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Let $H\subset \mathbb C^n$ be an irreducible hypersurface invariant under a diagonal $\mathbb C^*$-action with positive weights ($H$ is given by a quasi-homogeneous polynomial). Consider the Whitney stratification of $H$.

Question. Is it true that $0\subset \mathbb C^n$ is not a stratum of this stratification if an only if $H$ is a smooth hypersurface? If no, how to describe all such irreducible $\mathbb C^*$-invariant $H$ for which $0$ is not a stratum?

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    $\begingroup$ The answer to the first question is "no" as you can take $H$ to be the hypersurface defined by the equation $x^2+y^2+z^2=0$, which looks like $0$ is a Whitney stratum, but isn't, because it's secretly a hypersurface in the variables $x,y,z,w$. It might be reasonable to guess that all examples arise from this trick, but I don't see how to prove it. $\endgroup$
    – Will Sawin
    Commented May 6, 2021 at 23:11
  • $\begingroup$ Thanks Will! So one has to exclude some kind of products by a linear space... $\endgroup$
    – aglearner
    Commented May 6, 2021 at 23:17

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