Symmetrizer group of a projective hypersurface

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.