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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.

user867292
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1 Answers1

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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.

Alan Muniz
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