#### DOCUMENTA MATHEMATICA, Vol. 18 (2013), 1089-1136

I. Bauer and F. Catanese

Burniat Surfaces III: Deformations of Automorphisms and Extended Burniat Surfaces

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.

2010 Mathematics Subject Classification: 14J29, 14J25, 14J10, 14D22, 14H30, 32G05

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