Surfaces with $K^2 \lt 3\chi$ and finite fundamental group

In this paper we continue the study of $\pionealg(S)$ for minimal surfaces of general type $S$ satisfying $K_S^2 <3\chi(S)$. We show that, if $K_S^2= 3\chi(S)-1$ and $|\pionealg(S)|= 8$, then $S$ is a Campedelli surface. In view of the results of \cite{3chi} and \cite{3chi-2}, this implies that the fundamental group of a surface with $K^2<3\chi$ that has no irregular étale cover has order at most 9, and if it has order 8 or 9, then $S$ is a Campedelli surface. To obtain this result we establish some classification results for minimal surfaces of general type such that $K^2=3p_g-5$ and such that the canonical map is a birational morphism. We also study rational surfaces with a $\Z_2^3$-action.

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Published 1 January 2007