Let $C$ be a smooth projective curve and consider the Quot scheme $Quot$ of all quotients $\mathcal{O}^N_C \to F$ where the rank and $c_1$ of $F$ are fixed. Choosing an ample line bundle $\mathcal{O}_C(1)$ on $C$ we can consider for any $m$ the induced map on global sections $$ V(m) = H^0(C, \mathcal{O}^N_C(m)) \to H^0(C, F(m)). $$ By Grothendieck, for large enough $m$ this induces a closed embedding of the scheme $Quot$ into the Grassmannian of all quotient spaces of $V(m)$ with appropriate dimension. It follows that the line bundle $D(m)$ on $Quot$ computing the determinant of $H^0(C, F(m)))$, is ample for $m$ large enough.
Question: what is known about the global sections of $D(m)$? For example, does the embedding into the Grassmannian induce an isomorphism on global sections of the determinant line bundles?
Thank you.