Exceptional singular $\mathbb{Q}$-homology planes

Palka, Karol

Annales de l'Institut Fourier, Tome 61 (2011) no. 2, pp. 745-774.

On considère des surfaces $\mathbb{Q}$-acycliques singulières dont la partie lisse n’est pas de type général. On démontre que si les singularités sont topologiquement rationnelles, alors soit la partie lisse est réglée par ${ℂ}^1$ ou ${ℂ}^{*}$, soit la surface est l’une de deux surfaces exceptionnelles de dimension de Kodaira zéro. Pour les deux surfaces exceptionnelles, la dimension de Kodaira de la partie lisse est zéro, il n’y a qu’un seul point singulier et la singularité est de type $A_1$ ou $A_2$, respectivement.

We consider singular $\mathbb{Q}$-acyclic surfaces with smooth locus of non-general type. We prove that if the singularities are topologically rational then the smooth locus is ${ℂ}^1$- or ${ℂ}^{*}$-ruled or the surface is up to isomorphism one of two exceptional surfaces of Kodaira dimension zero. For both exceptional surfaces the Kodaira dimension of the smooth locus is zero and the singular locus consists of a unique point of type $A_1$ and $A_2$ respectively.

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