L\^e's polyhedron for line singularities

Abstract

We study the topology of a line singularity, which is a complex hypersurface with non-isolated singularity given by a complex line. We describe the degeneration of its Milnor fibre to the singular hypersurface by means of a pair of polyhedra, one in the Milnor fibre and other in the singular fibre, which are deformation retracts of the corresponding fibres; and a continuous map taking the Milnor fibre to the singular fibre and the first polyhedron to the second one, which restrict to a homeomorphism outside the polyhedra. In the same sense, we also study the topology of a complex isolated singularity hypersurface under a non-local viewpoint.