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I found the Luna's Slice Theorem very Technical. It will be helpful if someone illustrates the geometry involved in the theorem. Also why this theorem so useful?

This is Luna's Slice theorem from a lecture note of Jean-Mac Drezet Luna's Slice Theorem and applications- Lecture note by Jean Mac Drezet Luna_Slice And this is the Version of Luna's theorem when the point x is smooth- enter image description here

Also I will use the opportunity to attract attention to one of another question where I asked the meaning of a topological slice of a point- Slice of a point of a topological space

Is there any other resource to know more about Luna's Slice Theorem?

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    $\begingroup$ Many moduli spaces in algebraic geometry are of the form $X/G$; the slice theorem is very useful to understand their local structure. See for instance this paper by Y. Laszlo. $\endgroup$
    – abx
    Commented Nov 25, 2014 at 5:12
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    $\begingroup$ Think of the case where the action is free. Then the theorem says that $X$ is a $G$-torsor over $X/G$. $\endgroup$
    – Piotr Achinger
    Commented Nov 25, 2014 at 7:56
  • $\begingroup$ @abx could you please give the link to that paper again?! the link is broken. $\endgroup$
    – QGravity
    Commented May 26, 2022 at 18:46
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    $\begingroup$ Link to Laszlo's paper. $\endgroup$
    – abx
    Commented May 26, 2022 at 19:44

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