{'fraction_non_alphanumeric': 0.02799650043744532, 'fraction_numerical': 0.008748906386701663, 'mean_word_length': 4.5, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Algebraic group actions and quotients

This course is devoted to the proof of Luna’s étale slice theorem and to the study of some of its applications. A brief description of this theorem follows in 1.1 and 1.2. Our main source is the original paper [14]. The whole proof of this result is complicated, because the hypotheses are very general. We shall sometimes make some simplifications, for example by assuming that we consider only integral varieties or connected groups. Luna’s étale slice theorem is useful for the local study of good quotients by reductive groups. We will give here three applications. We will first give 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 [10]. There are also applications of Luna’s theorem to the study of moduli spaces of semi-stable bundles on surfaces (cf. for example [20]). In the third application we will study the factoriality of the local rings of the points of some quotients. Luna’s theorem will allow us to study the factoriality of the completions of