Numerical Campedelli surfaces with fundamental group of order 9

Abstract

We give explicit constructions of all the numerical Campedelli surfaces, i.e. the minimal surfaces of general type with $K^2=2$ and $p_g=0$, whose fundamental group has order 9. There are three families, one with ${\pi }^{\mathrm{alg}}={\mathbb{Z}}_9$ and two with ${\pi }^{\mathrm{alg}}={\mathbb{Z}}_3^2$.

We also determine the base locus of the bicanonical system of these surfaces. It turns out that for the surfaces with ${\pi }^{\mathrm{alg}}={\mathbb{Z}}_9$ and for one of the families of surfaces with ${\pi }^{\mathrm{alg}}={\mathbb{Z}}_3^2$ the base locus consists of two points. To our knowlegde, these are the only known examples of surfaces of general type with $K^2>1$ whose bicanonical system has base points.