# Universal extension using relative Ext sheaf

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.

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:

1. 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 $$X\times_BX \to X$$.
2. 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.