TITLE: A mistake in "Chtoucas de Drinfeld et correspondance de Langlands"
QUESTION [8 upvotes]: I have been told by some people that there is a mathematical mistake in the last section of "Chtoucas de Drinfeld et correspondance de Langlands" (L. Lafforgue) concerning the hyperplane section arguments which does not invalidate the main results. I was not able to identify the mistake myself. Could somebody equipped to answer this question tell me what the mistake was and how it could be fixed?

REPLY [6 votes]: I think that you refer to a rather small point, which is discussed by Deligne in Sections 0.7 and 1.5-1.9 of https://www.math.ias.edu/files/deligne/FrobTraces.pdf
The main results of the Lafforgue's paper are about curves over finite fields. In the last section, he discusses applications to higher dimensional varieties, reducing to the case of curves by hyperplane sections arguments. In one of such arguments (proof of Proposition VII.7), he cites some version of Bertini theorem stated in Hartshorne's book with projectivity assumption, whereas he applies it to some non-projective variety (the locus of smoothness of some l-adic sheaf). In his paper, Deligne writes down some clarification of the argument, with reference to some version of Bertini theorem due to Jouanolou.
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