Mixed quasi-étale surfaces, new surfaces of general type with $p_g=0$ and their fundamental group

Abstract

We call a projective surface $X$ mixed quasi-étale quotient if there exists a curve $C$ of genus $g(C)\ge 2$ and a finite group $G$ that acts on $C\times C$ exchanging the factors such that $X=(C\times C)/G$ and the map $C\times C\to X$ has finite branch locus. The minimal resolution of its singularities is called mixed quasi-étale surface. We study the mixed quasi-étale surfaces under the assumption that $(C\times C)/G^0$ has only nodes as singularities, where $G^0◃G$ is the index two subgroup of the elements that do not exchange the factors. We classify the minimal regular surfaces with $p_g=0$ whose canonical model is a mixed quasi-étale quotient as above. All these surfaces are of general type and as an important byproduct, we provide an example of a numerical Campedelli surface with topological fundamental group ${\mathbb{Z}}_4$, and we realize 2 new topological types of surfaces of general type. Three of the families we construct are $\mathbb{Q}$-homology projective planes.

18 pages, 3 tables, v2: change title, exposition improved; v3: minor corrections, final version to be published in Collectanea Mathematica

References in corpus (3)

• Some new surfaces with $p_g=q=0$
• On surfaces of general type with p_g=q=1 isogenous to a product of curves
• Involutions on a surface of general type with $p_g=q=0$, $K^2=7$

Cited by in corpus (1)

• Surfaces with p_g = 0: Constructions and Moduli spaces, Burniat surfaces and deformations of automorphisms