A Dimension Bound for Symmetrizer Groups of Projective Hypersurfaces

Abstract

Let $X$ be a projective hypersurface that is not a cone. The symmetrizer group of $X$ is an algebraic group that parametrizes hypersurfaces whose Jacobian ideal coincides with that of $X$. We prove that if the locus of points of multiplicity $d-1$ does not contain a line, then the nilpotent part of the Lie algebra of the symmetrizer group has dimension at most $2$, and consequently the symmetrizer group has dimension at most $\dim X+2$. Moreover, we show that if this locus has only finitely many lines, then the nilpotent part of the Lie algebra has dimension at most $4$, yielding the bound $\dim X+3$ for the symmetrizer group. To achieve this, we establish a connection between a class of singularities, called quasi-vertices, on $X$ with highly degenerate tangent cones and the unipotent part of its symmetrizer group.