If you use a Veronese map to reembed your variety, then the quadric section beomes a hyperplane section.
– ritaSep 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.
– IMeasySep 18 '12 at 10:36

6

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?
– ritaSep 18 '12 at 11:15

All you have to check for Lefschetz is that your hypersurface is an effective ample divisor...
– diveriettiSep 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 LoughranSep 18 '12 at 11:29