Stability of the Poincaré Bundle

Let C be a nonsingular projective curve of genus g ≥2 defined over the complex numbers, and let M ξ denote the moduli space of stable bundles of rank n and determinant ξ on C , where ξ is a line bundle of degree don C and n and d are coprime. It is shown that a universal bundle U ξ on C × M ξ is stable with respect to any polarisation on C × M ξ . Similar results are obtained for the case where the determinant is not fixed and for the bundles associated to the universal bundles by irreducible representations of GL( n , C). It is shown further that the connected component of the moduli space of bundles with the same Hilbert polynomial as U ξ on C × M ξ containing U ξ is isomorphic to the Jacobian of C .