Small resolutions of double solids, branched over a 13-nodal quartic surface
type: Item
instance of: scholarly article
title: Small resolutions of double solids, branched over a 13-nodal quartic surface
zbMATH Open document ID: 0699.32017
DOI: 10.1007/BF00128300
published in: Annals of Global Analysis and Geometry
publication date: 1989-01-01T00:00:00Z
review text: 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{\#}\({\mathbb{P}}^ 2({\mathbb{C}})\).
Mathematics Subject Classification ID: 32L25
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 57R20
zbMATH DE Number: 4146400
zbMATH Keywords: special 3-folds
zbMATH Keywords: selfduality
zbMATH Keywords: small resolutions
zbMATH Keywords: double solids
zbMATH Keywords: twistor spaces
author: Bernd Kreußler
reviewed by: Herbert Kurke
MaRDI profile type: Publication
cites work: Self-duality in four-dimensional Riemannian geometry
cites work: Double solids
cites work: Q5643279
cites work: The topology of four-dimensional manifolds
cites work: Q3924115
cites work: Compact four-dimensional self-dual EINSTEIN manifolds with positive scalar curvature
cites work: Q4143433
cites work: Linear field equations on self-dual spaces
cites work: Kählerian Twistor Spaces
cites work: Q5814388
cites work: The intersection of two quadrics in \({\mathbb{P}}_ 5({\mathbb{C}})\) as a twistor space
cites work: Q5521308
cites work: Q5270030
cites work: Q3962133
cites work: Singular Points of Complex Hypersurfaces. (AM-61)
cites work: Compact self-dual manifolds with positive scalar curvature
cites work: Q5522742
cites work: Q3804827
description: scientific article; zbMATH DE number 4146400