Smooth Three-Dimensional Canonical Thresholds

If $X$ is an algebraic variety with at most canonical singularities and $S$ is a $\mathbb{Q}$-Cartier hypersurface in $X$, then the canonical threshold of the pair $(X,S)$ is defined as the least upper bound of the reals $c$ for which the pair $(X,cS)$ is canonical. We show that the set of all possible canonical thresholds of the pairs $(X,S)$, where $X$ is smooth and three-dimensional, satisfies the ascending chain condition. We also derive a formula for the canonical threshold of the pair $(\mathbb{C}^3,S)$, where $S$ is a Brieskorn singularity.