Interpretation of the deformation space of a determinantal Barlow surface via smoothings (Q2781241)

In her famous paper ``A simply connected surface of general type with \(p_g=0\)'' [Invent. Math. 79, 293-301 (1985; Zbl 0561.14015)], \textit{R. Barlow} constructed a \(2\)-dimensional family of simply connected surfaces of general type with \(p_g=0\) and \(K^2=1\), which at the moment are still the only known examples of surfaces of general type with \(p_g=0\) and trivial fundamental group. The deformation space of these surfaces was shown to be smooth of dimension 8 by the author [\textit{Y. Lee}, Trans. Am. Math. Soc. 353, 893-905 (2001; Zbl 0967.14027)] and by \textit{F. Catanese} and \textit{C. Le Brun} [Math. Res. Lett. 4, 843-854 (1997; Zbl 0907.53035)], independently. NEWLINENEWLINENEWLINEHere the author gives a geometrical interpretation of this deformation space, by showing that 4 of the 6 ``missing'' deformation directions correspond to independent smoothings of the nodes of the Barlow surface, while the remaining 2 ones give global smoothings.