Burniat surfaces III: deformations of automorphisms and extended Burniat surfaces

TL;DR

This paper studies the deformation theory of Burniat surfaces, introduces extended Burniat surfaces, and explores their moduli space, revealing connected components, irreducibility, and automorphism group behaviors.

Contribution

It defines extended Burniat surfaces, proves their deformation relation to nodal Burniat surfaces, and analyzes the structure of their moduli space and automorphism group actions.

Findings

Extended Burniat surfaces form a deformation of nodal Burniat surfaces.

The union of extended and nodal Burniat surfaces forms a connected component in the moduli space.

Automorphism group actions differ between minimal models and canonical models, affecting deformation types.

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 = (\ZZ/2 \ZZ)^2) we have Def(S,G) \neq Def(S), whereas for the canonical model X it holds Def(X,G) = 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.