Quotients of products of curves, new surfaces with $p_g=0$ and their fundamental groups

TL;DR

This paper constructs 19 new complex surfaces of general type with p_g=0 and unknown fundamental groups, and classifies certain quotient surfaces, providing evidence for the Bloch conjecture.

Contribution

It introduces 19 new families of surfaces with p_g=0 and unknown fundamental groups, and classifies minimal resolutions of quotient surfaces with specific properties.

Findings

19 new families of surfaces with p_g=0 and unknown fundamental groups

Classification of quotient surfaces with q=p_g=0 and rational double points

Evidence supporting the Bloch conjecture

Abstract

The first main purpose of this paper is to contribute to the existing knowledge about the complex projective surfaces $S$ of general type with $p_g(S) = 0$ and their moduli spaces, constructing 19 new families of such surfaces with hitherto unknown fundamental groups. We also provide a table containing all the known such surfaces with K^2 <=7. Our second main purpose is to describe in greater generality the fundamental groups of smooth projective varieties which occur as the minimal resolutions of the quotient of a product of curves by the action of a finite group. We classify, in the two dimensional case, all the surfaces with q=p_g = 0 obtained as the minimal resolution of such a quotient, having rational double points as singularities. We show that all these surfaces give evidence to the Bloch conjecture.