Involutions on surfaces with $p_g=q=0$ and $K^2=3$

TL;DR

This paper investigates surfaces of general type with specific invariants, analyzing involutions and their impact on the bicanonical map, revealing conditions under which the quotient surface is birational to an Enriques surface or has Kodaira dimension 1.

Contribution

It classifies surfaces with $p_g=0$, $K^2=3$, and involutions where the bicanonical map isn't composed with the involution, providing explicit examples and descriptions of the quotient surfaces.

Findings

If the quotient surface is not rational, it is either birational to an Enriques surface or has Kodaira dimension 1.

Explicit example of such a surface with a hyperelliptic fibration of genus 3.

The bicanonical map in the example is degree 2 onto a rational surface.

Abstract

We study surfaces of general type $S$ with $p_g=0$ and $K^2=3$ having an involution $i$ such that the bicanonical map of $S$ is not composed with $i$. It is shown that, if $S/i$ is not rational, then $S/i$ is birational to an Enriques surface or it has Kodaira dimension $1$ and the possibilities for the ramification divisor of the covering map $S\rightarrow S/i$ are described. We also show that these two cases do occur, providing an example. In this example $S$ has a hyperelliptic fibration of genus $3$ and the bicanonical map of $S$ is of degree $2$ onto a rational surface.