A Dimension Bound for Symmetrizer Groups of Projective Hypersurfaces

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.