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 S ympl 2 n , L be the moduli stack of vector bundles E of rank 2 n on C endowed with a nowhere degenerate symplectic form b : E ⊗ E → L up to scalars. We prove that this stack is birational to B G 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 ξ of degree 0 with ξ ⊗ 2 ≇ O C . It follows that the corresponding coarse moduli scheme of Ramanathan-stable symplectic bundles is rational in this case.