Surfaces with $p_g = 0$, $K^2 = 5$ and bicanonical maps of degree 4

TL;DR

This paper classifies minimal surfaces of general type with specific invariants, focusing on their bicanonical maps of degree 4 and describing the geometric properties of their images.

Contribution

It characterizes surfaces with $p_g=0$, $K^2=5$, and bicanonical map degree 4, distinguishing cases based on the smoothness of the bicanonical image and describing their structure.

Findings

If the bicanonical image is smooth, the surface is a Burniat surface.

If the image is singular, it has at most one $(-2)$-curve.

The paper provides a detailed description of the singular case.

Abstract

Let $S$ be a minimal surface of general type with $p_g(S) = 0, K_S^2 = 5$ and bicanonical map of degree 4. Denote by $\Sigma$ the bicanonical image. If $\Sigma$ is smooth, then $S$ is a Burniat surface; and if $\Sigma$ is singular, then we reduced $\Sigma$ to one case and described it, furthermore $S$ has at most one $(-2)$-curve.