Optimal bounds for T-singularities in stable surfaces

TL;DR

This paper establishes explicit, optimal bounds on T-singularities for certain stable surfaces, depending only on their canonical divisor squared, and classifies surfaces that attain these bounds across different Kodaira dimensions.

Contribution

It provides the first explicit bounds for T-singularities on stable surfaces with one singularity, depending solely on $K_W^2$, and classifies surfaces reaching these bounds.

Findings

Bound on T-singularities depends only on $K_W^2$

Classification of surfaces attaining the bounds for each Kodaira dimension

Bounds are optimal when the surface is not rational

Abstract

We explicitly bound T-singularities on normal projective surfaces $W$ with one singularity, and $K_W$ ample. This bound depends only on $K_W^2$, and it is optimal when $W$ is not rational. We classify and realize surfaces attaining the bound for each Kodaira dimension of the minimal resolution of $W$. This answers effectiveness of bounds (see [Alexeev94], [Alexeev-Mori04], [Lee99]) for those surfaces.