Let $\pi: X \to B$ be an elliptic surface with a section $\sigma$. Assume that $\pi, X, B$ are all smooth and that $X,B$ are projective.
Let $E_{r_1,d_1}$ be a vector bundle of rank $r_1$ and degree $d_1$ on $X\times_B X$ such that $0<r_1<r$.
Let $E_{r_2,d_2}$ be a vector bundle of rank $r_2$ and degree $d_2$ on $X$ such that $E_{r_2,d_2}|_{\ell}$ is stable and such that $\operatorname{det}E_{r_2,d_2}|_{\ell} \cong \mathcal O(d_2\sigma) $ for every fiber $\ell$.
I want to understand two claims:
- by Atiyah's classification of stable vector bundles over elliptic curves, $\mathcal L:=\operatorname{Ext}^1_{p_X}(E_{r_2,d_2},E_{r_1,d_1} ) $ is a line bundle on X. Here $p_X$ is the projection $X\times_BX \to X$.
- There is a universal extension $0\to E_{r_1,d_1}\to E_{r,d}\to E_{r_2,d_2} \otimes p_{X}^*(\mathcal L)\to 0$.
I am mostly interested in the claim 2. The only reference I found on the relative Ext was this paper from H. Lange. but I couldn't obtain any hints for the universal extension.
Thank you.
Just note that \begin{align*} p_{X*}\mathcal{E}\mathit{xt}^1(E_{r_2,d_2} \otimes p_X^*(\mathcal{L}), E_{r_1,d_1}) &\cong p_{X*}(\mathcal{E}\mathit{xt}^1(E_{r_2,d_2}, E_{r_1,d_1}) \otimes p_X^*(\mathcal{L}^\vee)) \\ & \cong p_{X*}\mathcal{E}\mathit{xt}^1(E_{r_2,d_2}, E_{r_1,d_1}) \otimes \mathcal{L}^\vee \\ & \cong \mathcal{H}\mathit{om}(\mathcal{L}, p_{X*}\mathcal{E}\mathit{xt}^1(E_{r_2,d_2}, E_{r_1,d_1})) \\ & \cong \mathcal{H}\mathit{om}(\mathcal{L}, \mathcal{L}). \end{align*} The right-hand side has a natural global section (corresponding to the identity morphism), it gives a global section of the left-hand side, hence the required extension class.