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Let $G$ be a finite subgroup of $\mathrm{SL}(2,\mathbb{C})$ and $N \triangleleft G$ a normal subgroup. Let $x, y, z$ be the fundamental invariants for the standard action of $N$ on $\mathbb{C}^2$, related by the polynomial $f$. Then $G$ acts on the space $\langle x,y,z \rangle$ and on the Gorenstein hypersurface $\mathrm{Spec}\, \mathbb{C}[x,y,z]/f$. The polynomial $f$ is a semiinvariant for this action, with related character $\chi_f$ of $G$. It appears that this character $\chi_f$ is always trivial (that is, $f$ is fixed by all of $G$).

Question: Why?

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