On the rationality of $SU(r,d)$

Abstract

Let SU(r,d) be the moduli space of rank r, degree d vector bundles over a smooth projective curve of genus $g\ge 2$. If (r,d)=1 and d divides r+1, then SU is rational. Furthermore, if $0<δ< r$ and all prime divisors of $δ$ divide r, and if d divides $r-δ$, then SU is rational. The proof is a variation on a result of Newstead and modifications due to Ballico and then Boden and Yokogawa.