Nondegeneracy for Quotient Varieties under Finite Group Actions

Abstract

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

8 pages