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

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