Small resolutions of double solids, branched over a 13-nodal quartic surface
The reviewer and the author have shown that compact 1-connected 4- manifolds with second Betti number \(b_ 2=3\) are related via the twistor construction to double solids mentioned in the title or to certain conic bundles (according to presence or absence of base points in the half- canonical linear system on the twistor space). The case \(b_ 2=3\) was studied by Y. S. Poon. The question, which of such double solids have (suitable) small resolutions, that are twistor spaces, seems to be rather subtile, due to the difficulties to identify the family of twistor lines. Taking into account reality conditions there is exhibited explicitly a 6-parameter family of branch loci, which give rise to double solids that are strong candidates of being twistor spaces. The dimension 6 is also in accordance with theoretical computations of the number of moduli of self-dual metrics. The results of the present paper give strong support of this conjecture. We mention the following: 1. Computation of the Euler number of small resolutions of double solids (in terms of the degree of the branch locus, the Milnor number of its singularities and the Euler number of the exceptional curves). 2. The middle integral homology vanishes if the singularities of the branch locus are not contained in a quadric. 3. Small resolutions of double solids with nodes are 1-connected. 4. Real 13-nodal quartics with only one real point are classified and small resolutions of the corresponding double solids have the cohomology ring and Chern classes of a twistor space over 3{\#}\({\mathbb{P}}^ 2({\mathbb{C}})\).
scientific article; zbMATH DE number 4146400
scientific article