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Oct 12, 2013 at 4:39 comment added naf I assume that in positive characteristic you still take the quotient modulo $K(L)_0$. If this is not reduced then what you ask for is not true; for example, $L$ could be ample with $K(L)_0$ non-trivial and then sections will not be preserved.
Oct 11, 2013 at 18:48 history asked Marc CC BY-SA 3.0