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 $K_S^2=4$. Assume the bicanonical map $\phi$ of $S$ is a morphism of degree $4$ such that the image of $\phi$ is smooth. Then we prove that the surface $S$ is a Burniat surface with $K^2=4$ and of non nodal type.

The proof of Step 1 in Proof of Theorem 1.1 was changed. It has been accepted for publication in SCIENCE CHINA Mathematics