Optimal bounds for many T-singularities in stable surfaces

TL;DR

This paper establishes optimal bounds for T-singularities on stable surfaces with multiple singularities, generalizing previous results and analyzing the combinatorial complexity involved.

Contribution

It provides the first comprehensive bounds for multiple T-singularities on stable surfaces and classifies configurations that influence these bounds.

Findings

Bounded T-singularities on non-rational surfaces with ample canonical class.

Classification of combinatorial configurations affecting bounds.

Explicit analysis of the case with two singularities.

Abstract

We effectively bound T-singularities on non-rational projective surfaces with an arbitrary amount of T-singularities and ample canonical class. This fully generalizes the previous work for the case of one singularity, and illustrates the vast increase in combinatorial complexity as the number of singularities grows. We find that certain combinatorial configurations lead to relatively high bounds. We classify all such configurations, and show that their non-existence gives a strong and optimal bound. As an application, we work out in detail the case of two singularities.