A Dimension Bound for Symmetrizer Groups of Projective Hypersurfaces

TL;DR

This paper establishes a bound on the dimension of the symmetrizer group of a non-conical projective hypersurface, linking singularity properties to group structure.

Contribution

It introduces a new bound on the symmetrizer group's dimension based on singularity and tangent cone degeneracy conditions.

Findings

The nilpotent part of the Lie algebra has dimension at most 2.

The symmetrizer group's dimension is at most the hypersurface dimension plus 2.

Certain singularities relate to the unipotent part of the symmetrizer group.

Abstract

Let $X$ be a projective hypersurface that is not a cone. The symmetrizer group of $X$ is an algebraic group parametrizing hypersurfaces whose Jacobian ideal coincides with that of $X$. We show that if the locus of points in $X$ with multiplicity $d-1$ does not contain a line, then the dimension of the nilpotent part of the Lie algebra associated to the symmetrizer group is at most $2$, and the dimension of the symmetrizer group is bounded by $\dim X + 2$. To achieve this, we investigate the relation between a class of singularities on $X$ with highly degenerate tangent cones and the unipotent part of its symmetrizer group.