Optimal bounds for T-singularities in stable surfaces

Abstract: We explicitly bound T-singularities on normal projective surfaces $W$ with one singularity, and $K_W$ ample. This bound depends only on $K_W<sup>2$, 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.