Let $X$ be a surface and $Y$ be a curve over $\mathbb{C}$. Let $L$ and $L'$ be ample line bundles on $X$ and $Y$ respectively. Consider the product $X\times Y$. Let $p$ and $q$ be the projection from $X\times Y$ to $X$ and $Y$ respectively.


Then $E=:p^\*L\otimes q^\*L'$ is an ample line bundle on $X\times Y$. I want to compute the intersection product $(p^\*L).E.E$. Suppose $L=\mathcal{O}\_X(D)$ and $L'=\mathcal{O}\_Y(D')$, then 


$E=p^\*\mathcal{O}\_X(D)\otimes q^\*\mathcal{O}\_Y(D')=\mathcal{O}(D\times Y)\otimes\mathcal{O}(X\times D')$. Hence the required product is:


$p^\*L.E.E=\mathcal{O}(D\times Y).\mathcal{O}(D\times Y)\otimes\mathcal{O}(X\times D').\mathcal{O}(D\times Y)\otimes\mathcal{O}(X\times D')$.


So I would like to compute 


1) $\mathcal{O}(D\times Y)^3$, that is the product of $\mathcal{O}(D\times Y)$ with itself thrice, and


2) $\mathcal{O}(D\times Y)^2.\mathcal{O}(X\times D')$ . The other product can be computed similarly. 
How do we compute these products?


Thanks in advance!



Intersection product of pull back under projection
$(p^\*D)^3=p^\*D^3=0$ and $(p^\*D)^2\cdot q^\*D'= D^2\deg (D')$, hence
$$ p^\*D\cdot (p^\*D+q^\*D')^2= 2 (p^\*D)^2\cdot q^\*D'=2D^2\deg (D')\ .$$Note that the additive notation (say, in the Chow ring) is better adapted for this kind of calculations.


