Smooth 3-dimensional canonical thresholds

TL;DR

This paper investigates the set of canonical thresholds for pairs involving smooth 3-dimensional varieties, proving it satisfies the ascending chain condition and providing a formula for Brieskorn singularities.

Contribution

It establishes the ascending chain condition for canonical thresholds in smooth 3D varieties and derives a specific formula for Brieskorn singularities.

Findings

Set of canonical thresholds satisfies the ascending chain condition.

Derived a formula for canonical thresholds of Brieskorn singularities.

Extended understanding of singularity invariants in algebraic geometry.

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.