Classification of threefold canonical thresholds

TL;DR

This paper establishes that the set of smooth threefold canonical thresholds matches the 2-dimensional hypersurface log canonical thresholds and fully classifies the set of all threefold canonical thresholds, revealing their precise structure.

Contribution

It proves the equality of smooth threefold canonical thresholds with 2D hypersurface log canonical thresholds and classifies all threefold canonical thresholds explicitly.

Findings

Smooth threefold canonical thresholds equal 2D hypersurface log canonical thresholds.

The set of threefold canonical thresholds is explicitly classified.

The set includes 0, 4/5, and the smooth threefold canonical thresholds.

Abstract

We show that the set $\mathcal{T}_{3, \mathrm{sm}}^{\mathrm{can}}$ of smooth threefold canonical thresholds coincides with $\mathcal{T}_{2, \mathrm{sm}}^{\mathrm{lc}}=\mathcal{HT}_{2}$, where $\mathcal{HT}_{2}$ is the $2$-dimensional hypersurface log canonical thresholds characterized by Kuwata \cite{K99a, K99b}. We classify the set $\mathcal{T}_{3}^{\mathrm{can}}$ of threefold canonical thresholds. More precisely, we prove $\mathcal{T}_{3}^{\mathrm{can}}= \{0\} \cup \{\frac{4}{5}\} \cup \mathcal{T}_{3, \mathrm{sm}}^{\mathrm{can}}$.