Determinant of certain rank-2 bundle on product of curves

Question

Let $X_1,X_2\subset\mathbb{P}^n$ be two disjoint smooth projective and irreducible curves. Then we have a $\mathbb{P}^1$-bundle $B$ on the product $X_1\times X_2$ defined by $$B=\{(p,q,r)\in X_1\times X_2\times \mathbb{P}^{n}\mid r\in\textrm{Span}(p,q)\}.$$ The variety $B$ has natural maps $\pi_1:B\to X_1$, $\pi_2:B\to X_2$ and $\pi_0:B\to \mathbb{P}^n$. The preimage of $X_i$ under $\pi_0$ is a divisor $E_i$ on $B$ for $i=1,2$. We further have a divisor $D$ on $B$ corresponding to the pull-back of the determinant of the corresponding rank-2 vector bundle on $X_1\times X_2$. Can we express the divisor class of $D$ in terms of pull-backs of divisors under the $\pi_i$ ($i=0,1,2$) and the divisors $E_1,E_2$?