Let $X, Y$ be smooth, projective varieties, $L\_X$ and $L\_Y$ are very ample line bundles on $X$ and $Y$, respectively. If I understand correctly, $\mbox{pr}\_1^\*L\_X \otimes \mbox{pr}\_2^\*L\_Y$ is a very ample line bundle on $X \times Y$, where $\mbox{pr}\_1, \mbox{pr}\_2$ are the natural projections from $X \times Y$ to $X$ and $Y$, respectively. Is it possible to compute the degree of $\mbox{pr}\_1^\*L\_X$ or $\mbox{pr}\_2^\*L\_Y$ or $\mbox{pr}\_1^\*L\_X \otimes \mbox{pr}\_2^\*L\_Y$ with respect to the closed immersion defined by $\mbox{pr}\_1^\*L\_X \otimes \mbox{pr}\_2^\*L\_Y$ in terms of the degree of $L\_X$ and $L\_Y$ (computed with respect to the closed immersions given by $L\_X$ and $L\_Y$, respectively)?


Any reference/hint will be most welcome.



Fiber product of projective varieties and ample line bundles
Let $D\_X$ and $D\_Y$ be the Chern classes of $L\_X$ and $L\_Y$. Assume $X$ and $Y$ have dimension at least one. Thus $pr\_{1\*}pr\_1^\*D\_X=0$ and $pr\_{2\*}pr\_2^\*D\_Y=0$. Applying the projection formula we obtain:
$$pr\_1^\*D\_X^{\dim X+1}=D\_X^{\dim X}\cdot pr\_{1\*}pr\_1^\*D\_X=0$$
$$pr\_2^\*D\_Y^{\dim Y+1}=D\_X^{dim Y}\cdot pr\_{2\*}pr\_2^\*D\_Y=0$$
Now we compute the degree of $pr\_1^\*L\_X\otimes pr\_2^\*L\_Y$. It is same as the intersection number $(pr\_1^\*D\_X+pr\_2^\*D\_Y)^{\dim X+\dim Y}$. So, let us calculate: $$(pr\_1^\*D\_X+pr\_2^\*D\_Y)^{\dim X+\dim Y}=\sum\_{i+j=\dim X+\dim Y} {\dim X+\dim Y \choose i}pr\_1^\*D\_X^i\cdot pr\_2^\*D\_Y^j$$.
 By the formulas in the begining all summands except ${\dim X+\dim Y\choose \dim X}pr\_1^\*D\_X^{\dim X}\cdot pr\_2^\*D\_Y^{\dim Y}$ vanish. Note, $pr\_2^\*D\_Y^{\dim Y}=D\_Y^{\dim Y}[X]$ and $pr\_1^\*D\_X^{\dim X}=D\_X^{\dim
X}[Y]$, where $[X]$ and $[Y]$ are the classes of $X\times point$ and $point\times Y$ in the Chow ring of $X\times Y$. It is easy to see that $[X]\cdot[Y]=1$. Finaly we find: $$(pr\_1^\*D\_X\otimes pr\_2^\*D\_Y)^{\dim X + \dim Y}={\dim X+\dim Y\choose \dim X}D\_X^{\dim X}D\_Y^{\dim Y}.$$
In the same way we compute the degrees of $pr\_1^\*L\_X$ and $pr\_2^\*L\_Y$.
$$pr\_1^\*D\_X\cdot(pr\_1^\*D\_X+pr\_2^\*D\_Y)^{\dim X+\dim Y-1}={\dim X+ \dim Y -1 \choose \dim Y}D\_X^{\dim X}D\_Y^{\dim Y}$$ 
and $$pr\_2^\*D\_Y\cdot(pr\_1^\*D\_X+pr\_2^\*D\_Y)^{\dim X+\dim Y-1}={\dim X+ \dim Y -1 \choose \dim X}D\_X^{\dim X}D\_Y^{\dim Y}$$ 


