Stability of the poincare bundle

Let C be a nonsingular projective curve of genus g ≥ 2 defined over the complex numbers, and let Mξ 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 Uξ on C × Mξ is stable with respect to any polarisation on C × Mξ. 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 Uξ on C × Mξ containing Uξ is isomorphic to the Jacobian of C