Fiber product of projective varieties and ample line bundles

Question

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.

Answer 1

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 $$pr_2^*D_Y\cdot(pr_1^*D_X+pr_2^*D_Y)^{\dim X+\dim Y-1}={\dim X+ \dim Y -1 \choose \dim X}D_X^{\dim X}D_Y^{\dim Y}$$