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In the proof of Brieskorns lemma, see 3.3 here, Brieskorn mentions that we take "ouverts convenables" satisfying some properties, but, as far as I can tell, never specifies what these opens actually are, or why such opens even exist. None of the other sources that I am aware of discuss a prove of this lemma in more detail, so I wonder if anyone here knows what these opens are supposed to be?

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