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Does there exist an analogue of Lefschetz Hyperplane Theorem for cohomology that holds for intersections with (smooth) quadrics?

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If you use a Veronese map to reembed your variety, then the quadric section beomes a hyperplane section. –  rita Sep 18 '12 at 10:24
    
That's for sure! :) I should have added that, for some reason which it takes long to explain, I cannot do that Veronese-trick. –  IMeasy Sep 18 '12 at 10:36
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I think I'm missing a point here: isn'it it enough to know that a divisor is an hyperplane section for SOME embedding (namely that it is a very ample divisor) to apply Lefschetz theorem? –  rita Sep 18 '12 at 11:15
    
All you have to check for Lefschetz is that your hypersurface is an effective ample divisor... –  diverietti Sep 18 '12 at 11:22
    
The version of the Lefschetz Hyperplane Theorem that you want can probably be found in the book "Positivity In Algebraic Geometry" by Lazarsfeld. Unfortunately I don't have it to hand to give a more precise reference. As others have already noted, all you need is that the divisor is very ample. –  Daniel Loughran Sep 18 '12 at 11:29
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