Geometry of obstructed equisingular families of projective hypersurfaces

Abstract

We study geometric properties of certain obstructed equisingular families of projective hypersurfaces with emphasis on smoothness, reducibility, being reduced, and having expected dimension. In the case of minimal obstructness, 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 linear system |H| is complete with respect to one-parameter deformations.