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Let $X$ be an abelian variety of dimension $n>2$. Let $L$ be a very ample line bundle on $X$. Is it possible to find two divisors $D_1,D_2\in |L|$ which do not intersect or intersect in codimension 3 or higher codimension? Is there a reference for such a result?

This should be possible I feel. Looking forward to the answers. Thanks in advance!

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  • $\begingroup$ I would add tht if two divisors on a smooth variety have a non-empty intersection, then codimension of this intersection is at most 2. This has nothing to do with their ampleneess. $\endgroup$
    – Serge Lvovski
    Jan 1, 2016 at 7:16

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Of course not.

If $L$ is very ample, $D_1$ and $D_2$ are two hyperplane sections for some embedding in a projective space. Therefore their intersection is at most codimension 2 in $X$, intersection of $X$ with a linear subspace of codimension 2. This has nothing to do with $X$ being an abelian variety.

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