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!



Will any two linearly equivalent ample divisors on an abelian variety intersect?
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.


