Let $V$ be a fixed vector space. Let $X$ be a smooth curve. Consider the Quot scheme $Q$ of Quotients of $V\otimes \mathcal{O}\_X$ of degree $d$ and rank $r$. Let $R$ be the open subscheme of $Q$ consisiting only vector bundles with following property: $H^1$ vanishes and $H^0\cong V$ via the natural map. Its clear that every such quotient $E$ in $R$ is globally generated vector bundle. 


Now in Atiyah's Paper he shows that there is following exact sequence:


$0\rightarrow \mathcal{O}\_X^{\oplus (r-1)}\rightarrow E\rightarrow det ~E\rightarrow 0$, for $E$ globally generated.


My Question is does it hold in the described case i.e, is there a exact sequence like this:


$0\rightarrow \mathcal{O}\_{X\times R}^{\oplus (r-1)}\rightarrow \mathcal{E}\rightarrow det ~\mathcal{E}\rightarrow 0$, where $\mathcal{E} $ is the universal bundle on $R$?



On a result of Sir Michael Atiyah "Vector bundles on Elliptic curves" Theorem 2 Page 426
No, this is not true in general. Actually, the exact sequence on $X$ is not canonical, but is determined by an $(r-1)$-dimensional vector subspace in $V$. Moreover, to extend to such a sequence the first morphism should be a fiberwise monomorphism. For this, there is an obstruction given by $c\_2(E)$. Eventually, since $X$ is a curve, the obstruction vanishes, and such a sequence exists.


However, over the product $X \times R$ there is no reason for $c\_2(\mathcal{E})$ to be zero, so for a general morphism $\mathcal{O}\_{X \times R}^{\oplus(r-1)} \to \mathcal{E}$ the cokernel is not a line bundle.


