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Suppose you are given a smooth algebraic variety $X$ inside a projective space $\mathbb{P}$ and that there is a linear action of a finite cyclic group $G$ on $\mathbb{P}$ which restricts to an action on $X$. Does there exists an hyperplane $H$ on $\mathbb{P}$ such that $H \cap X$ is smooth and stable under $G$? I guess it should be possible to prove such a result by using a variant of Bertini's theorem, but I don't see how.

Thanks for your help

2 Answers 2