# Faithful action of stabiliser on group of automorphisms of a hypersurface

Let $$f\in\mathbb{C}[x_{1},\ldots,x_{n}]$$, and $$G_{f}\le\mathrm{GL}_{n}(\mathbb{C})$$ be the group of all matrices under which $$f$$ is invariant, i.e. the stabiliser of $$f$$ in $$\mathrm{GL}_{n}(\mathbb{C})$$ consisting of all $$\pi\in\mathrm{GL}_{n}(\mathbb{C})$$ such that $$f(\pi(x_{1},\ldots,x_{n}))=f$$. Then each matrix in $$G_{f}$$ induces an automorphism on the hypersurface $$V=\mathcal{V}(f)=\{p\in\mathbb{C}^{n}:f(p)=0\}.$$ Thus we get a group homomorphism $$\phi:G_{f}\to\mathrm{Aut}(V)$$, that assigns each matrix an automorphism of $$V$$.

My question is; under what conditions is the action of $$G_{f}$$ on $$V$$ faithful? That is, when is the mapping, $$\phi$$ injective?

There is a small discussion of this here, and it is mentioned that this question deserved its own thread.

An element $$\pi \in \mbox{GL}_n(\mathbb{C})$$ is the identity iff if fixes a basis of $$\mathbb{C}^n$$. Then we just need $$V$$ to have $$n$$ points in general position which is the same as $$V$$ not contained in a hyperplane.