Stability of the poincare bundle

Abstract

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.