma states that the quotient of the affine 3-space by a finite group is exceptional if an only if the group has no semiinvariants of degree 3 or less. It is also proved that for any positive ε, there are only finitely many ε-log terminal exceptional 3-dimensional quotient singularities.