On tangential deformations of homogeneous polynomials

TL;DR

This paper investigates whether linear deformations of homogeneous polynomials are trivial, revealing that they are generally non-trivial, and characterizes certain smoothable hypersurfaces with isolated singularities.

Contribution

It demonstrates that associated linear deformations are not always trivial and provides a characterization of tangentially smoothable hypersurfaces with isolated singularities.

Findings

Linear deformations are generally non-trivial.

Characterization of tangentially smoothable hypersurfaces with isolated singularities.

A general divisor does not belong to an isotrivial linear system of positive dimension.

Abstract

The Jacobian ideal provides the set of infinitesimally trivial deformations for a homogeneous polynomial, or for the corresponding complex projective hypersurface. In this article, we investigate whether the associated linear deformation is indeed trivial, and show that the answer is no in a general situation. We also give a characterization of tangentially smoothable hypersurfaces with isolated singularities. Our results have applications in the local study of variations of projective hypersurfaces, complementing the global versions given by J. Carlson and P. Griffiths, R. Donagi and the author, and in the study of isotrivial linear systems on the projective space, showing that a general divisor does not belong to an isotrivial linear system of positive dimension.