Catálogo de publicaciones - libros

We give examples of hypersurfaces of degree in (ℂ), whose singularities are versally deformed by the family () of all hypersurfaces of degree in (ℂ), and which are of minimal codimension with this property.

In the three cases (, ) = (2, 6), (3, 4) and (5, 3), such hypersurfaces necessarily have one-parameter symmetry. We list the possibilities. The singularities of these hypersurfaces are not all simple, and they are simultaneously versally deformed by ().

In less degenerate cases the examples we give are hypersurfaces with only simple singularities. The failure of versality can be expected to show itself in the geometry of (), either because the -constant stratum S containing the hypersurface is of codimension less than in (), or because is not smooth. We will see elsewhere that this is the case for the examples we consider here. In particular, the singularities of these hypersurfaces are topologically versally deformed by ().

In this paper we prove the following theorem: Let be the link space of a quasi-homogeneous hyperbolic ℚ-Gorenstein surface singularity. Then is diffeomorphic to a coset space , where is the 3-dimensional Lie group , while and are discrete subgroups of , the subgroup is co-compact and is cyclic. Conversely, if is diffeomorphic to a coset space as above, then is diffeomorphic to the link space of a quasi-homogeneous hyperbolic ℚ-Gorenstein singularity. We also prove the following characterisation of quasi-homogeneous ℚ-Gorenstein surface singularities: A quasi-homogeneous surface singularity is ℚ-Gorenstein of index if and only if for the corresponding automorphy factor (, Γ, ) some tensor power of the complex line bundle is Γ-equivariantly isomorphic to the th tensor power of the tangent bundle of the Riemannian surface .

In deformations of polynomial functions one may encounter “singularity exchange at infinity” when singular points disappear from the space and produce “virtual” singularities which have an influence on the topology of the limit polynomial. We find several rules of this exchange phenomenon, in which the total quantity of singularity turns out to be not conserved in general.

MacPherson’s Chern class transformation on complex algebraic varieties is a certain unique natural transformation from the constructible function covariant functor to the integral homology covariant functor, and it can be extended to a category of provarieties. In this paper, as further extensions of this we consider natural transformations among Mackey functors on provarieties and also on “indvarieties” and discuss some notions and examples related to these extensions.