Smooth 3-dimensional canonical thresholds

Abstract

If $X$ is an algebraic variety with at worst canonical singularities and $S$ is a $\Q$-Cartier hypersurface in $X$, the canonical threshold of the pair $(X,S)$ is the supremum of $c\in\R$ such that the pair $(X,cS)$ is canonical. We show that the set of all possible canonical thresholds of the pairs $(X,S)$, where $X$ is a germ of smooth 3-dimensional variety, satisfies the ascending chain condition. We also deduce a formula for the canonical threshold of $(\C^3,S)$, where S is a Brieskorn singularity.