Symmetrizer group of a projective hypersurface

TL;DR

This paper introduces the symmetrizer group for non-conical projective hypersurfaces, linking algebraic group properties to geometric features like singularities and the Sebastiani-Thom property.

Contribution

It defines the symmetrizer group for hypersurfaces and explains its role in understanding polynomial equivalences and geometric properties, extending prior results.

Findings

The symmetrizer group characterizes polynomials with identical Jacobian ideals.

Diagonalizable part detects the Sebastiani-Thom property.

Unipotent part relates to hypersurface singularities.

Abstract

To each projective hypersurface which is not a cone, we associate an abelian linear algebraic group called the symmetrizer group of the corresponding symmetric form. This group describes the set of homogeneous polynomials with the same Jacobian ideal and gives a conceptual explanation of results by Ueda--Yoshinaga and Wang. In particular, the diagonalizable part of the symmetrizer group detects Sebastiani-Thom property of the hypersurface and its unipotent part is related to the singularity of the hypersurface.