Nondegeneracy for Quotient Varieties under Finite Group Actions

Let V be a finite-dimensional representation of a finite group G of order n over a field k. Then by a classical theorem of Hermann Weyl [5] on polarizations one obtains a nice generating set for the invariant ring (SymV ) consisting of er(f), r = 1, 2, . . . , n, f ∈ V ∗ where er(f) denotes the r-th elementary symmetric polynomial in σ.f, σ ∈ G. Also, for any point x ∈ P(V ) there is a G-invariant section of the line bundle O(1) of the form s = en(f) = ∏ σ∈G σ.f, f ∈ V ∗ such that s(x) 6= 0. So, the line bundle O(1) descends to the quotient G\P(V ). (See [3], [4]). This leads one to ask the natural question whether the set { ∏