Rationality of moduli spaces of stable bundles on curves over $\mathbb{R}$

Abstract

Let $C$ be a smooth, projective, geometrically irreducible curve defined over $\mathbb{R}$ such that $C(\mathbb{R})=\varnothing$. Let $r>0$ and $d$ be integers which are coprime. Let $L$ be a line bundle on $C$ which corresponds to an $\mathbb{R}$ point of ${\mathrm{Pic}}_{C/\mathbb{R}}^d$. Let ${\mathcal{M}}_{r,L}$ be the moduli space of stable bundles on the complexification of $C$ of rank $r$ and determinant $L$. We classify birational types of ${\mathcal{M}}_{r,L}$ over $\mathbb{R}$.

The main result in section 3 of v1 is already contained in Hoffmann (arXiv:math/0511656v1 [math.AG]) and so this section has been removed. When the curve is real and has no rational points, a complete classification of the birational types of ${\mathcal{M}}_{r,L}$ is described in this version, improving the results of the previous version