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Projective normality of the moduli space of rank 2 vector bundles on a generic curve. (English) Zbl 1197.14033

Let \(C\) be a smooth projective curve, and \(\mathrm{SU}_C(2)\) be the moduli space of semi-stable bundles of rank \(2\) with trivial determinant on \(C\). Let \(\mathrm{Pic}(\mathrm{SU}_C(2))=\mathbb Z[\mathcal L]\). The main results of the paper under review are the following: If \(C\) is generic, then the multiplication maps \[ \mathrm{Sym}^{\ell}H^0(\mathrm{SU}_C(2), \mathcal L^2)\to H^0(\mathrm{SU}_C(2), \mathcal L^{2\ell}) \] and \[ H^0(\mathrm{SU}_C(2), \mathcal L)\otimes \mathrm{Sym}^{\ell}H^0(\mathrm{SU}_C(2), \mathcal L^2)\to H^0(\mathrm{SU}_C(2), \mathcal L^{2\ell+1}) \] are surjective for all \(\ell\geq 1\). The idea of proof is to study the degeneration of \(\mathrm{SU}_C(2)\) and \(H^0(\mathrm{SU}_C(2), \mathcal L^m)\) when \(C\) degenerates to a reducible nodal curve with only one node, which reduce the problem to lower genus curves.

MSC:

14H60 Vector bundles on curves and their moduli
14D20 Algebraic moduli problems, moduli of vector bundles
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