Invariants of D(q,p) singularities

Abstract

The basic examples of functions defining non-isolated hypersurface singularities are the A(d) singularities and the D(q,p) singularities. The A(d) singularities, up to analytic equivalence, are the product of a Morse function and the zero map, while the simplest D(q,p) singularity is the Whitney umbrella. These are the basic examples, because they correspond to stable germs of functions in the study of germs of functions with non-isolated singularities. Given a germ of a function which defines a non-isolated hypersurface singularity at the origin, which in the appropriate sense, has finite codimension in the set of such germs, the singularity type of such germs away from the origin is A(d) or D(q,p). In this note we calculate the homotopy type of the Milnor fiber of germs of type D(q,p), as well as their Lê numbers. The calculation of the Lê numbers involves the use of an incidence variety which may be useful for studying germs of finite codimension. The calculation shows that the set of symmetric matrices of kernel rank greater than or equal to 1 is an example of a hypersurface singularity with a Whitney stratification (given by the rank of the matrices) in which only one singular stratum gives a component of top dimension of the singular set of the conormal.