3

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 $E_1,E_2$?

Hans
  • 3,539
  • 1
    You will not need the $E_i$: Remember $\operatorname{Pic}(B)=\operatorname{Pic}(X_1\times X_2)\times\mathbb Z$. Since the $E_i$ are sections of the $\mathbb P^1$-bundle $B\to X_1\times X_2$, those correspond to the tautological quotient sheaf $\mathcal O(1)$ and generate $\mathbb Z\subset\operatorname{Pic}(B)$. On the other hand, the determinant you are interested in is a pull-back from $X_1\times X_2$, so lives in the other factor. This reduces to asking if your determinant bundle on $X_1\times X_2$ is a tensor product of pull-backs. Does https://math.stackexchange.com/questions/493005/ help? – red_trumpet Aug 24 '22 at 21:31
  • Ah thanks! The $E_i$ being section is of course a crucial observation - thanks. I will have a look at the other question you sent. – Hans Aug 26 '22 at 14:56

0 Answers0