# Rationality of moduli spaces of bundles over non-algebraically closed fields

Research paper by Souradeep Majumder, Ronnie Sebstian

Indexed on: 28 Nov '18Published on: 28 Nov '18Published in: arXiv - Mathematics - Algebraic Geometry

#### Abstract

Let $C$ be a smooth projective, geometrically irreducible curve defined over $k$ such that $C$ has a $k$ rational point. We assume that the Picard scheme ${\rm Pic}^0_{C/k}$ has a Zariski dense set of $k$ rational points. Let $r>0$ and $d$ be integers which are coprime and $L$ be a $k$ line bundle on $C$ of degree $d$. We show that the moduli space $\mathcal{M}_{r,L}$, of stable bundles of rank $r$ and determinant $L$, is $k$ rational. When $k=\mathbb{R}$ and $C(\mathbb{R})=\emptyset$, we give some results towards classifying all birational types over $\mathbb{R}$.