The classification of surfaces with p_g = q = 0 isogenous to a product of curves

TL;DR

This paper classifies surfaces with geometric genus and irregularity zero that are covered by a product of curves, identifying 17 families through combinatorial and computational group theory methods.

Contribution

It provides a complete classification of such surfaces, explicitly describing 17 families beyond the trivial case, using combinatorial group theory and computational tools.

Findings

Identified 17 families of surfaces with p_g = q = 0 covered by a product of curves

Reduced the classification problem to a combinatorial group theory problem

Utilized MAGMA's group library to solve the classification problem

Abstract

We classify all the surfaces with p_g = q = 0 which admit an unramified covering which is isomorphic to a product of curves. Beyond the trivial case \PP^1 x \PP^1 we find 17 families which we explicitly describe. We reduce the problem to a combinatorial description of certain generating systems for finite groups which we solve using also MAGMA's library of groups of small order.