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Let $V$ be an n-dimensional complex vector space, and $u\in S^nV$ be a polynomial, $G(u)$ be the stabilizer of $u$ in $GL(V)$. Let $[v]\in\overline{GL(V)\cdot[u]}\subset\mathbb{P}(S^nV)$, but $v\notin End(V)\cdot [u]$, and $GL(V)\cdot [v]$ is of codimension $1$ in $\overline{GL(V)\cdot [u]}$, where $\overline{GL(V)\cdot [u]}$ means the closure of $GL(V)$-orbit of $[u]$ in $\mathbb{P}S^nV$. Here $\overline{GL(V)\cdot [u]}$ need not be normal. Obviously $\dim G(v)=\dim G(u)+1$, I just wonder what the precise relationship with $G(u)$ and $G(v)$ is. Thanks a lot!

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