Keum-Naie-Mendes Lopes-Pardini surfaces yield an irreducible component of the moduli space (Q2854220)

The Keum-Naie surfaces are minimal surfaces of general type with \(K^2=3\) and \(p_g=0\) that were constructed by \textit{J. H. Keum} [``Some new surfaces of general type with \(p_g=0\)'', Preprint (1988)] and rediscovered by \textit{D. Naie} [Math. Z. 215, No. 2, 269--280 (1994; Zbl 0791.14016)]. A larger family \(\mathcal X\), containing the above mentioned examples, was described later by \textit{M. Mendes Lopes} and \textit{R. Pardini} [Ann. Sci. Éc. Norm. Supér. (4) 37, No. 4, 507--531 (2004; Zbl 1078.14054)], where the question whether the family \(\mathcal X\) were an irreducible component of the moduli space of surfaces of general type was left open. NEWLINENEWLINEThis paper answers the question in the affirmative. The author gives an alternative description of a subfamily \(\mathcal Y\subset \mathcal X\) and uses it to prove, by means of an infinitesimal computation, that the base of the Kuranishi family of a general surface of \(\mathcal Y\) is smooth of dimension 6. Since \(\mathcal X\) also has dimension 6, this is enough to prove that the closure of the image of \(\mathcal X\) in the moduli space of surfaces of general type is an irreducible component.