Mathematical Research Letters

Volume 14 (2007)

Number 6

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

Pages: 1081 – 1098

DOI: http://dx.doi.org/10.4310/MRL.2007.v14.n6.a14

Authors

Ciro Ciliberto (Università degli Studi di Roma Tor Vergatà)

Margarida Mendes Lopes (Universidade Técnica de Lisboa)

Rita Pardini (Università di Pisa)

Abstract

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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