Javier Fernandez de Bobadilla

Answers to some equisingularity questions

We give examples of families of hypersurface singularities with constant Lê numbers, constant Milnor fibration and non-constant topological type, answering negatively a question of D. Massey. On the other hand we prove that the constancy of the Lê numbers implies that the homotopy type of the link is constant along the family. As an application we give an example of a flat family of projective reduced and irreducible hypersurfaces having the same homotopy type but different topological type. Another application is an example of a Whitney-trivial family of isolated singularities such that the topological type of the projectivised tangent cone is not constant in the family. The last example answers negatively a question of O. Zariski.

A new geometric proof of Jung's theorem on factorisation of automorphisms of C^2

Building up on the classical theory of algebraic surfaces and their birational transformations we prove Jungs theorem on factorisation of automorphisms of C 2 reducing it to a simple combinatorial argument.

A reformulation of Le's conjecture

Abstract We give a new proof of Les conjecture on surface germs in ℂ 3 having as link a topological sphere for the case of surface singularities containing a smooth curve. Our proof leads to a reformulation of the general case of the conjecture into a problem of plane curve singularities and their relative polar curves.

Topology of hypersurface singularities with 3-dimensional critical set

In this paper, we prove that the Milnor fibre of a singularity over an i.c.i.s. of dimension 3 has the homotopy type of a bouquet of spheres, provided that the function that defines the singularity has finite extended codimension with respect to the ideal that defines the i.c.i.s.

Topological finite-determinacy of functions with non-isolated singularities

Abstract We introduce the concept of topological finite-determinacy for germs of analytic functions within a fixed ideal I, which provides a notion of topological finite-determinacy of functions with non-isolated singularities. We prove the following statement which generalizes classical results of Thom and Varchenko: let A be the complement in the ideal I of the space of germs whose topological type remains unchanged under a deformation within the ideal that only modifies sufficiently large order terms of the Taylor expansion. Then A has infinite codimension in I in a suitable sense. We also prove the existence of generic topological types of families of germs of I parametrized by an irreducible analytic set.