Luna's slice theorem and applications . Algebraic group actions and quotients, Notes of XXIII Autumn School in Algebraic Geometry (Wykno, Poland, 2000) Ed. J. Wisniewski, Hindawi Publ. Corp. (2004), 39-90.

This course is devoted to the proof of Luna's étale slice theorem and to the study of some of its applications. Our main source is the original paper of D. Luna. Luna's étale slice theorem is useful for the local study of good quotients by reductive groups. We give here three applications. We first obtain some general results on quotients by reductive groups. The second application is the local study of the moduli spaces of semi-stable vector bundles on curves, following a paper of Y. Laszlo. In the third application we study the factoriality of the local rings of the points of some quotients. Luna's theorem allows us to describe completions of these rings. This will give an example of a local ring of quotient which is factorial, but whose completion is not. This will show that some local properties of quotients cannot be derived from Luna's theorem.

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