A ℚ-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for ℚ-homology planes with smooth locus of non-general type. We show, in particular, that if a ℚ-homology plane contains a non-quotient singularity, then it is a quotient of an affine cone over a projective curve by an action of a finite group respecting the set of lines through the vertex. In particular, it is contractible, has negative Kodaira dimension and only one singular point. We describe minimal normal completions of such planes.
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The author was supported by Polish Grant NCN N N201 608640.
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Palka, K. Classification of singular ℚ-homology planes. I. Structure and singularities. Isr. J. Math. 195, 37–69 (2013). https://doi.org/10.1007/s11856-012-0123-z