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Vector bundles on curves and generalized theta functions: recent results and open problems
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Riemann surface carries a natural line bundle, the determinant bundle. The space of sections of this line bundle (or its multiples) constitutes a natural non-abelian generalization of the spaces of theta functions on the Jacobian. There has been much progress in the last few years towards a better understanding of these spaces, including a rigorous proof of the celebrated Verlinde formula which gives their dimension. This survey paper tries to explain what is now known and what remains open.
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A remark on a theorem of Narasimhan and Ramanan
The paper re-proves Narasimhan-Ramanan's identification of SU_X(2) with P^3 by showing the anticanonical Seshadri constant is 4, but the proof has a gap in the Quot-scheme construction.
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