Involutions on numerical Campedelli surfaces

Abstract

Numerical Campedelli surfaces are minimal surfaces of general type with vanishing geometric genus and canonical divisor with self-intersection 2. Although they have been studied by several authors,their complete classification is not known.

In this paper we classify numerical Campedelli surfaces with an involution, i.e., an automorphism of order 2. First we show that an involution on a numerical Campedelli surface $S$ has either four or six isolated fixed points, and the bicanonical map of $S$ is composed with the involution if and only if the involution has six isolated fixed points. Then we study in detail each of the possible cases, describing also several examples.