Some moduli stacks of symplectic bundles on a curve are rational

Abstract

Let C be a smooth projective curve of genus g≥ 2 over a field k. Given a line bundle L on C, let Sympl_2n,L be the moduli stack of vector bundles E of rank 2n on C endowed with a nowhere degenerate symplectic form b : E ⊗ E → L up to scalars. We prove that this stack is birational to BG_m × A_s for some s if deg(E)=n.deg(L) is odd and C admits a rational point P∈C(k) as well as a line bundle ζ of degree 0 with ζ ^⊗2 ≅ O_C. It follows that the corresponding coarse moduli scheme of Ramanathan-stable symplectic bundles is rational in this case.