Nondegeneracy for Quotient Varieties under Finite Group Actions

TL;DR

This paper investigates conditions under which a specific morphism related to quotient varieties under finite abelian group actions is nondegenerate, revealing it holds precisely when the group order is prime or four.

Contribution

It establishes a complete characterization of when the morphism is nondegenerate for abelian groups, extending understanding of quotient varieties in algebraic geometry.

Findings

The morphism is nondegenerate if and only if the group order is prime or four.

The result applies to all finite-dimensional representations of abelian groups.

Provides a criterion linking group order to geometric properties of quotient varieties.

Abstract

We prove that for an abelian group $G$ of order $n$ the morphism $ \varphi\colon \mathbf{P}(V^*)\longrightarrow \mathbf{P} ((\mathrm{sym}^n V^*)^G)$ defined by $\varphi([f]) = [\prod_{\sigma\in G} \sigma \cdot f ]$ is nondegenerate for every finite-dimensional representation $V$ of $G$ if and only if either $n$ is a prime number or $n=4$.