2010/12/16 by Ingrid Bauer, Fabrizio Catanese, Bauer, Ingrid +1
Mathematics · #14D22 #14H30 #14J10 #14J25 #14J29 #32G05 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14D22 #msc:14H30 #msc:14J10 #msc:14J25 #msc:14J29 #msc:32G05
paper · pdf · doi:10.48550/arxiv.1012.3770
40 pages plus an appendix of 4 pages
arxiv created 2010/12/16 · arxiv updated 2010/12/20
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 = (\ZZ/2 \ZZ)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.