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

I. Bauer ; F. Catanese

Documenta mathematica, Tome 18 (2013), pp. 1089-1136

Résumé

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 KS2​=4, resp. 3, and prove that they are a deformation of the family of nodal Burniat surfaces with KS2​=4, resp. 3. We show that the extended Burniat surfaces together with the nodal Burniat surfaces with KS2​=4 form a connected component of the moduli space. We prove that the extended Burniat surfaces together with the nodal Burniat surfaces with KS2​=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=(Z/2Z)2) we have Def(S,G)=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.

Détail

DOI : 10.4171/dm/422

Classification : 14D22, 14H30, 14J10, 14J25, 14J29, 32G05