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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.

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  • $\begingroup$ Do you mean "Choosing a very ample line bundle $O_C(1)$"? Otherwise it look strange to call it $O_C(1)$. $\endgroup$
    – Rami
    Mar 21, 2012 at 23:23
  • $\begingroup$ Let's assume it is very ample - since in the end we consider its $m$-th power anyway. $\endgroup$ Mar 22, 2012 at 18:18

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