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 fundamental group equal to Z_9 and two with fundamental group equal to Z_3^2. We also determine the base locus of the bicanonical system of these surfaces. It turns out that for the surfaces with fundamental group equal to Z_9 and for one of the families of surfaces with fundamental group equal to 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.