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A Dimension Bound for Symmetrizer Groups of Projective Hypersurfaces

Jegyeong Jung

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.

A Dimension Bound for Symmetrizer Groups of Projective Hypersurfaces

Abstract

Let be a projective hypersurface that is not a cone. The symmetrizer group of is an algebraic group parametrizing hypersurfaces whose Jacobian ideal coincides with that of . We show that if the locus of points in with multiplicity does not contain a line, then the dimension of the nilpotent part of the Lie algebra associated to the symmetrizer group is at most , and the dimension of the symmetrizer group is bounded by . To achieve this, we investigate the relation between a class of singularities on with highly degenerate tangent cones and the unipotent part of its symmetrizer group.
Paper Structure (6 sections, 27 theorems, 67 equations)

This paper contains 6 sections, 27 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hessian of Hypersurfaces
  4. Quasi-vertices
  5. Proof of Theorem \ref{['t_main']}
  6. Corollaries

Key Result

Theorem 1.1

Let $F\in \operatorname{Sym}^d_oV^*$. Then there is a connected abelian algebraic subgroup $G_F\subset \operatorname{GL}(V)$ canonically associated to $J(F)$, which contains $\mathbb{C}^\times\cdot \operatorname{Id}_V$, such that the fiber $J^{-1}(J(F))$ is a principal homogeneous space of the group

Theorems & Definitions (59)

  • Theorem 1.1: Hwang, Theorem 1.5
  • Theorem 1.2: Hwang; see also Mammana and Wang
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Definition 2.2: Hwang
  • Proposition 2.3: Hwang, Proposition 2.2
  • Proposition 2.4: Hwang, Proposition 2.3
  • Proposition 2.5
  • proof
  • ...and 49 more