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

• Autores: Ciro Ciliberto , Margarida Mendes Lopes , Rita Pardini
• Localización: Mathematical research letters, ISSN 1073-2780, Vol. 14, Nº 5, 2007, págs. 1081-1098
• Idioma: inglés
• DOI: 10.4310/mrl.2007.v14.n6.a14
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• Resumen

  • 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 \'etale 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