Consider a elliptically fibered variety $\pi: X\rightarrow B$ (with fiber $E$), and suppose we construct the fiber product $X\times_B X$. Then naturally there is a diagonal divisor $\Delta$ (which is Weil if I'm right), over each point $p\in B$ it reduces to the diagonal of $E\times E$. Because it's Weil, it's not possible to find a global algebraic description of $\Delta$.
Question1 : I just want to make sure what I just said is true. (probably it's wrong, and it's actually Cartier...)
Question2 : How to compute intersection numbers which involve $\Delta$, I know for example it's possible to compute self intersection by using GRR for $\delta: X \rightarrow X\times_B X$. But in other cases I don't know any clear/practical way to do that.
Thanks.