On accumulation points of volumes of stable surfaces with one cyclic quotient singularity

TL;DR

This paper investigates the accumulation points of volumes of stable surfaces with one cyclic quotient singularity, establishing inequalities and conditions for boundedness, and introducing generalized T-singularities to understand volume accumulation.

Contribution

It provides optimal inequalities for stable surfaces with one cyclic quotient singularity and introduces generalized T-singularities to analyze volume accumulation points.

Findings

Identifies conditions for boundedness of stable surfaces with one cyclic quotient singularity.

Derives optimal inequalities to control volumes of such surfaces.

Shows how generalized T-singularities influence accumulation points of volumes.

Abstract

The set of volumes of stable surfaces does have accumulation points. In this paper, we study this phenomenon for surfaces with one cyclic quotient singularity, towards answering the question under which conditions we can still have boundedness. Effective bounds allow listing singularities that might appear on a stable surface after fixing its invariants. We find optimal inequalities for stable surfaces with one cyclic quotient singularity, which can be used to prove boundedness under certain conditions. We also introduce the notion of generalized T-singularity, which is a natural generalization of the well-known T-singularities. By using our inequalities, we show how the accumulation points of volumes of stable surfaces with one generalized T-singularity are formed.