Point sets with nontrivial automorphism group

Abstract

Let $\mathbb{C}$ be the field of complex numbers and n and k natural numbers satisfying $k\ge 2$ and $n ⩾ k+3.$ Let ${\mathbb{P}}_n^k\left(\mathbb{C}\right)$ be the subset of nth symmetric product ${({\mathbb{P}}^k(\mathbb{C}))}^{(n)}$ of ${\mathbb{P}}^k(\mathbb{C})$ with itself consisting of n-point sets in general position and let $M_n^k(\mathbb{C})={\mathbb{P}}_n^k(\mathbb{C})/{\text{PGL}}_{k+1}(\mathbb{C}).$ We describe the dimension of the points in $M_n^k(\mathbb{C})$ which correspond to n-point sets whose automorphism group contains a subgroup of prime order. From this result, we prove that the points in $M_n^k(\mathbb{C})$ which correspond to n-point sets with nontrivial automorphism group form a proper subset $M_n^k(\mathbb{C}).$

Keywords:

• Point sets
• projective linear group
• cyclic subgroups
• dimension
• projective representations
• veronese emebedding

2010 Mathematical Subject Classification:

• Primary: 51N30
• 51N15
• Secondary: 20C25
• 51A05