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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 $X$?

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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.

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  • $\begingroup$ I see. So the "absolute" Grassmannian has dimension $k(r-k)$ and the extra 1 comes from $C$. What about the Quot scheme? There we don't have local triviality. $\endgroup$ Jun 17, 2022 at 11:55
  • $\begingroup$ For $\mathrm{Quot}$ schemes it is more tricky. But you can usually compute tangent spaces at points and thus bound the dimension. $\endgroup$
    – Sasha
    Jun 17, 2022 at 12:05

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