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!