Burniat surfaces. III: Deformations of automorphisms and extended Burniat surfaces

Abstract

We continue our investigation of the connected components of the moduli space of surfaces of general type containing the Burniat surfaces, correcting a mistake in part II. We define the family of extended Burniat surfaces with $K_S^2=4$, resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with $K_S^2=4$, resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with $K_S^2=4$ form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with $K_S^2=3$ form an irreducible open set in the moduli space. Finally we point out an interesting pathology of the moduli space of surfaces of general type given together with a group of automorphisms $G$. In fact, we show that for the minimal model $S$ of a nodal Burniat surface $(G=(\mathbb{Z}/2\mathbb{Z}{)}^2)$ we have $\mathrm{Def}(S,G)\ne \mathrm{Def}(S)$, whereas for the canonical model $X$ it holds $\mathrm{Def}(X,G)=\mathrm{Def}(X)$. All deformations of $S$ have a $G$-action, but there are different deformation types for the pairs $(S,G)$ of the minimal models $S$ together with the $G$-action, while the pairs $(X,G)$ have a unique deformation type.