Intersection product of pull back under projection

Question

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!

Answer 1

$(p^*D)^3=p^*D^3=0$ 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.