Rational configurations in \(K3\) surfaces and simply-connected \(p_g=1\) surfaces with \(K^2=1,2,3,4,5,6,7,8,9\)
One by now classical method for constructing simply connected surfaces of general type with given (small) invariants consists in constructing a normal surface \(Y\) with rational singularities with the given invariants and show that it admits a \(\mathbb Q\)-Gorenstein smoothing. This technique has been applied mainly to the case \(p_g=0\), in which the surface \(Y\) is obtained by blowing down configurations of rational curves on a rational or Enriques surface. To apply the same technique to the case \(p_g=1\) one can construct the surface \(Y\) by blowing down configurations of rational curves on a \(K3\) surface, as in [\textit{H. Park} et al., J. Korean Math. Soc. 50, No. 3, 493--507 (2013; Zbl 1293.14005)] where examples are constructed for \(K^2=3,4,5,6,8\). Here the authors make a theoretical analysis of the above situation, obtaining numerical restrictions on the singularities of \(Y\) and construct examples for all values of \(K^2=1, \dots 9\). It is worth noticing that no example of simply connected surface with \(p_g=1\) and \(K^2=7,9\) was previously known and that for these values of \(K^2\) the construction of the singular surface \(Y\) is highly non trivial.
scientific article; zbMATH DE number 7619455
scientific article