Some new surfaces with $p_g = q = 0$

TL;DR

This paper investigates the classification of certain minimal surfaces of general type with specific invariants, focusing on those constructed via Beauville's method, and provides a complete classification for cases with abelian symmetry groups.

Contribution

It offers a complete classification of minimal surfaces with $p_g=0, K^2=8$ constructed by Beauville's method when the symmetry group is abelian, including descriptions of their moduli spaces and homology groups.

Findings

Classified 5 cases with abelian groups G

Described moduli spaces and homology groups for these surfaces

Presented 5 examples with non-abelian groups, including 3 previously known

Abstract

Motivated by a question by D. Mumford : can a computer classify all surfaces with $p_g = 0$ ? we try to show the complexity of the problem. We restrict it to the classification of the minimal surfaces of general type with $p_g = 0, K^2 = 8$ which are constructed by the Beauville construction, namely, which are quotients of a product of curves by the free action of a finite group G acting separately on each component. We think that man and computer will soon solve this classification problem. In the paper we classify completely the 5 cases where the group G is abelian. For these surfaces, we describe the moduli space (sometimes it is just a real point), and the first homology group. We describe also 5 examples where the group G is non abelian. Three of the latter examples had been previously described by R. Pardini.