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

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.