Let $E$ be a $r$ rank vector bundle on a smooth projective curve $C$ of genus $g$. Let $G$ denote the Grassmannian of $(r-k)$ rank locally free quotients of $E$.
A paper by Mukai-Sakai seems to claim that $G$ has dimension = $k(r-k)+1$.
I would like to know an explanation behind the above claim.
More generally, what is the dimension of the Quot scheme $Quot_{E/X/S}$ of coherent quotients of $E$ which is a fixed coherent sheaf of an $S-$scheme $X$?
Since $E$ is Zariski locally trivial, the relative Grassmannian is covered by charts isomorphic to $U \times \mathrm{Gr}(k,r)$, where $U \subset C$ are open subsets.