Involutions on surfaces with $p_g=q=1$

TL;DR

This paper investigates surfaces with involutions, deriving numerical restrictions, and classifies those with specific properties, including constructing new examples such as a minimal surface with unique invariants and a birational bicanonical map.

Contribution

It provides new formulas for surfaces with involutions and classifies surfaces with $p_g=q=1$ under certain conditions, including the first example with $K^2=7$ and a birational bicanonical map.

Findings

Derived numerical restrictions for surfaces with involutions.

Classified surfaces with $p_g=q=1$ and specific involution properties.

Constructed new examples, including a minimal surface with $K^2=7$ and birational bicanonical map.

Abstract

In this paper some numerical restrictions for surfaces with an involution are obtained. These formulas are used to study surfaces of general type $S$ with $p_g=q=1$ having an involution $i$ such that $S/i$ is a non-ruled surface and such that the bicanonical map of $S$ is not composed with $i.$ A complete list of possibilities is given and several new examples are constructed, as bidouble covers of surfaces. In particular the first example of a minimal surface of general type with $p_g=q=1$ and $K^2=7$ having birational bicanonical map is obtained.