Mixed quasi-\'etale quotients with arbitrary singularities

TL;DR

This paper introduces an algorithm to classify mixed quasi-étale surfaces with specific invariants, proving minimality for irregular cases and constructing new examples of surfaces of general type with particular properties.

Contribution

It develops a method to compute and classify mixed quasi-étale surfaces with given invariants, including the first example of a minimal surface with p_g=q=1 and high Albanese fiber genus.

Findings

All irregular mixed quasi-étale surfaces of general type are minimal.

Classified all irregular mixed quasi-étale surfaces with genus equal to irregularity.

Constructed new examples of surfaces with al=1.

Abstract

A mixed quasi-\'etale quotient is the quotient of the product of a curve of genus at least 2 with itself by the action of a group which exchanges the two factors and acts freely out of a finite subset. A mixed quasi-\'etale surface is the minimal resolution of its singularities. We produce an algorithm computing all mixed quasi-\'etale surfaces with given geometric genus, irregularity, and self-intersection of the canonical class. We prove that all irregular mixed quasi-\'etale surfaces of general type are minimal. As application, we classify all irregular mixed quasi \'etale surfaces of general type with genus equal to the irregularity, and all the regular ones with K^2>0, thus constructing new examples of surfaces of general type with \chi=1. We mention the first example of a minimal surface of general type with p_g=q=1 and Albanese fibre of genus bigger than K^2.