0
$\begingroup$

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!

| cite | improve this question | | | | |
$\endgroup$
  • $\begingroup$ 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 '16 at 7:16
4
$\begingroup$

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.

| cite | improve this answer | | | | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.