Symmetrizer group of a projective hypersurface

TL;DR

This work provides a geometric description of the fibers of the Jacobian-image map $J$ for nondegenerate forms $F\in {\rm Sym}^d V^*$ by introducing the symmetrizer group $G_F$ and its Lie algebra ${\mathfrak g}_F$. Each fiber $J^{-1}(x)$ is a principal homogeneous space under a canonical abelian group $G_x\subset {\rm GL}(V)$, with a natural decomposition into diagonalizable and unipotent parts, $G_x/({\mathbb C}^{\times}\cdot {\rm Id}_V) = G_x^{\times} \times G_x^{+}$. The diagonalizable component detects Sebastiani–Thom type, while the unipotent component governs singularity data of the hypersurface $Z(F)$, yielding results that connect to the theorems of Ueda–Yoshinaga and Wang and to the structure of Sebastiani–Thom decompositions. Under certain finiteness conditions on multiplicity $d-2$ singularities, the unipotent part is tightly constrained, with $f^3=0$ for $f\in {\mathfrak g}_x^{+}$ and $\dim {\rm Im}(h)=1$ for nonzero $h\in {\mathfrak g}_x^{+}$. Overall, the paper provides a conceptual framework linking Jacobian ideals, symmetry groups of forms, and 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.

Theorems & Definitions (24)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Proposition 2.2
• proof