# Moduli spaces of rank 2 stable bundles over curves as projective varieties

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 $$g \geq 3$$?

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