A characterization of Burniat surfaces with $K^{2}=4$ and of non nodal type

TL;DR

This paper characterizes certain Burniat surfaces with specific invariants, showing that minimal surfaces with particular properties are indeed Burniat surfaces of non nodal type.

Contribution

It proves that minimal surfaces with $p_g=0$, $K^2=4$, and a degree 4 smooth bicanonical map are exactly Burniat surfaces of non nodal type.

Findings

Surfaces with the given properties are Burniat surfaces.

The bicanonical map degree and smoothness characterize the surface type.

Classifies a family of surfaces within the general type category.

Abstract

Let $S$ be a minimal surface of general type with $p_{g}(S)=0$ and $K^{2}_{S}=4$. Assume the bicanonical map $\varphi$ of $S$ is a morphism of degree $4$ such that the image of $\varphi$ is smooth. Then we prove that the surface $S$ is a Burniat surface with $K^{2}=4$ and of non nodal type.