Does there exist an analogue of Lefschetz Hyperplane Theorem for cohomology that holds for intersections with (smooth) quadrics?
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4$\begingroup$ If you use a Veronese map to reembed your variety, then the quadric section beomes a hyperplane section. $\endgroup$– ritaCommented Sep 18, 2012 at 10:24
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$\begingroup$ That's for sure! :) I should have added that, for some reason which it takes long to explain, I cannot do that Veronese-trick. $\endgroup$– IMeasyCommented Sep 18, 2012 at 10:36
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6$\begingroup$ 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? $\endgroup$– ritaCommented Sep 18, 2012 at 11:15
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$\begingroup$ All you have to check for Lefschetz is that your hypersurface is an effective ample divisor... $\endgroup$– diveriettiCommented Sep 18, 2012 at 11:22
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$\begingroup$ 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. $\endgroup$– Daniel LoughranCommented Sep 18, 2012 at 11:29
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