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Let $\Sigma$ be a Riemann surface of genus $g$. Let's consider the moduli space of rank $2$ stable vector bundles with determinant $L$ such that $\deg(L)$ is odd. Denote this space by $\mathcal{M}_{\Sigma}(2,L)$. It is well-known that $\mathcal{M}_{\Sigma}(2,L)$ is isomorphic to the intersection of quadrics in $\mathbb{P}^5$ when $g=2$ by Newstead and Narasimhan-Ramanan.

Question: do we have a similar "classical description" of $\mathcal{M}_{\Sigma}(2,L)$ as a projective variety when $g \geq 3$?

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    $\begingroup$ If the curve is hyperelliptic, there is an analogous description. $\endgroup$
    – Sasha
    Sep 2 at 19:21
  • $\begingroup$ That sounds interesting, what's the reference for hyperelliptic curves? $\endgroup$ Sep 2 at 20:20
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    $\begingroup$ U. V. Desale and S. Ramanan, ‘Classification of vector bundles of rank 2 on hyperelliptic curves’, Invent. Math. 38 (1976) 161–185. $\endgroup$
    – Sasha
    Sep 2 at 20:26
  • $\begingroup$ Thanks, that's very helpful! $\endgroup$ Sep 2 at 20:34

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