Picard Groups of Some Quot Schemes

Let $C$ be a smooth projective curve over the field of complex numbers ${\mathbb{C}}$ of genus $g(C)>0$. Let $E$ be a locally free sheaf on $C$ of rank $r$ and degree $e$. Let $\mathcal{Q}:=\textrm{Quot}_{C/{\mathbb{C}}}(E,k,d)$ denote the Quot scheme of quotients of $E$ of rank $k$ and degree $d$. For $k>0$ and $d\gg 0$, we compute the Picard group of $\mathcal{Q}$.