2026/03/31 by Jegyeong Jung, JeGyeong Jung
Mathematics · #math.AG
paper · pdf · doi:10.1016/j.jpaa.2026.108358
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.