Geometry of obstructed equisingular families of projective hypersurfaces

TL;DR

This paper investigates the geometric structure of obstructed equisingular families of projective hypersurfaces, focusing on properties like smoothness, reducibility, and expected dimension, with detailed analysis for minimally obstructed cases and stability under singularity equivalence.

Contribution

It provides a detailed description of obstructed equisingular families, especially for quasihomogeneous singularities, and explores how stabilization affects their geometric properties and irreducibility.

Findings

Minimally obstructed families are irreducible.

Stable equivalence can preserve reduced components of expected dimension.

Deformations induced by linear systems can be complete under certain conditions.

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.