We study geometric properties of certain obstructed equisingular families of projective hypersurfaces with quasihomogeneous singularity with emphasis on smoothness, reducibility, being reduced, and having expected dimension. In the case of minimal obstructedness, we give a detailed description of such families corresponding to quasihomogeneous singularities. Next we study the behavior of these properties with respect to stable equivalence of singularities. We show that under certain conditions, stabilization of singularities ensures the existence of a reduced component of expected dimension. For minimally obstructed families the whole family becomes irreducible. As an application we show that if the equisingular family of a projective hypersurface H has a reduced component of expected dimension then the deformation of H induced by the equisingular family | H | is complete with respect to one-parameter deformations.
Bibliographical noteFunding Information:
The first-named author was supported by Hermann-Minkowski-Minerva Center for Geometry at the Tel-Aviv University, by the grant no. 465/04 from the Israel Science Foundation. The second-named author was supported by GIF grant no. 861/05 and ISF grant no. 1438/06.